From b861a4533d73023987964e96f0cf97db7bba7768 Mon Sep 17 00:00:00 2001
From: Diego Restrepo <restrepo@udea.edu.co>
Date: Mon, 6 Feb 2023 19:49:18 -0500
Subject: [PATCH 1/9] lc

---
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diff --git a/_bibliography/references.bib b/_bibliography/references.bib
deleted file mode 100755
index cbf9b01..0000000
--- a/_bibliography/references.bib
+++ /dev/null
@@ -1,56 +0,0 @@
----
----
-
-@inproceedings{holdgraf_evidence_2014,
-	address = {Brisbane, Australia, Australia},
-	title = {Evidence for {Predictive} {Coding} in {Human} {Auditory} {Cortex}},
-	booktitle = {International {Conference} on {Cognitive} {Neuroscience}},
-	publisher = {Frontiers in Neuroscience},
-	author = {Holdgraf, Christopher Ramsay and de Heer, Wendy and Pasley, Brian N. and Knight, Robert T.},
-	year = {2014}
-}
-
-@article{holdgraf_rapid_2016,
-	title = {Rapid tuning shifts in human auditory cortex enhance speech intelligibility},
-	volume = {7},
-	issn = {2041-1723},
-	url = {http://www.nature.com/doifinder/10.1038/ncomms13654},
-	doi = {10.1038/ncomms13654},
-	number = {May},
-	journal = {Nature Communications},
-	author = {Holdgraf, Christopher Ramsay and de Heer, Wendy and Pasley, Brian N. and Rieger, Jochem W. and Crone, Nathan and Lin, Jack J. and Knight, Robert T. and Theunissen, Frédéric E.},
-	year = {2016},
-	pages = {13654},
-	file = {Holdgraf et al. - 2016 - Rapid tuning shifts in human auditory cortex enhance speech intelligibility.pdf:C\:\\Users\\chold\\Zotero\\storage\\MDQP3JWE\\Holdgraf et al. - 2016 - Rapid tuning shifts in human auditory cortex enhance speech intelligibility.pdf:application/pdf}
-}
-
-@inproceedings{holdgraf_portable_2017,
-	title = {Portable learning environments for hands-on computational instruction using container-and cloud-based technology to teach data science},
-	volume = {Part F1287},
-	isbn = {978-1-4503-5272-7},
-	doi = {10.1145/3093338.3093370},
-	abstract = {© 2017 ACM. There is an increasing interest in learning outside of the traditional classroom setting. This is especially true for topics covering computational tools and data science, as both are challenging to incorporate in the standard curriculum. These atypical learning environments offer new opportunities for teaching, particularly when it comes to combining conceptual knowledge with hands-on experience/expertise with methods and skills. Advances in cloud computing and containerized environments provide an attractive opportunity to improve the effciency and ease with which students can learn. This manuscript details recent advances towards using commonly-Available cloud computing services and advanced cyberinfrastructure support for improving the learning experience in bootcamp-style events. We cover the benets (and challenges) of using a server hosted remotely instead of relying on student laptops, discuss the technology that was used in order to make this possible, and give suggestions for how others could implement and improve upon this model for pedagogy and reproducibility.},
-	booktitle = {{ACM} {International} {Conference} {Proceeding} {Series}},
-	author = {Holdgraf, Christopher Ramsay and Culich, A. and Rokem, A. and Deniz, F. and Alegro, M. and Ushizima, D.},
-	year = {2017},
-	keywords = {Teaching, Bootcamps, Cloud computing, Data science, Docker, Pedagogy}
-}
-
-@article{holdgraf_encoding_2017,
-	title = {Encoding and decoding models in cognitive electrophysiology},
-	volume = {11},
-	issn = {16625137},
-	doi = {10.3389/fnsys.2017.00061},
-	abstract = {© 2017 Holdgraf, Rieger, Micheli, Martin, Knight and Theunissen. Cognitive neuroscience has seen rapid growth in the size and complexity of data recorded from the human brain as well as in the computational tools available to analyze this data. This data explosion has resulted in an increased use of multivariate, model-based methods for asking neuroscience questions, allowing scientists to investigate multiple hypotheses with a single dataset, to use complex, time-varying stimuli, and to study the human brain under more naturalistic conditions. These tools come in the form of “Encoding” models, in which stimulus features are used to model brain activity, and “Decoding” models, in which neural features are used to generated a stimulus output. Here we review the current state of encoding and decoding models in cognitive electrophysiology and provide a practical guide toward conducting experiments and analyses in this emerging field. Our examples focus on using linear models in the study of human language and audition. We show how to calculate auditory receptive fields from natural sounds as well as how to decode neural recordings to predict speech. The paper aims to be a useful tutorial to these approaches, and a practical introduction to using machine learning and applied statistics to build models of neural activity. The data analytic approaches we discuss may also be applied to other sensory modalities, motor systems, and cognitive systems, and we cover some examples in these areas. In addition, a collection of Jupyter notebooks is publicly available as a complement to the material covered in this paper, providing code examples and tutorials for predictive modeling in python. The aimis to provide a practical understanding of predictivemodeling of human brain data and to propose best-practices in conducting these analyses.},
-	journal = {Frontiers in Systems Neuroscience},
-	author = {Holdgraf, Christopher Ramsay and Rieger, J.W. and Micheli, C. and Martin, S. and Knight, R.T. and Theunissen, F.E.},
-	year = {2017},
-	keywords = {Decoding models, Encoding models, Electrocorticography (ECoG), Electrophysiology/evoked potentials, Machine learning applied to neuroscience, Natural stimuli, Predictive modeling, Tutorials}
-}
-
-@book{ruby,
-  title     = {The Ruby Programming Language},
-  author    = {Flanagan, David and Matsumoto, Yukihiro},
-  year      = {2008},
-  publisher = {O'Reilly Media}
-}
\ No newline at end of file
diff --git a/_build/features/Intro_clases.html b/_build/features/Intro_clases.html
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-    <div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/Intro_clases.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<h1 id="Clases">Clases<a class="anchor-link" href="#Clases"> </a></h1><p>En Python, todo es un objeto, es decir, una instancia o realización de un clase. Cada objeto tiene unos <em>métodos</em>:</p>
-<div class="highlight"><pre><span></span><span class="n">objeto</span><span class="o">.</span><span class="n">metodo</span><span class="p">(</span><span class="o">...</span><span class="p">)</span>
-<span class="n">que</span> <span class="n">corresponde</span>  <span class="n">a</span> <span class="n">funciónes</span> <span class="n">internas</span> <span class="k">del</span> <span class="n">objeto</span><span class="p">,</span> <span class="n">y</span> <span class="n">también</span> <span class="n">puede</span> <span class="n">tener</span> <span class="n">_atributos_</span><span class="p">:</span>
-</pre></div>
-<div class="highlight"><pre><span></span><span class="n">objeto</span><span class="o">.</span><span class="n">atributo</span>
-</pre></div>
-<p>que son variables internas dentro del objeto: <code>self.atributo=....</code>.</p>
-<p>Algunos objetos en Python también pueden recibir parámetros de entrada a modo de claves de un diccionario interno del objeto</p>
-<div class="highlight"><pre><span></span><span class="n">objeto</span><span class="p">[</span><span class="s1">&#39;key&#39;</span><span class="p">]</span><span class="o">=</span><span class="s1">&#39;value&#39;</span>
-</pre></div>
-<p>entro otras muchas propiedades</p>
-<p>Ejemplos de objetos son:</p>
-<ul>
-<li><code>int</code>: Enteros</li>
-<li><code>float</code>: Números punto flotante (floating point numbers)</li>
-<li><code>str</code>: Cadenas de caracteres</li>
-<li><code>list</code>: Listas </li>
-<li><code>dict</code>: Diccionarios.</li>
-</ul>
-<p>Si el tipo de objeto es conocido, uno pude comprobar si un objeto determinado corresponde a ese tipo con <code>isinstance</code>:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span><span class="o">=</span><span class="s1">&#39;hola mundo&#39;</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Is `s` an string?: </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">s</span><span class="p">,</span><span class="nb">str</span><span class="p">))</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Is `s` a float?: </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">s</span><span class="p">,</span><span class="nb">float</span><span class="p">))</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>Is `s` an string?: True
-Is `s` a float?: False
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Dentro del paradigma de objetos es posible agrupar diferentes conjuntos de variables de modo que el nombre de un atributo a método adquiere dos partes, una parte principal, que en el análogo con el <em>nombre completo</em> de una persona podríamos asimilar a su <em>apellido</em>, y el método o atributo, que sería como el primer <em>nombre</em>, separados por un punto:</p>
-<ul>
-<li>Método:  <code>last_name.first_name()</code></li>
-<li>Atributo: <code>last_name.first_name</code>.</li>
-</ul>
-<p>Esto nos permite tener objetos dentro de un programa con igual <em>nombre</em>, pero diferente <em>appellido</em>. Como por ejemplo, los diferentes $\cos(x)$ que vienen en los diferentes módulos matématicos implementados en Python. Por eso es recomendado cargar los módulos manteniendo el espacio de nombres (que en nuestra analogía sería el espacio de <em>apellidos</em>). Cuando el modulo tenga un nombre muy largo (más de cuatro caracteres), se puede usar una abreviatura lo suficientemente original para evitar que pueda ser sobreescrita por un nuevo objeto:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">math</span>
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="n">k</span><span class="o">=</span><span class="mi">3</span> <span class="c1">#N/m</span>
-<span class="n">m</span><span class="o">=</span><span class="mi">2</span> <span class="c1">#Kg</span>
-<span class="n">A</span><span class="o">=</span><span class="mi">3</span> <span class="c1">#m</span>
-<span class="n">t</span><span class="o">=</span><span class="mi">2</span> <span class="c1">#s</span>
-<span class="n">ω</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">k</span><span class="o">/</span><span class="n">m</span><span class="p">)</span> <span class="c1">#rad/s</span>
-<span class="c1">#ver https://pyformat.info/</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;x(t)=</span><span class="si">{:.2f}</span><span class="s1"> m&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> 
-    <span class="n">A</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span> <span class="n">ω</span><span class="o">*</span><span class="n">t</span> <span class="p">)</span>
-     <span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;x(t)=</span><span class="si">{:.2f}</span><span class="s1"> m&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> 
-    <span class="n">A</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span> <span class="n">ω</span><span class="o">*</span><span class="n">t</span> <span class="p">)</span>
-     <span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>x(t)=-2.31 m
-x(t)=-2.31 m
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Note que <code>import math as m</code> entraría en conflicto con la definición de <code>m</code> en <code>m=2</code></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>La forma recomendada de importar los diferentes módulos y el uso de sus métodos y atributos suele resumirse en <em>Cheat Sheets</em>. Para Python científico recomendamos las elaboradas por <a href="https://learn.datacamp.com/">Data Camp</a>, que pueden consultarse <a href="https://drive.google.com/drive/folders/11ReN5mXiYGsBjNfdj3zEj2Z1E3bZ7TRk?usp=sharing">aquí</a></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Para programación de Python en general se recomienda el estándar <a href="https://www.python.org/dev/peps/pep-0008/">PEP 8</a> de Python</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Antes de comenzar con las clases, es conveniente resumir el paradigma de <em>programación funcional</em>:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Programaci&#243;n-funcional">Programaci&#243;n funcional<a class="anchor-link" href="#Programaci&#243;n-funcional"> </a></h2><p>En cálculo científico el paradigma funcional, en el cual el programa se escribe en términos de funciones, suele ser suficiente.</p>
-<p>El esqueleto de un programa funcional  es típicamente del siguiente tipo</p>
-<div class="highlight"><pre><span></span><span class="ch">#!/usr/bin/user/env python3</span>
-<span class="kn">import</span> <span class="nn">somemodule</span> <span class="k">as</span> <span class="nn">sm</span>
-<span class="k">def</span> <span class="nf">func1</span><span class="p">(</span><span class="o">...</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Ayuda func1</span>
-<span class="sd">   &#39;&#39;&#39;</span>
-    <span class="o">.....</span>
-
-<span class="k">def</span> <span class="nf">func2</span><span class="p">(</span><span class="o">...</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Ayuda func2</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="o">.....</span>
-
-
-<span class="k">def</span> <span class="nf">main</span><span class="p">(</span><span class="o">...</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Ayuda función principal</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="n">x</span><span class="o">=</span><span class="n">func1</span><span class="p">(</span><span class="o">...</span><span class="p">)</span>
-    <span class="n">y</span><span class="o">=</span><span class="n">func2</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-<span class="k">if</span> <span class="vm">__name__</span><span class="o">=</span><span class="s1">&#39;__main__&#39;</span><span class="p">:</span>
-    <span class="n">z</span><span class="o">=</span><span class="n">main</span><span class="p">(</span><span class="o">...</span><span class="p">)</span>
-    <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;El resultado final es: </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-</pre></div>
-<p>Para su diseño, el programa se debe separar  sus partes independientes y para cada una de ellas se debe definir una función.</p>
-<p><em>La función ideal es una  que se pueda reutilizar facilamente en otro contexto.</em></p>
-<p>La última función, <code>main(...)</code> combina todas las anteriores para entregar el resultado final del programa.</p>
-<p>La instrucción</p>
-<div class="highlight"><pre><span></span><span class="k">if</span> <span class="vm">__name__</span><span class="o">=</span><span class="s1">&#39;__main__&#39;</span>
-</pre></div>
-<p>permite que el programa pueda ser usado también como un módulo de Python, es decir que se pueda cargar desde otro programa con el <code>import</code>. En tal caso, la variable 
-interna de Python <code>__name__</code>  es diferente a la cadena de caracteres <code>'__main__'</code> y esa parte del programa no se ejecuta. Dentro de Jupyter:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="vm">__name__</span>
-</pre></div>
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-<pre>&#39;__main__&#39;</pre>
-</div>
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-<p>Ejemplo módulo</p>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%%writefile</span> example.py
-<span class="ch">#!/usr/bin/env python3</span>
-<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;check __name__: </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="vm">__name__</span><span class="p">))</span>
-
-<span class="k">def</span> <span class="nf">hola</span><span class="p">():</span>
-    <span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;check __name__ como módulo: </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="vm">__name__</span><span class="p">))</span>
-    <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;mundo&#39;</span><span class="p">)</span>
-    
-<span class="k">if</span> <span class="vm">__name__</span><span class="o">==</span><span class="s1">&#39;__main__&#39;</span><span class="p">:</span>
-    <span class="n">hola</span><span class="p">()</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<div class="output_wrapper">
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-<pre>Overwriting example.py
-</pre>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p><code>ls</code> funciona</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ls</span> <span class="o">-</span><span class="n">l</span> <span class="n">example</span><span class="o">.</span><span class="n">py</span>
-</pre></div>
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-<pre>-rwxr-xr-x 1 restrepo restrepo 205 Jun 16 12:20 <span class="ansi-green-intense-fg ansi-bold">example.py</span>*
-</pre>
-</div>
-</div>
-</div>
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-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p><code>cat</code> funciona</p>
-
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">cat</span> <span class="n">example</span><span class="o">.</span><span class="n">py</span> 
-</pre></div>
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-<pre>#!/usr/bin/env python3
-print( &#39;check __name__: {}&#39;.format(__name__))
-
-def hola():
-    print( &#39;check __name__ como módulo: {}&#39;.format(__name__))
-    print(&#39;mundo&#39;)
-    
-if __name__==&#39;__main__&#39;:
-    hola()
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Cambia los permisos a ejecución</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">!</span> chmod a+x example.py
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ls</span> <span class="o">-</span><span class="n">l</span> <span class="n">example</span><span class="o">.</span><span class="n">py</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>-rwxr-xr-x 1 restrepo restrepo 205 Jun 16 12:20 <span class="ansi-green-intense-fg ansi-bold">example.py</span>*
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Corre el programa desde la consola</p>
-
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">!</span>./example.py
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
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-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>check __name__: __main__
-check __name__ como módulo: __main__
-mundo
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Uso como módulo</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">example</span>
-</pre></div>
-
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-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>check __name__: example
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">example</span><span class="o">.</span><span class="n">hola</span><span class="p">()</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>check __name__ como módulo: example
-mundo
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Ejecutarlo desde una celda de Jupyter es equivalente a ejecutarlo desde la consola</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="ch">#!/usr/bin/env python3</span>
-<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;check __name__: </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="vm">__name__</span><span class="p">))</span>
-
-<span class="k">def</span> <span class="nf">hola</span><span class="p">():</span>
-    <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;mundo&#39;</span><span class="p">)</span>
-    
-<span class="k">if</span> <span class="vm">__name__</span><span class="o">==</span><span class="s1">&#39;__main__&#39;</span><span class="p">:</span>
-    <span class="n">hola</span><span class="p">()</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>check __name__: __main__
-mundo
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Clases">Clases<a class="anchor-link" href="#Clases"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Aunque en Python se puede trabajar directamente con objetos, en general un objeto es una instancia de un clase. Es decir, debe incializarse a partir de una Clase. Esto típicamente involucra ejecutar varios métodos e inicializar varios atributos bajo el espacio de nombres del objeto incializado. Por ejemplo, para inicializar las clases 'int', 'float', 'str', 'list','dict'</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">n</span><span class="o">=</span><span class="nb">int</span><span class="p">()</span> <span class="c1"># ⬄ to n=0</span>
-<span class="n">n</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>0</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span><span class="o">=</span><span class="nb">float</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="c1"># ⬄ x=3.</span>
-<span class="n">x</span>
-</pre></div>
-
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-</div>
-</div>
-
-<div class="output_wrapper">
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-
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-
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-<div class="output_text output_subarea output_execute_result">
-<pre>3.0</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">l</span><span class="o">=</span><span class="nb">list</span><span class="p">()</span> <span class="c1"># ⬄ l=[]</span>
-<span class="n">l</span>
-</pre></div>
-
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-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
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-
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-<div class="output_text output_subarea output_execute_result">
-<pre>[]</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">d</span><span class="o">=</span><span class="nb">dict</span><span class="p">()</span> <span class="c1"># ⬄ d={}</span>
-<span class="n">d</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
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-<div class="output_text output_subarea output_execute_result">
-<pre>{}</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>La principal motivación para escribir una clase en lugar de una función, es que la clase puede ser la base para generar nuevas clases que hereden los métodos y atributos de la clase original.</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Por ejemplo el DataFrame de Pandas es una clase que puede ser inicializada de múltiples formas. Además, está diseñada para que pueda ser extendida facilmente: <a href="https://pandas.pydata.org/pandas-docs/stable/development/extending.html">https://pandas.pydata.org/pandas-docs/stable/development/extending.html</a></p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="input">
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-<div class="inner_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<div class="highlight"><pre><span></span><span class="n">In</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="err">??</span>
-<span class="o">...</span>
-<span class="k">class</span> <span class="nc">DataFrame</span><span class="p">(</span><span class="n">NDFrame</span><span class="p">):</span>
-    <span class="o">...</span>
-    <span class="c1"># ----------------------------------------------------------------------</span>
-    <span class="c1"># Constructors</span>
-
-    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
-                 <span class="n">copy</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">data</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
-            <span class="o">...</span>
-</pre></div>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Como puede verse, un <code>DataFrame</code> es una subclase (es decir, un caso especial) de <code>NDFrame</code>.</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Una <em>instancia</em> u objeto de la clase <code>DataFrame</code>, <code>df</code> a continuación, se puede inicializar de diferentes maneras</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">()</span>
-<span class="n">df</span>
-</pre></div>
-
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-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
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-
-<div class="output_text output_subarea output_execute_result">
-<pre>(0, 0)</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">([{</span><span class="s1">&#39;A&#39;</span><span class="p">:</span><span class="mi">1</span><span class="p">,</span><span class="s1">&#39;B&#39;</span><span class="p">:</span><span class="mi">2</span><span class="p">}])</span>
-<span class="n">df</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
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-<span class="n">df</span>
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-<h2 id="Programaci&#243;n-por-clases">Programaci&#243;n por clases<a class="anchor-link" href="#Programaci&#243;n-por-clases"> </a></h2>
-</div>
-</div>
-</div>
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-<p>Una clase se puede pensar como un conjuto de funciones y atributos que comparten algo común.</p>
-<p>Algunas veces, cuando la complejidad del problema se puede descomponer en una estructura de capas, donde la capa interna es la mas simple y las capas más externas van aumentando la complejidad, pude ser conveniente pensar en una estructura de clases.</p>
-<p>De hecho, la clase básica puede heredar todas sus propiedades  a subclases basadas en ella.</p>
-<p>El espacio de nombres asociado a la clase se define con el nombre génerico de <code>self</code> el cual toma el nombre de las instancia (objeto) asociada a la inicialización de la clase. 
-Las variables globales de la clase pasán a ser automáticamente atributos del objeto, y nuevo atributos de pueden definir dentro del espacio de nombres <code>self</code> dentro de cada función de la clase.</p>
-<p>Esqueleto:</p>
-<div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">clasenueva</span><span class="p">:</span>
-     <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">     Ayuda de la clase</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-
-    <span class="n">var1</span><span class="o">=</span><span class="s1">&#39;valor&#39;</span>
-    <span class="k">def</span> <span class="nf">func1</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="o">...</span><span class="p">):</span>
-        <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">        Ayuda del método</span>
-<span class="sd">        &#39;&#39;&#39;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">var2</span><span class="o">=</span><span class="s1">&#39;hola mundo&#39;</span>
-        <span class="o">....</span>
-    <span class="k">def</span> <span class="nf">func2</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="o">....</span><span class="p">):</span>
-        <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">        Ayuda del método</span>
-<span class="sd">        &#39;&#39;&#39;</span>        
-        <span class="nb">print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">var1</span><span class="p">)</span>
-        <span class="nb">print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">var2</span><span class="p">)</span>
-        <span class="o">....</span>
-
-    <span class="k">def</span> <span class="nf">main</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="o">....</span><span class="p">):</span>
-        <span class="o">....</span>
-<span class="k">if</span> <span class="vm">__name__</span><span class="o">==</span><span class="s1">&#39;__main__&#39;</span><span class="p">:</span>
-         <span class="n">objetonuevo</span><span class="o">=</span><span class="n">clasenueva</span><span class="p">()</span>
-         <span class="c1"># self → objetonuevo</span>
-        <span class="n">objetonuevo</span><span class="o">.</span><span class="n">main</span><span class="p">()</span>
-</pre></div>
-
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Cada una de las funciones pasan a ser métodos de la clase, mientras que cada una de la variables globales y con espacio de nombre <code>self</code> pasan a ser atributos de la clase.</p>
-<p><strong>Ejemplo</strong>:</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">class</span> <span class="nc">clasenueva</span><span class="p">:</span>
-    <span class="n">var</span><span class="o">=</span><span class="s1">&#39;hola&#39;</span>
-    <span class="n">var2</span><span class="o">=</span><span class="p">[]</span>
-    <span class="k">def</span> <span class="nf">main</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">var</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">var</span><span class="o">+</span><span class="s1">&#39; mundo&#39;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">var3</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">var2</span><span class="o">+</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
-        <span class="nb">print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">var</span><span class="p">)</span>
-</pre></div>
-
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-<p>Creación del objeto <code>c</code>, de manera que <code>self</code> → <code>c</code>. <code>c</code> es una instancia de la clase <code>clasenueva</code> a continuación:</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">c</span><span class="o">=</span><span class="n">clasenueva</span><span class="p">()</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">c</span><span class="o">.</span><span class="n">var</span>
-</pre></div>
-
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-<pre>&#39;hola&#39;</pre>
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-<p><code>main</code> es un método de la clase</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">c</span><span class="o">.</span><span class="n">main</span><span class="p">()</span>
-</pre></div>
-
-    </div>
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-<div class="output_subarea output_stream output_stdout output_text">
-<pre>hola mundo
-</pre>
-</div>
-</div>
-</div>
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-<p><code>var</code> es un atributo de la clase</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">c</span><span class="o">.</span><span class="n">var</span>
-</pre></div>
-
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-<pre>&#39;hola mundo&#39;</pre>
-</div>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">c</span><span class="o">.</span><span class="n">var3</span>
-</pre></div>
-
-    </div>
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-<pre>[3]</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Para la creación de clases disponemos de métodos especiales que podemos definir para adicionar "magia" a nuestras clases. Estas están simpre rodeadas de un doble guión bajo, por ejemplo: <code>__init__</code> o <code>__lt__</code>. Una lista completa de ellos con su explicación de uso se puede encontrar en <a href="#References">1</a>.</p>
-<p>Resaltamos a continuación algunos de ellos</p>
-<ul>
-<li><code>__init__</code>: Se ejecuta automáticamente al inicializar la clase</li>
-<li><code>__call__</code>: Se ejecuta cuando el objeto creado es usado como función</li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
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-<h3 id="Herencia">Herencia<a class="anchor-link" href="#Herencia"> </a></h3><p>Los métodos especiales se heredan automáticamente desde la clase inicial, que llamaremos super clase inicializamos la clase con la super clase como argumento, es decir, en forma genérica como:</p>
-<div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">subclass</span><span class="p">(</span><span class="n">superclass</span><span class="p">):</span>
-    <span class="o">...</span>
-</pre></div>
-
-</div>
-</div>
-</div>
-</div>
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-<h2 id="Clase-vector">Clase vector<a class="anchor-link" href="#Clase-vector"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Comenzaremos definiendo un alias de clase <code>list</code> que llameremos vector y que funcione exactamente igual que la clase lista. Todas los métodos especiales se heredan automáticamente</p>
-
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">class</span> <span class="nc">vector</span><span class="p">(</span><span class="nb">list</span><span class="p">):</span>
-    <span class="k">pass</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">l</span><span class="o">=</span><span class="nb">list</span><span class="p">()</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">v</span><span class="o">=</span><span class="n">vector</span><span class="p">()</span>
-</pre></div>
-
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-</div>
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">v</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="n">v</span>
-</pre></div>
-
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-</div>
-</div>
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-<div class="output_text output_subarea output_execute_result">
-<pre>[1]</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">l1</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
-<span class="n">l2</span><span class="o">=</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">]</span>
-<span class="n">l1</span><span class="o">+</span><span class="n">l2</span>
-</pre></div>
-
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-</div>
-</div>
-
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-<pre>[1, 2, 3, 4]</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Vamos a inicializar dos instancias de la clase <code>vector</code> y comprobar que suman como listas</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">v1</span><span class="o">=</span><span class="n">vector</span><span class="p">(</span><span class="n">l1</span><span class="p">)</span>
-<span class="n">v2</span><span class="o">=</span><span class="n">vector</span><span class="p">(</span><span class="n">l2</span><span class="p">)</span>
-<span class="n">v1</span><span class="o">+</span><span class="n">v2</span>
-</pre></div>
-
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-<pre>[1, 2, 3, 4]</pre>
-</div>
-
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-</div>
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Ahora <em>reemplazaremos</em> el método especial <code>__add__</code> de la lista para que el operador <code>+</code> realice la suma vectorial:</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">class</span> <span class="nc">vector</span><span class="p">(</span><span class="nb">list</span><span class="p">):</span>
-    <span class="k">def</span> <span class="fm">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">        __add__ asocia la operación del símbolo &#39;+&#39;</span>
-<span class="sd">        &#39;&#39;&#39;</span>
-        <span class="k">return</span> <span class="n">vector</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">))</span>    
-</pre></div>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">v1</span><span class="o">=</span><span class="n">vector</span><span class="p">(</span><span class="n">l1</span><span class="p">)</span>
-<span class="n">v2</span><span class="o">=</span><span class="n">vector</span><span class="p">(</span><span class="n">l2</span><span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">v1</span><span class="o">+</span><span class="n">v2</span>
-</pre></div>
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-<pre>[4, 6]</pre>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Si lo que queremos es <em>modificar</em> el comportomiento de un método especial entonces debemos usar la función <code>super</code>.</p>
-<p>Para mantener la herencia el comportamiento de los métodos de una clase inicial, ls super clase, se usa la función <code>super</code> que mantiene  la <strong>herencia</strong>. Ver:</p>
-<ul>
-<li><a href="https://realpython.com/python-super/">https://realpython.com/python-super/</a> (<a href="https://web.archive.org/web/20190922043302/https://realpython.com/python-super/">Backup</a>)</li>
-<li><a href="https://rhettinger.wordpress.com/2011/05/26/super-considered-super/">https://rhettinger.wordpress.com/2011/05/26/super-considered-super/</a> (<a href="https://web.archive.org/web/20191129114558/https://rhettinger.wordpress.com/2011/05/26/super-considered-super/">Backup</a>)</li>
-</ul>
-<blockquote><p><code>super()</code> gives you access to methods in a superclass from the subclass that inherits from it.</p>
-<p><code>super()</code> is the same as <code>super(__class__, &lt;first argument&gt;)</code>: the first is the subclass, and the second parameter is an object that is an instance of that subclass.</p>
-</blockquote>
-<p>La estructura general es como la siguiente, basada en <a href="http://stackoverflow.com/questions/13404476/inherited-class-variable-modification-in-python">Inherited class variable modification in Python</a>:</p>
-<div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">subclass</span><span class="p">(</span><span class="n">superclass</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__special__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="c1">#Modifications to __especial__ here</span>
-        <span class="o">...</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">subclass</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__special__</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="o">...</span>
-</pre></div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Como ejemplo, vamos a implemetar el atributo <code>modulus</code> a la clase <code>vector</code> dentro del método especial <code>__init__</code>:</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">math</span>
-<span class="k">class</span> <span class="nc">vector</span><span class="p">(</span><span class="nb">list</span><span class="p">):</span>
-    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">        Add the modulus of a vector at initialization </span>
-<span class="sd">        &#39;&#39;&#39;</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">l</span><span class="o">=</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">modulus</span><span class="o">=</span><span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span> <span class="nb">sum</span><span class="p">(</span> <span class="nb">map</span><span class="p">(</span> <span class="k">lambda</span> <span class="n">l</span><span class="p">:</span> <span class="n">l</span><span class="o">*</span><span class="n">l</span><span class="p">,</span><span class="n">l</span> <span class="p">))</span> <span class="p">)</span>
-        <span class="k">except</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">modulus</span><span class="o">=</span><span class="mi">0</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">vector</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="k">def</span> <span class="fm">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">        __add__ asocia la operación del símbolo &#39;+&#39;</span>
-<span class="sd">        &#39;&#39;&#39;</span>
-        <span class="k">return</span> <span class="n">vector</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">))</span>        
-</pre></div>
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-<span class="n">v1</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">v1</span><span class="o">.</span><span class="n">modulus</span>
-</pre></div>
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-<pre>3.605551275463989</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">type</span><span class="p">(</span><span class="n">v1</span><span class="p">)</span>
-</pre></div>
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-<pre>__main__.vector</pre>
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-<p>Una implementación completa de la clase vector, adapta de <a href="http://code.activestate.com/recipes/52272-vector-a-list-based-vector-class-supporting-elemen/">vector: a list based vector class supporting elementwise operations (python recipe)</a>, se puede encontrar <a href="./vector.ipynb">aquí</a></p>
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-<h2 id="Otro-ejemplo">Otro ejemplo<a class="anchor-link" href="#Otro-ejemplo"> </a></h2>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">class</span> <span class="nc">veterinaria</span><span class="p">:</span>
-    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">x</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">tipo</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;perro&#39;</span><span class="p">:</span><span class="s1">&#39;guau&#39;</span><span class="p">,</span><span class="s1">&#39;gato&#39;</span><span class="p">:</span><span class="s1">&#39;miau&#39;</span><span class="p">}</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">sonido</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">tipo</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">nombre</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">nombre</span><span class="o">==</span><span class="s1">&#39;greco&#39;</span><span class="p">:</span>
-            <span class="nb">print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">sonido</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;grrr!!&quot;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">color</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">nombre</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">nombre</span><span class="o">==</span><span class="s1">&#39;greco&#39;</span><span class="p">:</span>
-            <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;blanco&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<ul>
-<li>Cundo se inicializa la clase la función llamada <code>__init__</code>, se ejecuta automáticamente.</li>
-<li>La función <code>__call__</code> permite usar el objeto directamente como función, sin hacer referencia al método, que por supuesto es   <code>__call__</code></li>
-</ul>
-<p>Hay muchos otros métodos especiales, que comienzan con un <code>__</code>.... Usar <code>&lt;TAB&gt;</code> a continuación para ver algunos</p>
-
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">veterinaria</span><span class="o">.</span><span class="n">__</span>
-</pre></div>
-
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-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Para utilizar la clase, primero se debe incializar como un objeto. Crear la instancia de la clase.</p>
-<p><strong>Ejemplo</strong>: Crea la instancia <code>mascotafeliz</code></p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">mascotafeliz</span><span class="o">=</span><span class="n">veterinaria</span><span class="p">(</span><span class="s1">&#39;perro&#39;</span><span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">mascotafeliz</span><span class="p">(</span><span class="s1">&#39;greco&#39;</span><span class="p">)</span>
-</pre></div>
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-<pre>guau
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">mascotafeliz</span><span class="o">.</span><span class="n">color</span><span class="p">(</span><span class="s1">&#39;greco&#39;</span><span class="p">)</span>
-</pre></div>
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-<pre>blanco
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">perro</span><span class="o">.</span><span class="n">tipo</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;vaca&#39;</span><span class="p">)</span>
-</pre></div>
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-<h2 id="Herencia">Herencia<a class="anchor-link" href="#Herencia"> </a></h2>
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">class</span> <span class="nc">finanzas</span><span class="p">(</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">):</span>
-    <span class="k">pass</span>
-</pre></div>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f</span><span class="o">=</span><span class="n">finanzas</span><span class="p">()</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">type</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-</pre></div>
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-<pre>__main__.finanzas</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f</span><span class="p">([{</span><span class="s1">&#39;A&#39;</span><span class="p">:</span><span class="mi">1</span><span class="p">}])</span>
-</pre></div>
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-<pre>
-<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
-<span class="ansi-red-fg">TypeError</span>                                 Traceback (most recent call last)
-<span class="ansi-green-fg">&lt;ipython-input-47-d585aedd1152&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span><span class="ansi-blue-fg">()</span>
-<span class="ansi-green-fg">----&gt; 1</span><span class="ansi-red-fg"> </span>f<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">[</span><span class="ansi-blue-fg">{</span><span class="ansi-blue-fg">&#39;A&#39;</span><span class="ansi-blue-fg">:</span><span class="ansi-cyan-fg">1</span><span class="ansi-blue-fg">}</span><span class="ansi-blue-fg">]</span><span class="ansi-blue-fg">)</span>
-
-<span class="ansi-red-fg">TypeError</span>: &#39;finanzas&#39; object is not callable</pre>
-</div>
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-    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">hkdict</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">has_key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">k</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">k</span> <span class="ow">in</span> <span class="bp">self</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">d</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;perro&#39;</span><span class="p">:</span><span class="s1">&#39;guau&#39;</span><span class="p">,</span><span class="s1">&#39;gato&#39;</span><span class="p">:</span><span class="s1">&#39;miau&#39;</span><span class="p">}</span>
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-<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
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-<span class="ansi-green-fg">&lt;ipython-input-52-4598d173b43c&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span><span class="ansi-blue-fg">()</span>
-<span class="ansi-green-fg">----&gt; 1</span><span class="ansi-red-fg"> </span>d<span class="ansi-blue-fg">.</span>has_key<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">&#39;gato&#39;</span><span class="ansi-blue-fg">)</span>
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-<span class="ansi-red-fg">AttributeError</span>: &#39;dict&#39; object has no attribute &#39;has_key&#39;</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">dd</span><span class="o">.</span><span class="n">has_key</span><span class="p">(</span><span class="s1">&#39;vaca&#39;</span><span class="p">)</span>
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-<pre>False</pre>
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-<h1 id="Implementation-of-arXiv:1905.13729">Implementation of <a href="https://arxiv.org/abs/1905.13729">arXiv:1905.13729</a><a class="anchor-link" href="#Implementation-of-arXiv:1905.13729"> </a></h1><p>See also: DOI: <a href="http://doi.org/10.1103/PhysRevD.101.095032">10.1103/PhysRevD.101.095032</a></p>
-<h2 id="General-solution-to-the-U(1)-anomaly-equations">General solution to the U(1) anomaly equations<a class="anchor-link" href="#General-solution-to-the-U(1)-anomaly-equations"> </a></h2><p>Let a vector $\boldsymbol{z}$ with $N$ integer entries such that
-$$ \sum_{i=1}^N z_i=0\,,\qquad \sum_{i=1}^N z_i^3=0\,.$$
-We like to build this set of of $N$ from two subsets $\boldsymbol{\ell}$ and $\boldsymbol{k}$ with sizes</p>
-\begin{align}
-\operatorname{dim}(\boldsymbol{\ell})=&amp;
-\begin{cases}
-\alpha=\frac{N}{2}-1\,, &amp; \text{ if $N$  even } \\
-\beta=\frac{N-3}{2}\,, &amp; \text{ if $N$  odd }\\
-\end{cases};&amp;
-\operatorname{dim}(\boldsymbol{k})=&amp;
-\begin{cases}
-\alpha=\frac{N}{2}-1\,, &amp; \text{ if $N$  even } \\
-\beta+1=\frac{N-1}{2}\,, &amp; \text{ if $N$  odd }\\
-\end{cases};
-\end{align}<ul>
-<li>$N$ even: Consider the following two examples of $\boldsymbol{z}$ with vector-like solutions, i.e, with opposite integeres which automatically satisfy the equations 
-\begin{align}
-\boldsymbol{x}=&amp;\left(\ell_{1}, {k}_{1}, \ldots, k_{\alpha},-\ell_{1},-k_{1}, \ldots,-k_{\alpha}\right)\\
-\boldsymbol{y}=&amp;\left(0,0,\ell_{1},  \ldots, \ell_{\alpha},-\ell_1, \ldots,-\ell_{\alpha}\right)\,.
-\end{align}</li>
-<li>$N$ odd: Consider the two vector-like solutions
-\begin{align}
-\begin{array}
-\boldsymbol{x}=&amp;\left(0, k_{1}, \ldots, k_{\beta+1},-k_{1}, \ldots,-k_{\beta+1}\right) \\
-\boldsymbol{y}=&amp;\left(\ell_{1}, \ldots, \ell_{\beta}, k_{1}, 0,-\ell_{1}, \ldots,-\ell_{\beta},-k_{1}\right)
-\end{array}
-\end{align}</li>
-</ul>
-<p>From any of this, we can build a final $\boldsymbol{z}$ which can includes <em>chiral</em> solutions, i.e, non vector-like solutions</p>
-$$
-\boldsymbol{x} \oplus \boldsymbol{y} \equiv\left(\sum_{i=1}^{N} x_{i} y_{i}^{2}\right)\boldsymbol{x}-\left(\sum_{i=1}^{N} x_{i}^{2} y_{i}\right)\boldsymbol{y}\,.
-$$
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-<h2 id="Python-implmentation">Python implmentation<a class="anchor-link" href="#Python-implmentation"> </a></h2><p>Obtain a numpy array <code>z</code> of <code>N</code> integers which satisfy the Diophantine equations</p>
-<div class="highlight"><pre><span></span><span class="o">&gt;&gt;&gt;</span> <span class="n">z</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
-<span class="mi">0</span>
-<span class="o">&gt;&gt;&gt;</span> <span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
-<span class="mi">0</span>
-</pre></div>
-<p>The input is two lists <code>l</code> and <code>k</code> with any <code>(N-3)/2</code> and <code>(N-1)/2</code> integers for <code>N</code> odd, or <code>N/2-1</code> and <code>N/2-1</code> for <code>N</code> even (<code>N&gt;4</code>).
-The function is implemented below under the name: <code>free(l,k)</code></p>
-<p><a href="https://colab.research.google.com/github/restrepo/anomaly/blob/main/anomalysolutions-class.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<p><a href="https://stackoverflow.com/a/43793179/2268280">https://stackoverflow.com/a/43793179/2268280</a></p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">from</span> <span class="nn">astropy.table</span> <span class="kn">import</span> <span class="n">Table</span>
-<span class="kn">import</span> <span class="nn">itertools</span>
-<span class="kn">import</span> <span class="nn">sys</span>
-<span class="kn">import</span> <span class="nn">os</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="n">reduce</span>
-<span class="kn">import</span> <span class="nn">warnings</span>
-<span class="n">warnings</span><span class="o">.</span><span class="n">filterwarnings</span><span class="p">(</span><span class="s2">&quot;ignore&quot;</span><span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">!</span>pip install anomalies <span class="m">2</span>&gt;/dev/null &gt; /dev/null
-</pre></div>
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-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">anomaly</span><span class="o">.</span><span class="n">free</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],[</span><span class="mi">4</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">])</span>
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-<pre>array([  3,   3,   3, -12, -12,  15])</pre>
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-</pre></div>
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-<pre>3</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">anomaly</span><span class="o">.</span><span class="n">free</span><span class="o">.</span><span class="n">simplified</span>
-</pre></div>
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-<pre>array([ 1,  1,  1, -4, -4,  5])</pre>
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-<h2 id="Analysis">Analysis<a class="anchor-link" href="#Analysis"> </a></h2><p><code>solutions</code> class → Initialize the object to obtain anomaly free solutions for any set of <code>N</code> integers</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#TODO: inherit from free class</span>
-<span class="kn">import</span> <span class="nn">sys</span>
-<span class="k">def</span> <span class="nf">_get_chiral</span><span class="p">(</span><span class="n">q</span><span class="p">,</span><span class="n">q_max</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">inf</span><span class="p">):</span>
-    <span class="c1">#Normalize to positive minimum</span>
-    <span class="k">if</span> <span class="n">q</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">&lt;</span><span class="mi">0</span> <span class="ow">or</span> <span class="p">(</span><span class="n">q</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">==</span><span class="mi">0</span> <span class="ow">and</span> <span class="n">q</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">&lt;</span><span class="mi">0</span><span class="p">):</span>
-        <span class="n">q</span><span class="o">=-</span><span class="n">q</span>
-    <span class="c1">#Divide by GCD</span>
-    <span class="n">GCD</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">gcd</span><span class="o">.</span><span class="n">reduce</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-    <span class="n">q</span><span class="o">=</span><span class="p">(</span><span class="n">q</span><span class="o">/</span><span class="n">GCD</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-    <span class="k">if</span> <span class="p">(</span> <span class="c1">#not 0 in z and </span>
-          <span class="mi">0</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">[</span> <span class="nb">sum</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">permutations</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="p">]</span> <span class="ow">and</span> <span class="c1">#avoid vector-like and multiple 0&#39;s</span>
-          <span class="c1">#q.size &gt; np.unique(q).size and # check for at least a duplicated entry</span>
-          <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">()</span><span class="o">&lt;=</span><span class="n">q_max</span>
-           <span class="p">):</span>
-        <span class="k">return</span> <span class="n">q</span><span class="p">,</span><span class="n">GCD</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="kc">None</span><span class="p">,</span><span class="kc">None</span>
-<span class="k">class</span> <span class="nc">solutions</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Obtain anomaly free solutions with N chiral fields</span>
-<span class="sd">    </span>
-<span class="sd">    Call the initialize object with N and get the solutions:</span>
-<span class="sd">    Example:</span>
-<span class="sd">    &gt;&gt;&gt; s=solutions()</span>
-<span class="sd">    &gt;&gt;&gt; s(6) # N = 6</span>
-<span class="sd">    </span>
-<span class="sd">    Redefine the self.chiral function to implement further restrictions:</span>
-<span class="sd">    inherit from this class and define the new chiral function</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">nmin</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span><span class="n">nmax</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span><span class="n">zmax</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">inf</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">nmin</span><span class="o">=</span><span class="n">nmin</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">nmax</span><span class="o">=</span><span class="n">nmax</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">zmax</span><span class="o">=</span><span class="n">zmax</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">CALL</span><span class="o">=</span><span class="kc">False</span>
-
-    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">N</span><span class="p">,</span><span class="o">*</span><span class="n">args</span><span class="p">,</span><span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">CALL</span><span class="o">=</span><span class="kc">True</span>
-        <span class="k">if</span> <span class="n">N</span><span class="o">%</span><span class="k">2</span>!=0: #odd
-            <span class="n">N_l</span><span class="o">=</span><span class="p">(</span><span class="n">N</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span>
-            <span class="n">N_k</span><span class="o">=</span><span class="p">(</span><span class="n">N</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span>
-        <span class="k">else</span><span class="p">:</span> <span class="c1">#even</span>
-            <span class="n">N_l</span><span class="o">=</span><span class="n">N</span><span class="o">//</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span>
-            <span class="n">N_k</span><span class="o">=</span><span class="n">N_l</span>
-        <span class="n">r</span><span class="o">=</span><span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">nmin</span><span class="p">,</span><span class="bp">self</span><span class="o">.</span><span class="n">nmax</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ls</span><span class="o">=</span><span class="nb">list</span><span class="p">(</span><span class="n">itertools</span><span class="o">.</span><span class="n">product</span><span class="p">(</span> <span class="o">*</span><span class="p">(</span><span class="n">r</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_l</span><span class="p">))</span> <span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ks</span><span class="o">=</span><span class="nb">list</span><span class="p">(</span><span class="n">itertools</span><span class="o">.</span><span class="n">product</span><span class="p">(</span> <span class="o">*</span><span class="p">(</span><span class="n">r</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_k</span><span class="p">))</span> <span class="p">))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">chiral</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        
-        
-    <span class="k">def</span> <span class="nf">chiral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="o">*</span><span class="n">args</span><span class="p">,</span><span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">CALL</span><span class="p">:</span>
-            <span class="n">sys</span><span class="o">.</span><span class="n">exit</span><span class="p">(</span><span class="s1">&#39;Call the initialized object first:</span><span class="se">\n</span><span class="s1">&gt;&gt;&gt; s=solutions()</span><span class="se">\n</span><span class="s1">&gt;&gt;&gt; self(5)&#39;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">list</span><span class="o">=</span><span class="p">[]</span>
-        <span class="n">solt</span><span class="o">=</span><span class="p">[]</span>
-        <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">ls</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">ks</span><span class="p">:</span>
-                <span class="n">l</span><span class="o">=</span><span class="nb">list</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-                <span class="n">k</span><span class="o">=</span><span class="nb">list</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-                <span class="n">q</span><span class="p">,</span><span class="n">gcd</span><span class="o">=</span><span class="n">_get_chiral</span><span class="p">(</span> <span class="n">z</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">k</span><span class="p">)</span> <span class="p">)</span>
-                <span class="c1">#print(z(l,k))</span>
-                <span class="k">if</span> <span class="n">q</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="ow">and</span> <span class="nb">list</span><span class="p">(</span><span class="n">q</span><span class="p">)</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">list</span> <span class="ow">and</span> <span class="nb">list</span><span class="p">(</span><span class="o">-</span><span class="n">q</span><span class="p">)</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">list</span><span class="p">:</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
-                    <span class="n">solt</span><span class="o">.</span><span class="n">append</span><span class="p">({</span><span class="s1">&#39;l&#39;</span><span class="p">:</span><span class="n">l</span><span class="p">,</span><span class="s1">&#39;k&#39;</span><span class="p">:</span><span class="n">k</span><span class="p">,</span><span class="s1">&#39;z&#39;</span><span class="p">:</span><span class="nb">list</span><span class="p">(</span><span class="n">q</span><span class="p">),</span><span class="s1">&#39;gcd&#39;</span><span class="p">:</span><span class="n">gcd</span><span class="p">})</span>
-        <span class="k">return</span> <span class="n">solt</span>
-</pre></div>
-
-    </div>
-</div>
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-<p>Chiral solutions for l and k in the range [-2,2]</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span><span class="o">=</span><span class="n">solutions</span><span class="p">()</span>
-</pre></div>
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-    </div>
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-<p>solutions for $N=5$ integers</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-</pre></div>
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-<pre>[{&#39;l&#39;: [-2], &#39;k&#39;: [-1, 2], &#39;z&#39;: [2, 4, -7, -9, 10], &#39;gcd&#39;: 1},
- {&#39;l&#39;: [-2], &#39;k&#39;: [2, -1], &#39;z&#39;: [1, 5, -7, -8, 9], &#39;gcd&#39;: 4}]</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span>  <span class="n">s</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>  <span class="p">)</span>
-</pre></div>
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-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
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-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
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-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>l</th>
-      <th>k</th>
-      <th>z</th>
-      <th>gcd</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>[-2]</td>
-      <td>[-1, 2]</td>
-      <td>[2, 4, -7, -9, 10]</td>
-      <td>1</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>[-2]</td>
-      <td>[2, -1]</td>
-      <td>[1, 5, -7, -8, 9]</td>
-      <td>4</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
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-<p>To filter solutions with duplicate or triplicate integers, let us create a class <code>dark</code> that inherits from <code>solutions</code>. Therefore, in the argument of the new class is the old class instead of just <code>object</code></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
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-<div class="inner_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">class</span> <span class="nc">dark</span><span class="p">(</span><span class="n">solutions</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Modify the self.chiral function to obtain solutions</span>
-<span class="sd">    with either duplicate or triplicate integers</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="k">def</span> <span class="nf">chiral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">X</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span><span class="n">verbose</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span><span class="n">print_step</span><span class="o">=</span><span class="mi">100000</span><span class="p">):</span>
-        <span class="n">m</span><span class="o">=</span><span class="mi">2</span>
-        <span class="k">if</span> <span class="n">X</span><span class="p">:</span>
-            <span class="n">m</span><span class="o">=</span><span class="mi">3</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">list</span><span class="o">=</span><span class="p">[]</span>
-        <span class="n">solt</span><span class="o">=</span><span class="p">[]</span>
-        <span class="n">tot</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ls</span><span class="p">)</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ks</span><span class="p">)</span>
-        <span class="n">i</span><span class="o">=</span><span class="mi">0</span>
-        <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">ls</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">ks</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-                    <span class="n">i</span><span class="o">=</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span>
-                    <span class="k">if</span> <span class="n">i</span><span class="o">%</span><span class="k">print_step</span>==0:
-                        <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;</span><span class="si">{}</span><span class="s1">/</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">tot</span><span class="p">))</span>
-                <span class="n">l</span><span class="o">=</span><span class="nb">list</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-                <span class="n">k</span><span class="o">=</span><span class="nb">list</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-                <span class="n">q</span><span class="p">,</span><span class="n">gcd</span><span class="o">=</span><span class="n">_get_chiral</span><span class="p">(</span> <span class="n">z</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">k</span><span class="p">)</span> <span class="p">)</span>
-                <span class="c1">#print(z(l,k))</span>
-                <span class="k">if</span> <span class="p">(</span><span class="n">q</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="ow">and</span> 
-                    <span class="nb">list</span><span class="p">(</span><span class="n">q</span><span class="p">)</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">list</span> <span class="ow">and</span> <span class="nb">list</span><span class="p">(</span><span class="o">-</span><span class="n">q</span><span class="p">)</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">list</span> <span class="ow">and</span>
-                    <span class="mi">1</span> <span class="ow">in</span> <span class="p">[</span> <span class="nb">len</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">p</span><span class="p">))</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">permutations</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span> <span class="p">]</span> <span class="ow">and</span>
-                    <span class="c1">#q.size-np.unique(q).size&gt;m</span>
-                    <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">()</span><span class="o">&lt;=</span><span class="bp">self</span><span class="o">.</span><span class="n">zmax</span>
-                   <span class="p">):</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
-                    <span class="n">solt</span><span class="o">.</span><span class="n">append</span><span class="p">({</span><span class="s1">&#39;l&#39;</span><span class="p">:</span><span class="n">l</span><span class="p">,</span><span class="s1">&#39;k&#39;</span><span class="p">:</span><span class="n">k</span><span class="p">,</span><span class="s1">&#39;z&#39;</span><span class="p">:</span><span class="nb">list</span><span class="p">(</span><span class="n">q</span><span class="p">),</span><span class="s1">&#39;gcd&#39;</span><span class="p">:</span><span class="n">gcd</span><span class="p">})</span>
-        <span class="k">return</span> <span class="n">solt</span>        
-</pre></div>
-
-    </div>
-</div>
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-<p>Chiral solutions with repeated integers</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span><span class="o">=</span><span class="n">dark</span><span class="p">()</span>
-</pre></div>
-
-    </div>
-</div>
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-<p>Example: Force solutions with triplicate integers</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-</pre></div>
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-<pre>[]</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span>   <span class="n">s</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="n">X</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>   <span class="p">)</span>
-</pre></div>
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-<style scoped>
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-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>l</th>
-      <th>k</th>
-      <th>z</th>
-      <th>gcd</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>[-2, 2]</td>
-      <td>[-2, -1]</td>
-      <td>[1, 1, 1, -4, -4, 5]</td>
-      <td>8</td>
-    </tr>
-  </tbody>
-</table>
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-1000000/3418801
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-<h2 id="References">References<a class="anchor-link" href="#References"> </a></h2><p>[1] <a href="https://rszalski.github.io/magicmethods/">A Guide to Python's Magic Methods</a></p>
-<p>[2] <a href="https://realpython.com/python3-object-oriented-programming/">https://realpython.com/python3-object-oriented-programming/</a></p>
-<p>[3] <a href="https://drive.google.com/file/d/0BxoOXsn2EUNIWkt5Ni1vV2E4QlU/view?usp=sharing">Building Skills in Object-Oriented Design</a></p>
-<p>[4]  Ver también: <a href="https://gist.github.com/mcleonard/5351452">https://gist.github.com/mcleonard/5351452</a></p>
-<p>[5] Bhasin, Harsh. Python Basics: A Self-teaching Introduction. Stylus Publishing, LLC, 2018. <a href="https://drive.google.com/file/d/1DNXeb5UmcL_rNl6lHFbmmRT5jodtNoc0/view?usp=sharing">[PDF]</a> <a href="https://scholar.google.co.in/scholar_lookup?title=Python+Basics:+A+Self-teaching-Introduction&amp;auhtor=H+Bhasin">[Google Scholar]</a></p>
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-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/LagrangePoly.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<h1 id="Interpolation-polynomial-in-the-Lagrange-form">Interpolation polynomial in the Lagrange form<a class="anchor-link" href="#Interpolation-polynomial-in-the-Lagrange-form"> </a></h1><p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/LagrangePoly.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
-<p>Based on <a href="https://gist.github.com/folkertdev/084c53887c49a6248839">this code</a></p>
-$$f(x)\approx P_n(x)\,,$$$$P_n(x) = \sum_{i=0}^n f(x_i)L_{n,i}(x) = \sum_{i=0}^n y_iL_{n,i}(x)$$<p><strong>References:</strong>
-<a href="https://en.wikipedia.org/wiki/Lagrange_polynomial">Wikipedia</a></p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">simplify</span><span class="p">(</span><span class="s1">&#39;2/3+5/6&#39;</span><span class="p">)</span>
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-$\displaystyle \frac{3}{2}$
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-<span class="n">expand</span><span class="p">(</span><span class="s1">&#39;(x-1)*(x+1)&#39;</span><span class="p">)</span>
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-$\displaystyle x^{2} - 1$
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-$\displaystyle \left(x - 1\right) \left(x + 1\right)$
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-<h2 id="Implementation-of-the-Lagrange-interpolating-polynomials-and--Lagrange-polynomials-in-SymPy">Implementation of the Lagrange interpolating polynomials and  Lagrange polynomials in SymPy<a class="anchor-link" href="#Implementation-of-the-Lagrange-interpolating-polynomials-and--Lagrange-polynomials-in-SymPy"> </a></h2>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%%writefile</span> LagrangePolynomial.py
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">From: https://gist.github.com/folkertdev/084c53887c49a6248839</span>
-<span class="sd">A sympy-based Lagrange polynomial constructor. </span>
-
-<span class="sd">Implementation of Lagrangian interpolating polynomial.</span>
-<span class="sd">See:</span>
-
-<span class="sd">   def lagrangePolynomial(xs, ys):</span>
-
-<span class="sd">Given two 1-D arrays `xs` and `ys,` returns the Lagrange interpolating</span>
-<span class="sd">polynomial through the points ``(xs, ys)``</span>
-
-
-<span class="sd">Given a set 1-D arrays of inputs and outputs, the lagrangePolynomial function </span>
-<span class="sd">will construct an expression that for every input gives the corresponding output. </span>
-<span class="sd">For intermediate values, the polynomial interpolates (giving varying results </span>
-<span class="sd">based  on the shape of your input). </span>
-
-<span class="sd">The Lagrangian polynomials can be obtained explicitly with (see below):</span>
-<span class="sd">   </span>
-<span class="sd">   def polyL(xs,j):</span>
-<span class="sd">   </span>
-<span class="sd">as sympy polynomial, and </span>
-
-<span class="sd">    def L(xs,j):</span>
-
-<span class="sd">as Python functions.</span>
-
-
-<span class="sd">This is useful when the result needs to be used outside of Python, because the </span>
-<span class="sd">expression can easily be copied. To convert the expression to a python function </span>
-<span class="sd">object, use sympy.lambdify.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">expand</span><span class="p">,</span> <span class="n">lambdify</span><span class="p">,</span> <span class="n">solve_poly_system</span>
-<span class="c1">#Python library for arithmetic with arbitrary precision</span>
-<span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="n">tan</span><span class="p">,</span> <span class="n">e</span>
-
-<span class="kn">import</span> <span class="nn">math</span>
-
-<span class="kn">from</span> <span class="nn">operator</span> <span class="kn">import</span> <span class="n">mul</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="n">reduce</span><span class="p">,</span> <span class="n">lru_cache</span>
-<span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">chain</span>
-
-<span class="c1"># sympy symbols</span>
-<span class="n">x</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
-
-<span class="c1"># convenience functions</span>
-<span class="n">product</span> <span class="o">=</span> <span class="k">lambda</span> <span class="o">*</span><span class="n">args</span><span class="p">:</span> <span class="n">reduce</span><span class="p">(</span><span class="n">mul</span><span class="p">,</span> <span class="o">*</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">+</span> <span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-<span class="c1"># test data</span>
-<span class="n">labels</span> <span class="o">=</span> <span class="p">[(</span><span class="o">-</span><span class="mi">3</span><span class="o">/</span><span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">/</span><span class="mi">4</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="o">/</span><span class="mi">2</span><span class="p">]</span>
-<span class="n">points</span> <span class="o">=</span> <span class="p">[</span><span class="n">math</span><span class="o">.</span><span class="n">tan</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">labels</span><span class="p">]</span>
-
-<span class="c1"># this product may be reusable (when creating many functions on the same domain)</span>
-<span class="c1"># therefore, cache the result</span>
-<span class="nd">@lru_cache</span><span class="p">(</span><span class="mi">16</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">l</span><span class="p">(</span><span class="n">labels</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">gen</span><span class="p">(</span><span class="n">labels</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">labels</span><span class="p">)</span>
-        <span class="n">current</span> <span class="o">=</span> <span class="n">labels</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">labels</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="n">current</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">yield</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">m</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">current</span> <span class="o">-</span> <span class="n">m</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">expand</span><span class="p">(</span><span class="n">product</span><span class="p">(</span><span class="n">gen</span><span class="p">(</span><span class="n">labels</span><span class="p">,</span> <span class="n">j</span><span class="p">)))</span>
-
-<span class="k">def</span> <span class="nf">polyL</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span><span class="n">j</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Lagrange polynomials as sympy polynomial</span>
-<span class="sd">    xs: the n+1 nodes of the intepolation polynomial in the Lagrange Form</span>
-<span class="sd">    j: Is the j-th Lagrange polinomial for the specific xs.</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="n">xs</span><span class="o">=</span><span class="nb">tuple</span><span class="p">(</span><span class="n">xs</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">l</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span><span class="n">j</span><span class="p">)</span>
-
-<span class="k">def</span> <span class="nf">L</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span><span class="n">j</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Lagrange polynomials as python function</span>
-<span class="sd">    xs: the n+1 nodes of the intepolation polynomial in the Lagrange Form</span>
-<span class="sd">    j: Is the j-th Lagrange polinomial for the specific xs.</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="k">return</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">polyL</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span><span class="n">j</span><span class="p">)</span> <span class="p">)</span>
-
-<span class="k">def</span> <span class="nf">lagrangePolynomial</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">ys</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Given two 1-D arrays `x` and `w,` returns the Lagrange interpolating</span>
-<span class="sd">    polynomial through the points ``(x, w)``.</span>
-
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="c1"># based on https://en.wikipedia.org/wiki/Lagrange_polynomial#Example_1</span>
-    <span class="n">k</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">xs</span><span class="p">)</span>
-    <span class="n">total</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="c1"># use tuple, needs to be hashable to cache</span>
-    <span class="n">xs</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">xs</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">current</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ys</span><span class="p">):</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">current</span> <span class="o">*</span> <span class="n">l</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-        <span class="n">total</span> <span class="o">+=</span> <span class="n">t</span>
-
-    <span class="k">return</span> <span class="n">total</span>
-
-
-
-
-<span class="k">def</span> <span class="nf">x_intersections</span><span class="p">(</span><span class="n">function</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-    <span class="s2">&quot;Finds all x for which function(x) = 0&quot;</span>
-    <span class="c1"># solve_poly_system seems more efficient than solve for larger expressions</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">var</span> <span class="k">for</span> <span class="n">var</span> <span class="ow">in</span> <span class="n">chain</span><span class="o">.</span><span class="n">from_iterable</span><span class="p">(</span><span class="n">solve_poly_system</span><span class="p">([</span><span class="n">function</span><span class="p">],</span> <span class="o">*</span><span class="n">args</span><span class="p">))</span> <span class="k">if</span> <span class="p">(</span><span class="n">var</span><span class="o">.</span><span class="n">is_real</span><span class="p">)]</span>
-
-<span class="k">def</span> <span class="nf">x_scale</span><span class="p">(</span><span class="n">function</span><span class="p">,</span> <span class="n">factor</span><span class="p">):</span>
-    <span class="s2">&quot;Scale function on the x-axis&quot;</span>
-    <span class="k">return</span> <span class="n">functions</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span> <span class="o">/</span> <span class="n">factor</span><span class="p">)</span>
-
-<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">&#39;__main__&#39;</span><span class="p">:</span>
-    <span class="n">func</span> <span class="o">=</span> <span class="n">lagrangePolynomial</span><span class="p">(</span><span class="n">labels</span><span class="p">,</span> <span class="n">points</span><span class="p">)</span>
-
-    <span class="n">pyfunc</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">labels</span><span class="p">,</span> <span class="n">points</span><span class="p">):</span>
-        <span class="k">assert</span><span class="p">(</span><span class="n">pyfunc</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">-</span> <span class="n">b</span> <span class="o">&lt;</span> <span class="mf">1e-6</span><span class="p">)</span>
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-<pre>Overwriting LagrangePolynomial.py
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%</span><span class="k">pylab</span> inline
-<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">import</span> <span class="nn">LagrangePolynomial</span> <span class="k">as</span> <span class="nn">LP</span>
-<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">interpolate</span>
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-<pre>/usr/local/lib/python3.5/dist-packages/IPython/core/magics/pylab.py:160: UserWarning: pylab import has clobbered these variables: [&#39;cos&#39;, &#39;sin&#39;]
-`%matplotlib` prevents importing * from pylab and numpy
-  &#34;\n`%matplotlib` prevents importing * from pylab and numpy&#34;
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-<h2 id="Example-of-interpolation-of-tree-points-with-a-polynomial-of-degree-2">Example of interpolation of tree points with a polynomial of degree 2<a class="anchor-link" href="#Example-of-interpolation-of-tree-points-with-a-polynomial-of-degree-2"> </a></h2>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">({</span> <span class="s1">&#39;X&#39;</span><span class="p">:[</span><span class="mi">3</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mf">21.3</span><span class="p">],</span><span class="s1">&#39;Y&#39;</span><span class="p">:[</span><span class="mf">8.</span><span class="p">,</span><span class="mf">6.5</span><span class="p">,</span><span class="mf">3.</span><span class="p">]}</span>  <span class="p">)</span>
-<span class="n">df</span>
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-<table border="1" class="dataframe">
-  <thead>
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-      <th></th>
-      <th>X</th>
-      <th>Y</th>
-    </tr>
-  </thead>
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-      <th>0</th>
-      <td>3.0</td>
-      <td>8.0</td>
-    </tr>
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-      <th>1</th>
-      <td>10.0</td>
-      <td>6.5</td>
-    </tr>
-    <tr>
-      <th>2</th>
-      <td>21.3</td>
-      <td>3.0</td>
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-<h3 id="Numpy-implementation:"><code>Numpy</code> implementation:<a class="anchor-link" href="#Numpy-implementation:"> </a></h3>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">coeffs</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">polyfit</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="n">deg</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-
-<span class="n">P</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">30</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">(</span> <span class="n">x</span><span class="p">),</span><span class="s1">&#39;b-&#39;</span><span class="p">)</span>
-<span class="c1">#plt.grid()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">12</span><span class="p">)</span>
-
-<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">coeffs</span><span class="p">))</span>
-</pre></div>
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-<pre>           2
--0.005216 x - 0.1465 x + 8.486
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->
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-<h3 id="Scipy-implementation-of-the-Interpolation-polynomial-in-the-Lagrange-form"><code>Scipy</code> implementation of the Interpolation polynomial in the Lagrange form<a class="anchor-link" href="#Scipy-implementation-of-the-Interpolation-polynomial-in-the-Lagrange-form"> </a></h3>
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-<div class=" highlight hl-ipython3"><pre><span></span>interpolate.lagrange<span class="o">?</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">P</span><span class="o">=</span><span class="n">interpolate</span><span class="o">.</span><span class="n">lagrange</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">P</span><span class="p">)</span>
-</pre></div>
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-<pre>           2
--0.005216 x - 0.1465 x + 8.486
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-<h3 id="Sympy-implementation-of-the-Interpolation-polynomial-in-the-Lagrange-form"><code>Sympy</code> implementation of the Interpolation polynomial in the Lagrange form<a class="anchor-link" href="#Sympy-implementation-of-the-Interpolation-polynomial-in-the-Lagrange-form"> </a></h3>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">LP</span><span class="o">.</span><span class="n">lagrangePolynomial</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">)</span>
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-$\displaystyle - 0.00521578136549848 x^{2} - 0.146480556534234 x + 8.48638370189219$
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-<p>With this simpy implementation we can check expliclty that:
-$$P_2(x) =  L_{2,0}(x)f(x_0)+L_{2,1}(x)f(x_1)+L_{2,2}(x)f(x_2)$$</p>
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-<p>a) By using <code>sympy</code> polynomials:  <code>LP.polyL</code>:</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">LP</span><span class="o">.</span><span class="n">polyL</span><span class="p">(</span> <span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span><span class="o">*</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">LP</span><span class="o">.</span><span class="n">polyL</span><span class="p">(</span> <span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">LP</span><span class="o">.</span><span class="n">polyL</span><span class="p">(</span> <span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-</pre></div>
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-$\displaystyle - 0.00521578136549848 x^{2} - 0.146480556534234 x + 8.48638370189219$
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">LP</span><span class="o">.</span><span class="n">polyL</span><span class="p">(</span> <span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-</pre></div>
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-$\displaystyle 0.0078064012490242 x^{2} - 0.244340359094457 x + 1.66276346604215$
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-<p>As a function</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">P_2</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">xs</span><span class="p">,</span><span class="n">ys</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">LP</span><span class="o">.</span><span class="n">L</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span><span class="mi">0</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">ys</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">LP</span><span class="o">.</span><span class="n">L</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span><span class="mi">1</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">ys</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">LP</span><span class="o">.</span><span class="n">L</span><span class="p">(</span> <span class="n">xs</span><span class="p">,</span><span class="mi">2</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">ys</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">30</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P_2</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">),</span><span class="s1">&#39;b-&#39;</span><span class="p">)</span>
-<span class="c1">#plt.grid()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">12</span><span class="p">)</span>
-</pre></div>
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-<pre>(0, 12)</pre>
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-<img src="../images/features/LagrangePoly_26_1.png"
->
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-</div>
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-<h2 id="Lagrange-polynomial-properties">Lagrange polynomial properties<a class="anchor-link" href="#Lagrange-polynomial-properties"> </a></h2>$$L_{n,i}(x_i) = 1\,,\qquad\text{and}\,,\qquad L_{n,i}(x_j) = 0\quad\text{for $i\neq j$}$$<p></p>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>As <code>sympy</code> objects</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">L2_0</span><span class="o">=</span><span class="n">LP</span><span class="o">.</span><span class="n">polyL</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="n">L2_0</span>
-</pre></div>
-
-    </div>
-</div>
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-$\displaystyle 0.0078064012490242 x^{2} - 0.244340359094457 x + 1.66276346604215$
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">L2_0</span><span class="o">.</span><span class="n">as_poly</span><span class="p">()</span>
-</pre></div>
-
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-$\displaystyle \operatorname{Poly}{\left( 0.0078064012490242 x^{2} - 0.244340359094457 x + 1.66276346604215, x, domain=\mathbb{R} \right)}$
-</div>
-
-</div>
-</div>
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-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">L2_0</span><span class="o">.</span><span class="n">as_poly</span><span class="p">()(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
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-$\displaystyle 1.0$
-</div>
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-</div>
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-</div>
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-</div>
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-
-<div class="jb_cell">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">L2_0</span><span class="o">.</span><span class="n">as_poly</span><span class="p">()(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-</pre></div>
-
-    </div>
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-$\displaystyle -4.44089209850063 \cdot 10^{-16}$
-</div>
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-</div>
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-</div>
-</div>
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-<p>As <code>python</code> function</p>
-
-</div>
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-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span> <span class="n">LP</span><span class="o">.</span><span class="n">L</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="mi">0</span><span class="p">)(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span><span class="n">LP</span><span class="o">.</span><span class="n">L</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="mi">1</span><span class="p">)(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span><span class="n">LP</span><span class="o">.</span><span class="n">L</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="mi">2</span><span class="p">)(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<div class="output_wrapper">
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-<pre>0.9999999999999968 0.9999999999999951 0.9999999999999998
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">LP</span><span class="o">.</span><span class="n">L</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="mi">0</span><span class="p">)(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
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-<pre>0    1.000000e+00
-1   -4.440892e-16
-2    5.329071e-15
-Name: X, dtype: float64</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">LP</span><span class="o">.</span><span class="n">L</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="mi">1</span><span class="p">)(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">)</span>
-</pre></div>
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-<pre>0   -4.302114e-16
-1    1.000000e+00
-2   -2.042810e-14
-Name: X, dtype: float64</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">LP</span><span class="o">.</span><span class="n">L</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="mi">2</span><span class="p">)(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">)</span>
-</pre></div>
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-<pre>0    1.665335e-16
-1    0.000000e+00
-2    1.000000e+00
-Name: X, dtype: float64</pre>
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-<p><strong>Actividad</strong> Fit a cuatro puntos, comprobando las propiedades del polinomio de Lagrange</p>
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-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/Minimization.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<h1 id="Minimization">Minimization<a class="anchor-link" href="#Minimization"> </a></h1><p>Find the minumum of a function</p>
-<hr>
-<h2 id="Bibliography">Bibliography<a class="anchor-link" href="#Bibliography"> </a></h2><p>[1] <a href="https://scipy-lectures.org/intro/scipy.html#optimization-and-fit-scipy-optimize">Optimization and fit scipy optimize</a></p>
-<ul>
-<li><a href="https://scipy-lectures.org/intro/scipy/auto_examples/plot_optimize_example2.html">Example</a></li>
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-`%matplotlib` prevents importing * from pylab and numpy
-  &#34;\n`%matplotlib` prevents importing * from pylab and numpy&#34;
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-<span class="kn">import</span> <span class="nn">scipy.optimize</span> <span class="k">as</span> <span class="nn">optimize</span>
-<span class="kn">import</span> <span class="nn">scipy.interpolate</span> <span class="k">as</span> <span class="nn">interpolate</span>
-<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
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-              <span class="s1">&#39;Y&#39;</span><span class="p">:[</span><span class="mi">3</span><span class="p">,</span><span class="o">-</span><span class="mf">1.01</span><span class="p">,</span><span class="mf">1.5</span><span class="p">,</span><span class="mf">0.7</span><span class="p">,</span><span class="mf">2.8</span><span class="p">,</span><span class="mf">1.5</span><span class="p">]})</span>
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-      <th>0</th>
-      <td>2.5</td>
-      <td>3.00</td>
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-      <th>1</th>
-      <td>3.1</td>
-      <td>-1.01</td>
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-      <th>2</th>
-      <td>4.5</td>
-      <td>1.50</td>
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-      <th>3</th>
-      <td>5.0</td>
-      <td>0.70</td>
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-      <th>4</th>
-      <td>5.9</td>
-      <td>2.80</td>
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-      <th>5</th>
-      <td>6.2</td>
-      <td>1.50</td>
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-<h3 id="Laggrange-interpolation">Laggrange interpolation<a class="anchor-link" href="#Laggrange-interpolation"> </a></h3>
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--0.8941 x + 19.97 x - 174.7 x + 746.7 x - 1557 x + 1264
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-<pre>0    2.5
-1    3.1
-2    4.5
-3    5.0
-4    5.9
-5    6.2
-Name: X, dtype: float64</pre>
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-<p>To obtain some specific value by using slices, we must use <code>.iloc</code></p>
-<p>Note that <code>df.X[3]=df.X.loc[3]=df.X.iloc[3]</code></p>
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-<p>works!</p>
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-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">pol</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;X&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;Y&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
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-<h2 id="Hermite-interpolation">Hermite interpolation<a class="anchor-link" href="#Hermite-interpolation"> </a></h2><p>The recommend degree for the Hermite polynomial is $n-1$ where $n$ is the number of data of the dataset</p>
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-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">pol</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="s1">&#39;c--&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;$x$&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;$H(x)$&#39;</span><span class="p">)</span>
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-<h2 id="Finding-the-local-minimum-of-a-function">Finding the local minimum of a function<a class="anchor-link" href="#Finding-the-local-minimum-of-a-function"> </a></h2>
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-<p>Finding the first minimum close to  <span style="color:red">3</span>  (which corresponds to the global minimum), and the second close to <span style="color:red">5</span>  (a  local minimum)</p>
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-<span class="n">min2</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">fmin_powell</span><span class="p">(</span><span class="n">pol</span><span class="p">,</span><span class="mf">5.2</span><span class="p">,</span><span class="n">full_output</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
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-<pre>Optimization terminated successfully.
-         Current function value: -1.374113
-         Iterations: 2
-         Function evaluations: 28
-Optimization terminated successfully.
-         Current function value: 0.699898
-         Iterations: 2
-         Function evaluations: 50
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;The global minimum is f(x)=</span><span class="si">{}</span><span class="s1"> for x=</span><span class="si">{}</span><span class="s1">;</span><span class="se">\n</span><span class="s1"> the local minimum is f(x)=</span><span class="si">{}</span><span class="s1"> for x=</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
-    <span class="n">min1</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span><span class="n">min1</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">min2</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span><span class="n">min2</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
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-<pre>The global minimum is f(x)=-1.3741131581291484 for x=2.9278495523433894;
- the local minimum is f(x)=0.6998981514389016 for x=5.004929292406532
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-<p><strong>Activity</strong> Find the maximum values of the Hermite interpolation function of degree 5 to the set of points: <a href="https://github.com/restrepo/ComputationalMethods/blob/master/data/hermite.csv">https://github.com/restrepo/ComputationalMethods/blob/master/data/hermite.csv</a></p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;https://raw.githubusercontent.com/restrepo/ComputationalMethods/master/data/hermite.csv&#39;</span><span class="p">)</span>
-<span class="n">H</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">polynomial</span><span class="o">.</span><span class="n">hermite</span><span class="o">.</span><span class="n">Hermite</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
-<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span><span class="mi">100</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="o">-</span><span class="n">H</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="s1">&#39;k-&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;$x$&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;$H(x)$&#39;</span><span class="p">)</span>
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-<p><code>fmin_powell</code> try to search the global minimum</p>
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-<pre>Optimization terminated successfully.
-         Current function value: -2.803764
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-<h3 id="Find-a-local-minumum">Find a local minumum<a class="anchor-link" href="#Find-a-local-minumum"> </a></h3>
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-<p>close minimum</p>
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-<span class="n">min1</span><span class="p">[</span><span class="s1">&#39;x&#39;</span><span class="p">]</span>
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-<span class="n">min1</span><span class="p">[</span><span class="s1">&#39;x&#39;</span><span class="p">]</span>
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-<p>minimum in a range</p>
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-<span class="n">min1</span><span class="p">[</span><span class="s1">&#39;x&#39;</span><span class="p">]</span>
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-<span class="n">xmin_local</span>
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-<h3 id="Find-a-global-minumum-(alternative)">Find a global minumum (alternative)<a class="anchor-link" href="#Find-a-global-minumum-(alternative)"> </a></h3>
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-<span class="n">min1</span><span class="p">[</span><span class="s1">&#39;x&#39;</span><span class="p">]</span>
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-<h2 id="The-Higgs-potential">The Higgs potential<a class="anchor-link" href="#The-Higgs-potential"> </a></h2><p>To write greek letter inside a cell use the $\rm \LaTeX$ macro and the tab, e.g: <code>\mu&lt;TAB&gt;</code>, to produce μ</p>
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-<span class="n">G_F</span><span class="o">=</span><span class="mf">1.1663787E-5</span> <span class="c1">#GeV^-2</span>
-<span class="n">v</span><span class="o">=</span><span class="mi">1</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mf">2.</span><span class="p">)</span><span class="o">*</span><span class="n">G_F</span><span class="p">)</span> <span class="c1"># GeV</span>
-<span class="n">μ</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">m_H</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="n">λ</span><span class="o">=</span><span class="n">m_H</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="mf">2.</span><span class="o">*</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="n">μ</span><span class="p">,</span><span class="n">λ</span><span class="p">,</span><span class="n">v</span>
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-<span class="n">Vp</span><span class="o">=</span><span class="k">lambda</span> <span class="n">ϕ</span><span class="p">:</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">μ</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">ϕ</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="mf">0.25</span><span class="o">*</span><span class="n">λ</span><span class="o">*</span><span class="n">ϕ</span><span class="o">**</span><span class="mi">4</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ϕ</span><span class="p">,</span> <span class="n">Vp</span><span class="p">(</span><span class="n">ϕ</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$\phi$ [GeV]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">20</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$V(\phi)$ [GeV]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$\phi$ [GeV]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">20</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$V(\phi)$ [GeV$^4$]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
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-<span class="n">V</span><span class="o">=</span><span class="k">lambda</span> <span class="n">ϕ</span><span class="p">:</span> <span class="n">Vp</span><span class="p">(</span><span class="n">ϕ</span><span class="p">)</span><span class="o">.</span><span class="n">real</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Vp</span><span class="p">(</span><span class="n">ϕ</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">real</span>
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-<pre>-92060568.64037189</pre>
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-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ϕ</span><span class="p">,</span> <span class="n">V</span><span class="p">(</span><span class="n">ϕ</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$\phi$ [GeV]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">20</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$V(\phi)$ [GeV]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$\phi$ [GeV]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">20</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$V(\phi)$ [GeV$^4$]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
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-<img src="../images/features/Minimization_48_0.png"
->
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">fp</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">fmin_powell</span><span class="p">(</span><span class="n">V</span><span class="p">,</span><span class="mi">200</span><span class="p">,</span><span class="n">ftol</span><span class="o">=</span><span class="mf">1E-16</span><span class="p">,</span><span class="n">full_output</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-</pre></div>
-
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-<pre>Optimization terminated successfully.
-         Current function value: -120308559.069597
-         Iterations: 4
-         Function evaluations: 74
-</pre>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ϕ_min</span><span class="o">=</span><span class="n">fp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">ϕ_min</span><span class="p">,</span><span class="n">v</span><span class="p">)</span>
-</pre></div>
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-<pre>246.21964987858152 246.21965079413738
-</pre>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Minimization-in-higher-dimensions">Minimization in higher dimensions<a class="anchor-link" href="#Minimization-in-higher-dimensions"> </a></h2><div style="float: right;" markdown="1">
-    <img src="https://raw.githubusercontent.com/restrepo/ComputationalMethods/master/material/figures/mexicanhat.svg?sanitize=true" width="300">
-</div><p>For a complex scalar field with potential
-\begin{equation}
-V(\phi)=\mu^2\phi^*\phi + \lambda (\phi^*\phi)^2
-\end{equation}
-with 
-\begin{equation}
-\phi=\frac{\phi_1+i\phi_2  }{\sqrt{2} }
-\end{equation}
-and $\mu^2&lt;0$, and $\lambda&gt;0$, find some of the infinite number of minimum values of $\phi$, as illustrated in the figure, with the plane, $\phi_1-\phi_2$, moved to the minimum to easy the visualization. Expanding in terms of the real and imaginary part of $\phi$
-\begin{equation}
-V(\phi)=\frac{\mu^2}{2}\left(\phi_1^2+\phi_2^2 \right) + \frac{\lambda}{4}\left( \phi_1^2+\phi_2^2\right)^2
-\end{equation}</p>
-
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%</span><span class="k">pylab</span> inline
-<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">optimize</span>
-<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-</pre></div>
-
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-<pre>Populating the interactive namespace from numpy and matplotlib
-</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">ϕ</span><span class="p">,</span><span class="n">m_H</span><span class="o">=</span><span class="mi">126</span><span class="p">,</span><span class="n">G_F</span><span class="o">=</span><span class="mf">1.1663787E-5</span><span class="p">):</span>
-    <span class="n">v</span><span class="o">=</span><span class="mi">1</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mf">2.</span><span class="p">)</span><span class="o">*</span><span class="n">G_F</span><span class="p">)</span> <span class="c1"># GeV</span>
-    <span class="n">μ</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">m_H</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="mi">1</span><span class="n">j</span> <span class="c1">#imaginary mass</span>
-    <span class="n">λ</span><span class="o">=</span><span class="n">m_H</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="mf">2.</span><span class="o">*</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-    <span class="c1">#print(μ,λ)</span>
-    <span class="k">return</span> <span class="p">(</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">μ</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">ϕ</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">ϕ</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span><span class="mf">0.25</span><span class="o">*</span><span class="n">λ</span><span class="o">*</span><span class="p">(</span><span class="n">ϕ</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">ϕ</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span><span class="o">.</span><span class="n">real</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Check a point of the function</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">])</span>
-</pre></div>
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-</div>
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-<pre>-396572.6550230127</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Check the minimim obtained when an inizialization point at $\phi_0=(0,0)$</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">fmin</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">fmin_powell</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">x0</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">],</span><span class="n">ftol</span><span class="o">=</span><span class="mf">1E-16</span><span class="p">,</span><span class="n">full_output</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-<span class="n">fmin</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</pre></div>
-
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-</div>
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-<pre>Optimization terminated successfully.
-         Current function value: -120308559.069597
-         Iterations: 3
-         Function evaluations: 111
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
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-<div class="output_text output_subarea output_execute_result">
-<pre>array([246.21914011,  -0.50137284])</pre>
-</div>
-
-</div>
-</div>
-</div>
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-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Check the proyection of the minimum in the plane $\phi_1-\phi_2$</p>
-
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-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;V(ϕ)=</span><span class="si">{}</span><span class="s1"> GeV^4&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">fmin</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">1</span><span class="p">)))</span>
-</pre></div>
-
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-</div>
-</div>
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-<pre>V(ϕ)=-120308559.1 GeV^4
-</pre>
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span> <span class="n">fmin</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">fmin</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<pre>246.21965057683713</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>For random initialization points, we can get several minima</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">300</span><span class="p">,</span><span class="mi">300</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
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-<pre>array([-164.99567257,  -65.7636795 ])</pre>
-</div>
-
-</div>
-</div>
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-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">()</span>
-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000</span><span class="p">):</span>
-    <span class="n">ϕ0</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">300</span><span class="p">,</span><span class="mi">300</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-    <span class="n">ϕmin</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">fmin_powell</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">x0</span><span class="o">=</span><span class="n">ϕ0</span><span class="p">,</span><span class="n">ftol</span><span class="o">=</span><span class="mf">1E-16</span><span class="p">,</span><span class="n">disp</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
-    <span class="n">df</span><span class="o">=</span><span class="n">df</span><span class="o">.</span><span class="n">append</span><span class="p">({</span><span class="s1">&#39;ϕ1&#39;</span><span class="p">:</span><span class="n">ϕmin</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="s1">&#39;ϕ2&#39;</span><span class="p">:</span><span class="n">ϕmin</span><span class="p">[</span><span class="mi">1</span><span class="p">]},</span><span class="n">ignore_index</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Projection of the minima in the plane, $\phi_1,\phi_2$</p>
-
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-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="p">[:</span><span class="mi">3</span><span class="p">]</span>
-</pre></div>
-
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-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
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-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>ϕ1</th>
-      <th>ϕ2</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>118.814666</td>
-      <td>-215.655260</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>239.086165</td>
-      <td>-58.838100</td>
-    </tr>
-    <tr>
-      <th>2</th>
-      <td>228.276742</td>
-      <td>-92.270501</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;ϕ1&#39;</span><span class="p">],</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;ϕ2&#39;</span><span class="p">],</span><span class="s1">&#39;r.&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s1">&#39;$\phi_{{\rm min}}=</span><span class="si">{}</span><span class="s1">$ GeV&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
-    <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="s1">&#39;ϕ1&#39;</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">df</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="s1">&#39;ϕ2&#39;</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">1</span><span class="p">)))</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="s1">&#39;ϕ1&#39;</span><span class="p">],</span><span class="n">df</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="s1">&#39;ϕ2&#39;</span><span class="p">],</span><span class="s1">&#39;k*&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;Our Universe&#39;</span><span class="p">,</span><span class="n">markersize</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;$\phi_1$&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;$\phi_2$&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">&#39;best&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-</pre></div>
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-<h2 id="Least-action">Least action<a class="anchor-link" href="#Least-action"> </a></h2><p>See <a href="./least_action_minimization.ipynb">Least action minimization</a></p>
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-<h3 id="Further-material">Further material<a class="anchor-link" href="#Further-material"> </a></h3><p><a href="http://www.damtp.cam.ac.uk/user/nsm10/PrincLeaAc.pdf">http://www.damtp.cam.ac.uk/user/nsm10/PrincLeaAc.pdf</a>
-<a href="https://jfuchs.hotell.kau.se/kurs/amek/prst/14_hpvp.pdf">https://jfuchs.hotell.kau.se/kurs/amek/prst/14_hpvp.pdf</a></p>
-<p><a href="https://www.coursera.org/learn/general-relativity/lecture/8UHtE/the-least-action-or-minimal-action-principle-part-1">https://www.coursera.org/learn/general-relativity/lecture/8UHtE/the-least-action-or-minimal-action-principle-part-1</a></p>
-<p><a href="https://jfi.uchicago.edu/~tten/from.panza/Physics185/Handouts/rubber-band%20analogy.pdf">https://jfi.uchicago.edu/~tten/from.panza/Physics185/Handouts/rubber-band%20analogy.pdf</a></p>
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-  Data analysis with Pandas
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-    <div id="page-info"><div id="page-title">Data analysis with Pandas</div>
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-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/Pandas.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<h1 id="Pandas">Pandas<a class="anchor-link" href="#Pandas"> </a></h1><p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/Pandas.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<p>From <a href="http://pandas.pydata.org/pandas-docs/stable/">http://pandas.pydata.org/pandas-docs/stable/</a></p>
-<p>pandas is a Python package providing fast, flexible, and expressive data structures designed to make working with “relational” or “labeled” data both easy and intuitive. It aims to be the fundamental high-level building block for doing practical, real world data analysis in Python. Additionally, it has the broader goal of becoming the most powerful and flexible open source data analysis / manipulation tool available in any language. It is already well on its way toward this goal.</p>
-<p>See also:</p>
-<ul>
-<li><a href="https://github.com/restrepo/data-analysis">https://github.com/restrepo/data-analysis</a><ul>
-<li><a href="https://classroom.github.com/g/sSMBdBqN">https://classroom.github.com/g/sSMBdBqN</a></li>
-<li><a href="https://classroom.github.com/a/PcbQBE7F">https://classroom.github.com/a/PcbQBE7F</a></li>
-</ul>
-</li>
-<li><a href="https://github.com/restrepo/PythonTipsAndTricks">https://github.com/restrepo/PythonTipsAndTricks</a></li>
-<li><a href="https://pbpython.com/excel-pandas-comp.html">https://pbpython.com/excel-pandas-comp.html</a> [<a href="https://web.archive.org/web/20201126143453/https://pbpython.com/excel-pandas-comp.html">archive.org</a>]</li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>A good and practice book about <code>Pandas</code> possibilities is:</p>
-<p><a href="https://drive.google.com/open?id=0BxoOXsn2EUNIWExXbVc4SDN0YTQ"><strong>Python for Data Analysis</strong></a><br/>
-Data Wrangling with Pandas, NumPy, and IPython<br/>
-<em>By William McKinney</em></p>
-<p>This other is about aplications based on <code>Pandas</code>:
-<img src="https://covers.oreillystatic.com/images/0636920030515/cat.gif" alt="image.png"> <a href="https://drive.google.com/open?id=0BxoOXsn2EUNISGhrdEZ3S29fS3M">Introduction to Machine Learning with Python</a><br/>
-A Guide for Data Scientists
-By Sarah Guido, Andreas Müller</p>
-<p><code>Pandas</code> can be used in a similar way to <code>R</code>, which is based on similar data structures. <code>Pandas</code> also can replace the use of graphical interfaces to access spreadsheets like Excel</p>
-
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-<h2 id="Standard-way-to-load-the-module">Standard way to load the module<a class="anchor-link" href="#Standard-way-to-load-the-module"> </a></h2>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-</pre></div>
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-<p>An flat <em>spreadsheet</em> can be seen in terms of the types of variables of <code>Python</code> just as dictionary of lists, where each column of the spreadsheet is a pair key-list of the dictionary</p>
-<table>
-<thead><tr>
-<th></th>
-<th style="text-align:center">A</th>
-<th style="text-align:center">B</th>
-</tr>
-</thead>
-<tbody>
-<tr>
-<td>1</td>
-<td style="text-align:center">even</td>
-<td style="text-align:center">odd</td>
-</tr>
-<tr>
-<td>2</td>
-<td style="text-align:center">0</td>
-<td style="text-align:center">1</td>
-</tr>
-<tr>
-<td>3</td>
-<td style="text-align:center">2</td>
-<td style="text-align:center">3</td>
-</tr>
-<tr>
-<td>4</td>
-<td style="text-align:center">4</td>
-<td style="text-align:center">5</td>
-</tr>
-<tr>
-<td>5</td>
-<td style="text-align:center">6</td>
-<td style="text-align:center">7</td>
-</tr>
-<tr>
-<td>6</td>
-<td style="text-align:center">8</td>
-<td style="text-align:center">9</td>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">numbers</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;even&quot;</span><span class="p">:</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span>   <span class="c1">#  First  key-list</span>
-         <span class="s2">&quot;odd&quot;</span> <span class="p">:</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">9</span><span class="p">]</span> <span class="p">}</span>  <span class="c1">#  Second key-list</span>
-</pre></div>
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-<h2 id="Data-structures">Data structures<a class="anchor-link" href="#Data-structures"> </a></h2>
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-<p><code>Pandas</code> has two new data structures:</p>
-<ol>
-<li><code>DataFrame</code> which are similar to numpy arrays but with some assigned key. For example, for the previous case<div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
-       <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span>
-       <span class="p">[</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span>
-       <span class="p">[</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">],</span>
-       <span class="p">[</span><span class="mi">8</span><span class="p">,</span><span class="mi">9</span><span class="p">]</span> 
-      <span class="p">])</span>
-</pre></div>
-</li>
-<li><code>Series</code> which are enriched  to dictionaries, as the ones defined for the rows of the previous example: <code>{'even':0,'odd':1}</code>.</li>
-</ol>
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-<p>The rows in a two-dimensional <code>DataFrame</code> corresponds to <code>Series</code> with similar keys, while the columns are also Series with the indices as keys.</p>
-<p>An example of a  <code>DataFrame</code> is a spreadsheet, as the one before.</p>
-
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-<h3 id="DataFrame"><code>DataFrame</code><a class="anchor-link" href="#DataFrame"> </a></h3><p><code>Pandas</code> can convert a dictionary of lists, like the <code>numbers</code> dictionary before, into a <code>DataFrame</code>, which is just an spreadsheet but interpreted at the programming level:</p>
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-<pre>{&#39;even&#39;: [0, 2, 4, 6, 8], &#39;odd&#39;: [1, 3, 5, 7, 9]}</pre>
-</div>
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-<span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">numbers</span><span class="p">)</span>
-<span class="n">df</span>
-</pre></div>
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-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>even</th>
-      <th>odd</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>0</td>
-      <td>1</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>2</td>
-      <td>3</td>
-    </tr>
-    <tr>
-      <th>2</th>
-      <td>4</td>
-      <td>5</td>
-    </tr>
-    <tr>
-      <th>3</th>
-      <td>6</td>
-      <td>7</td>
-    </tr>
-    <tr>
-      <th>4</th>
-      <td>8</td>
-      <td>9</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
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-</div>
-</div>
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-<p>It is equivalent to:</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">to_dict</span><span class="p">()</span>
-</pre></div>
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-<pre>{&#39;even&#39;: {0: 0, 1: 2, 2: 4, 3: 6, 4: 8}, &#39;odd&#39;: {0: 1, 1: 3, 2: 5, 3: 7, 4: 9}}</pre>
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-<p>See below for other possibilities of <a href="https://fisica.udea.edu.co:4443/user/restrepo/notebooks/prog/cursos/data-analysis/Pandas.ipynb#Intialization-from-lists-and-dictionaries">creating Pandas DataFrames from lists and dictionaries</a></p>
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-<p>The main advantage of the <code>DataFrame</code>, <code>df</code>, is that it can be managed without a graphical interface.</p>
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-<p>We can check the shape of the <code>DataFrame</code></p>
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-<h4 id="Export-DataFrame-to-other-formats">Export DataFrame to other formats<a class="anchor-link" href="#Export-DataFrame-to-other-formats"> </a></h4><ul>
-<li>To export to excel:</li>
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-      <td>4</td>
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-      <td>6</td>
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-<p><strong>Activity</strong>: Open the resulting spreadsheet in Google Drive, publish it and open from the resulting link with Pandas in the next cell</p>
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-<span class="n">df</span>
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-<h3 id="Series"><code>Series</code><a class="anchor-link" href="#Series"> </a></h3>
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-<p>Each column of the DataFrame is now an augmented dictionary called <code>Series</code>, with the indices as the keys of the <code>Series</code></p>
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-<p>A <code>Pandas</code> <code>Series</code> object can be just initialized from a <code>Python</code> dictionary:</p>
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-<pre>pandas.core.series.Series</pre>
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-<p>The keys are the index of the <code>DataFrame</code></p>
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-<span class="n">df</span><span class="o">.</span><span class="n">even</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span>
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-<p>Each row is also a series</p>
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-odd     1
-Name: 0, dtype: int64</pre>
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-<p>with keys: <code>'even'</code> and <code>'odd'</code></p>
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-<p>or attributes <code>even</code> and <code>odd</code></p>
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-<p>One specific cell value can be reached with the index and the key:</p>
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-<span class="n">s</span>
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-<pre>Name           Juan Valdez
-Nacionality       Colombia
-Age                     23
-dtype: object</pre>
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-<p><em>Note</em> that the key name can be used also as an attribute.</p>
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-1    3
-2    5
-3    7
-4    9
-Name: odd, dtype: int64</pre>
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-<blockquote><p>The <strong>power</strong> of Pandas rely in that their main data structures: <code>DataFrames</code> and <code>Series</code>, are enriched with many useful methods and attributes.</p>
-</blockquote>
-<h3 id="Official-definition-of-Pandas"><a href="http://pandas.pydata.org/pandas-docs/stable/">Official definition of Pandas</a><a class="anchor-link" href="#Official-definition-of-Pandas"> </a></h3><blockquote><p>Pandas is a Python package providing <strong>fast</strong>, <strong>flexible</strong>, and <strong>expressive</strong> <em>data structures</em> designed to make working with “relational” or “labeled” data both easy and intuitive. It aims to be the fundamental high-level building block for doing practical, real world data analysis in Python. Additionally, it <em>has the broader goal of becoming the most powerful and flexible open source data analysis / manipulation tool</em> available in any language. It is already well on its way toward this goal.</p>
-</blockquote>
-<ul>
-<li>"relational": the list of data is identified with some unique index (like a <code>SQL</code> table)</li>
-<li>"labeled": the list is identified with a key, like the previous <code>odd</code> or <code>even</code> keys.</li>
-</ul>
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-<p>For example. A double bracket <code>[[...]]</code>, can be used to filter data.</p>
-<p>A row in a two-dimensional <code>DataFrame</code> corresponds to <code>Series</code> with the same keys of the <code>DataFrame</code>, but with single values instead of a list</p>
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-      <th>even</th>
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-    </tr>
-  </thead>
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-      <th>0</th>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;the row has&#39;</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;                  keys: </span><span class="si">{}</span><span class="s1"> and values: </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> <span class="nb">list</span><span class="p">(</span> <span class="n">df</span><span class="o">.</span><span class="n">loc</span><span class="p">[[</span><span class="mi">0</span><span class="p">]]</span><span class="o">.</span><span class="n">keys</span><span class="p">()</span> <span class="p">),</span><span class="n">df</span><span class="o">.</span><span class="n">loc</span><span class="p">[[</span><span class="mi">0</span><span class="p">]]</span><span class="o">.</span><span class="n">values</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>  <span class="p">)</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span> <span class="s2">&quot;like the dictionay:&quot;</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span> <span class="s2">&quot;                      { &#39;even&#39; : 0, &#39;odd&#39; : 1 }&quot;</span><span class="p">)</span>
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-<pre>the row has
-                  keys: [&#39;even&#39;, &#39;odd&#39;] and values: [0 1]
-like the dictionay:
-                      { &#39;even&#39; : 0, &#39;odd&#39; : 1 }
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-<p>To filter a column:</p>
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-      <th></th>
-      <th>odd</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>1</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>3</td>
-    </tr>
-    <tr>
-      <th>2</th>
-      <td>5</td>
-    </tr>
-    <tr>
-      <th>3</th>
-      <td>7</td>
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-<h3 id="More-on-Series">More on <code>Series</code><a class="anchor-link" href="#More-on-Series"> </a></h3>
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-<p>A <code>Pandas</code> <code>Series</code> object can be just initialized from a <code>Python</code> dictionary:</p>
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-<span class="n">s</span>
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-<pre>Name           Juan Valdez
-Nacionality       Colombia
-Age                     23
-dtype: object</pre>
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-<p>but also as containers of name spaces!</p>
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-<h2 id="DataFrame-initialization"><code>DataFrame</code> initialization<a class="anchor-link" href="#DataFrame-initialization"> </a></h2>
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-<h3 id="Initization-from-an-existing-spreadsheet.">Initization from an existing spreadsheet.<a class="anchor-link" href="#Initization-from-an-existing-spreadsheet."> </a></h3><p>This can be locally in your computer o from some downloadable  link</p>
-
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-<span class="n">df</span>
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-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>Nombre</th>
-      <th>Edad</th>
-      <th>Compañia</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>Juan Valdez</td>
-      <td>23</td>
-      <td>Café de Colombia</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>Álvaro Uribe Vélez</td>
-      <td>65</td>
-      <td>Senado de la República</td>
-    </tr>
-  </tbody>
-</table>
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-<p>To make a downloadable link for any spread sheet in Google Drive, follow the sequence:</p>
-
-<pre><code>File → Publish to the web...→ Entire Document → Web page → Microsoft excel (xlsx)</code></pre>
-<p>as illustrated in the figure:
-<img src="https://github.com/restrepo/data-analysis/blob/master/img/img1.png?raw=1" alt="GS"></p>
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-<span class="c1">#df.at[0,&#39;Edad&#39;]=32</span>
-<span class="n">df</span>
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-      <th>Nombre</th>
-      <th>Edad</th>
-      <th>Compañia</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>Juan Valdez</td>
-      <td>32</td>
-      <td>Café de Colombia</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>Álvaro Uribe Vélez</td>
-      <td>65</td>
-      <td>Senado de la República</td>
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-<p><em>After</em> some modification</p>
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-<p>it can be saved again as an <code>excel file</code> with the option to not create a column of indices: <code>index=False</code></p>
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-<h3 id="Intialization-from-lists-and-dictionaries">Intialization from lists and dictionaries<a class="anchor-link" href="#Intialization-from-lists-and-dictionaries"> </a></h3>
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-<h4 id="Inizialization-from-Series">Inizialization from Series<a class="anchor-link" href="#Inizialization-from-Series"> </a></h4><p>We start with an empty <code>DataFrame</code>:</p>
-
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-<p>Creating Pandas DataFrame from list and dictionaries <a href="http://pbpython.com/pandas-list-dict.html">offers many alternatives</a></p>
-<p><img src="http://pbpython.com/images/pandas-dataframe-shadow.png" alt="creating dataframes"></p>
-<h4 id="Row-oriented-way">Row oriented way<a class="anchor-link" href="#Row-oriented-way"> </a></h4><ul>
-<li>In addition to the dictionary of lists <a href="">already illustrated at the beginning</a> that in this case corresponds to:</li>
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-              <span class="s1">&#39;Edad&#39;</span>     <span class="p">:</span> <span class="p">[</span><span class="mi">32</span><span class="p">,</span>            <span class="mi">65</span>                 <span class="p">],</span>
-              <span class="s1">&#39;Compañia&#39;</span> <span class="p">:</span> <span class="p">[</span><span class="s1">&#39;Café de Colombia&#39;</span><span class="p">,</span><span class="s1">&#39;Senado de la República&#39;</span><span class="p">]})</span>
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-      <th>0</th>
-      <td>Juan Valdez</td>
-      <td>32</td>
-      <td>Café de Colombia</td>
-    </tr>
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-      <th>1</th>
-      <td>Álvaro Uribe Vélez</td>
-      <td>65</td>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="o">.</span><span class="n">from_items</span><span class="p">([</span> <span class="p">[</span> <span class="s1">&#39;Nombre&#39;</span>  <span class="p">,</span> <span class="p">[</span><span class="s1">&#39;Juan Valdez&#39;</span><span class="p">,</span><span class="s1">&#39;Álvaro Uribe Vélez&#39;</span><span class="p">]],</span>
-                          <span class="p">[</span> <span class="s1">&#39;Edad&#39;</span>    <span class="p">,</span> <span class="p">[</span>  <span class="mi">32</span><span class="p">,</span>            <span class="mi">65</span>               <span class="p">]],</span>
-                          <span class="p">[</span> <span class="s1">&#39;Compañia&#39;</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;Café de Colombia&#39;</span><span class="p">,</span><span class="s1">&#39;Senado de la República&#39;</span><span class="p">]]</span> <span class="p">])</span>
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-              <span class="p">{</span><span class="s1">&#39;Nombre&#39;</span><span class="p">:</span><span class="s1">&#39;Álvaro Uribe Vélez&#39;</span><span class="p">,</span> <span class="s1">&#39;Edad&#39;</span><span class="p">:</span> <span class="mi">65</span>   <span class="p">,</span><span class="s1">&#39;Compañia&#39;</span><span class="p">:</span><span class="s1">&#39;Senado de la República&#39;</span><span class="p">}]</span>
-            <span class="p">)</span>
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-      <td>32</td>
-      <td>Juan Valdez</td>
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-      <th>1</th>
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-      <td>65</td>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">()</span>
-<span class="n">df</span>
-</pre></div>
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-      <th></th>
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-  </thead>
-  <tbody>
-  </tbody>
-</table>
-</div>
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-<h3 id="Initialization-from-sequential-rows-as--Series">Initialization from sequential rows as  Series<a class="anchor-link" href="#Initialization-from-sequential-rows-as--Series"> </a></h3><p>We start with an empty <code>DataFrame</code>:</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">()</span>
-<span class="n">df</span><span class="o">.</span><span class="n">empty</span>
-</pre></div>
-
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-</div>
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-<pre>True</pre>
-</div>
-
-</div>
-</div>
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-</div>
-
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-</div>
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-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>We can append a dictionary (or Series) as a row of the <code>DataFrame</code>, provided that we always use the option: <code>ignore_index=True</code></p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">d</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;Name&#39;</span><span class="p">:</span><span class="s1">&#39;Juan Valdez&#39;</span><span class="p">,</span><span class="s1">&#39;Nacionality&#39;</span><span class="p">:</span><span class="s1">&#39;Colombia&#39;</span><span class="p">,</span><span class="s1">&#39;Age&#39;</span><span class="p">:</span><span class="mi">23</span><span class="p">}</span>
-<span class="n">df</span><span class="o">=</span><span class="n">df</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">d</span><span class="p">,</span><span class="n">ignore_index</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-<span class="n">df</span>
-</pre></div>
-
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-</div>
-</div>
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-<div class="output">
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-<div class="output_area">
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-<div>
-<style scoped>
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-
-    .dataframe tbody tr th {
-        vertical-align: top;
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-
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-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>Age</th>
-      <th>Nacionality</th>
-      <th>Name</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>23.0</td>
-      <td>Colombia</td>
-      <td>Juan Valdez</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>We can fix the type of data of the <code>'Age'</code> column</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">type</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">Age</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-</pre></div>
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-<pre>numpy.float64</pre>
-</div>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Age&#39;</span><span class="p">]</span><span class="o">=</span><span class="n">df</span><span class="o">.</span><span class="n">Age</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-<span class="n">df</span>
-</pre></div>
-
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-</div>
-</div>
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-<div class="output">
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-<div class="output_area">
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-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>Age</th>
-      <th>Nacionality</th>
-      <th>Name</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>23</td>
-      <td>Colombia</td>
-      <td>Juan Valdez</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>To add a second file we build another <code>dict</code></p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">d</span><span class="o">=</span><span class="p">{}</span>
-<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;Name&#39;</span><span class="p">,</span><span class="s1">&#39;Nacionality&#39;</span><span class="p">,</span><span class="s1">&#39;Age&#39;</span><span class="p">,</span><span class="s1">&#39;Company&#39;</span><span class="p">]:</span>
-    <span class="n">var</span><span class="o">=</span><span class="nb">input</span><span class="p">(</span><span class="s1">&#39;</span><span class="si">{}</span><span class="s1">:</span><span class="se">\n</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">k</span><span class="p">))</span>
-    <span class="n">d</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">=</span><span class="n">var</span>
-</pre></div>
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-<div class="output_subarea output_stream output_stdout output_text">
-<pre>Name:
-Diego Restrepo
-Nacionality:
-Colombia
-Age:
-33
-Company:
-Universidad de Antioquia
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h4 id="Exercises">Exercises<a class="anchor-link" href="#Exercises"> </a></h4><ul>
-<li>Display the resulting <code>Series</code> in the screen:</li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<ul>
-<li>Append to the previous <code>DataFrame</code> and visualize it:</li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">=</span><span class="n">df</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">d</span><span class="p">,</span><span class="n">ignore_index</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-<span class="n">df</span>
-</pre></div>
-
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-</div>
-</div>
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-<div class="output_area">
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-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
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-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>Age</th>
-      <th>Nacionality</th>
-      <th>Name</th>
-      <th>Company</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>23</td>
-      <td>Colombia</td>
-      <td>Juan Valdez</td>
-      <td>NaN</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>33</td>
-      <td>Colombia</td>
-      <td>Diego Restrepo</td>
-      <td>Universidad de Antioquia</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<ul>
-<li>Fill NaN with empty strings</li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">=</span><span class="n">df</span><span class="o">.</span><span class="n">fillna</span><span class="p">(</span><span class="s1">&#39;&#39;</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<ul>
-<li>Save <code>Pandas</code> <code>DataFrame</code> as an Excel file</li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">to_excel</span><span class="p">(</span><span class="s1">&#39;prof.xlsx&#39;</span><span class="p">,</span><span class="n">index</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
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-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<ul>
-<li>Load pandas DataFrame from the saved file in Excel</li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">pd</span><span class="o">.</span><span class="n">read_excel</span><span class="p">(</span><span class="s1">&#39;prof.xlsx&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
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-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
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-        vertical-align: top;
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-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>Age</th>
-      <th>Nacionality</th>
-      <th>Name</th>
-      <th>Company</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>23</td>
-      <td>Colombia</td>
-      <td>Juan Valdez</td>
-      <td>NaN</td>
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-      <th>1</th>
-      <td>33</td>
-      <td>Colombia</td>
-      <td>Diego Restrepo</td>
-      <td>Universidad de Antioquia</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Common-operations-upon-DataFrames">Common operations upon <code>DataFrames</code><a class="anchor-link" href="#Common-operations-upon-DataFrames"> </a></h3><p>See <a href="https://github.com/restrepo/PythonTipsAndTricks">https://github.com/restrepo/PythonTipsAndTricks</a></p>
-
-</div>
-</div>
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<ul>
-<li><strong>To fill a specific cell</strong></li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="s1">&#39;Company&#39;</span><span class="p">]</span><span class="o">=</span><span class="s1">&#39;Federación de Caferos&#39;</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span>
-</pre></div>
-
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-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>Age</th>
-      <th>Nacionality</th>
-      <th>Name</th>
-      <th>Company</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>23</td>
-      <td>Colombia</td>
-      <td>Juan Valdez</td>
-      <td>Federación de Caferos</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>33</td>
-      <td>Colombia</td>
-      <td>Diego Restrepo</td>
-      <td>Universidad de Antioquia</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Other-formats-to-saving-and-read-files">Other formats to saving and read files<a class="anchor-link" href="#Other-formats-to-saving-and-read-files"> </a></h2><p>We are interested in format which keeps the tags of the columns, like <code>'Nombre', 'Edad', 'Compañia'</code></p>
-
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">read_excel</span><span class="p">(</span><span class="s1">&#39;http://bit.ly/spreadsheet_xlsx&#39;</span><span class="p">)</span>
-<span class="n">df</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>Nombre</th>
-      <th>Edad</th>
-      <th>Compañia</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>Juan Valdez</td>
-      <td>23</td>
-      <td>Café de Colombia</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>Álvaro Uribe Vélez</td>
-      <td>65</td>
-      <td>Senado de la República</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">type</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="s1">&#39;Edad&#39;</span><span class="p">])</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>numpy.int64</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h4 id="CSV">CSV<a class="anchor-link" href="#CSV"> </a></h4>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">to_csv</span><span class="p">(</span><span class="s1">&#39;hoja.csv&#39;</span><span class="p">,</span><span class="n">index</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>We can check the explicit file format with</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">to_csv</span><span class="p">(</span><span class="kc">None</span><span class="p">,</span><span class="n">index</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>&#39;Nombre,Edad,Compañia\nJuan Valdez,23,Café de Colombia\nÁlvaro Uribe Vélez,65,Senado de la República\n&#39;</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">to_csv</span><span class="p">(</span><span class="kc">None</span><span class="p">,</span><span class="n">index</span><span class="o">=</span><span class="kc">False</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>Nombre,Edad,Compañia
-Juan Valdez,23,Café de Colombia
-Álvaro Uribe Vélez,65,Senado de la República
-
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;hoja.csv&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>Nombre</th>
-      <th>Edad</th>
-      <th>Compañia</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>Juan Valdez</td>
-      <td>23</td>
-      <td>Café de Colombia</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>Álvaro Uribe Vélez</td>
-      <td>65</td>
-      <td>Senado de la República</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h4 id="JSON">JSON<a class="anchor-link" href="#JSON"> </a></h4>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>This format keeps the Python lists and dictionaries at the storage level</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">([{</span><span class="s2">&quot;Name&quot;</span><span class="p">:</span><span class="s2">&quot;Donald Trump&quot;</span><span class="p">,</span><span class="s2">&quot;Age&quot;</span><span class="p">:</span><span class="mi">74</span><span class="p">},</span>
-                 <span class="p">{</span><span class="s2">&quot;Name&quot;</span><span class="p">:</span><span class="s2">&quot;Barak Obama&quot;</span><span class="p">,</span> <span class="s2">&quot;Age&quot;</span><span class="p">:</span><span class="mi">59</span><span class="p">}])</span>
-<span class="n">df</span>
-</pre></div>
-
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-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>Name</th>
-      <th>Age</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>Donald Trump</td>
-      <td>74</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>Barak Obama</td>
-      <td>59</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>This format allow us to keep exactly the very same list of dictionaries structure!</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">to_json</span><span class="p">(</span><span class="kc">None</span><span class="p">,</span><span class="n">orient</span><span class="o">=</span><span class="s1">&#39;records&#39;</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>[{&#34;Name&#34;:&#34;Donald Trump&#34;,&#34;Age&#34;:74},{&#34;Name&#34;:&#34;Barak Obama&#34;,&#34;Age&#34;:59}]
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Activity</strong>:</p>
-<ul>
-<li>Save to a file instead of <code>None</code> and open the file with some editor. </li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">to_json</span><span class="p">(</span><span class="s1">&#39;presidents.json&#39;</span><span class="p">,</span><span class="n">orient</span><span class="o">=</span><span class="s1">&#39;records&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<ul>
-<li>Add a break-line at the end of the first dictionary and try to
-load the resulting file with <code>pd.read_json</code></li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">pd</span><span class="o">.</span><span class="n">read_json</span><span class="p">(</span><span class="s1">&#39;presidents.json&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>Name</th>
-      <th>Age</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>Donald Trump</td>
-      <td>74</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>Barak Obama</td>
-      <td>59</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>JSON allows for some flexibility in the break-lines structure:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">hm</span><span class="o">=</span><span class="s1">&#39;&#39;&#39;</span>
-<span class="s1">hola</span>
-<span class="s1">mundo</span>
-<span class="s1">&#39;&#39;&#39;</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">hm</span>
-</pre></div>
-
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-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>&#39;\nhola\nmundo\n&#39;</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">pd</span><span class="o">.</span><span class="n">read_json</span><span class="p">(</span><span class="s1">&#39;&#39;&#39;</span>
-<span class="s1">             [{&quot;Name&quot;:&quot;Donald Trump&quot;,&quot;Age&quot;:73},</span>
-<span class="s1">              {&quot;Name&quot;:&quot;Barak Obama&quot;, &quot;Age&quot;:58}]</span>
-<span class="s1">            &#39;&#39;&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>Name</th>
-      <th>Age</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>Donald Trump</td>
-      <td>73</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>Barak Obama</td>
-      <td>58</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>For large databases it is convinient just to accumulate dictionaries in a sequential form:</p>
-
-</div>
-</div>
-</div>
-</div>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">to_json</span><span class="p">(</span><span class="kc">None</span><span class="p">,</span><span class="n">orient</span><span class="o">=</span><span class="s1">&#39;records&#39;</span><span class="p">,</span><span class="n">lines</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
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-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>{&#34;Name&#34;:&#34;Donald Trump&#34;,&#34;Age&#34;:74}
-{&#34;Name&#34;:&#34;Barak Obama&#34;,&#34;Age&#34;:59}
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">pd</span><span class="o">.</span><span class="n">read_json</span><span class="p">(</span><span class="s1">&#39;&#39;&#39;</span>
-<span class="s1">             {&quot;Name&quot;:&quot;Donald Trump&quot;,&quot;Age&quot;:73}</span>
-<span class="s1">             {&quot;Name&quot;:&quot;Barak Obama&quot;, &quot;Age&quot;:58}</span>
-<span class="s1">            &#39;&#39;&#39;</span><span class="p">,</span><span class="n">orient</span><span class="o">=</span><span class="s1">&#39;records&#39;</span><span class="p">,</span><span class="n">lines</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>Name</th>
-      <th>Age</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>Donald Trump</td>
-      <td>73</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>Barak Obama</td>
-      <td>58</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Activity</strong>:</p>
-<ul>
-<li>Save to a file instead of <code>None</code>, with options: <code>orient='records',lines=True</code>, and open the file with some editor. </li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">to_json</span><span class="p">(</span><span class="s1">&#39;presidents.json&#39;</span><span class="p">,</span><span class="n">orient</span><span class="o">=</span><span class="s1">&#39;records&#39;</span><span class="p">,</span><span class="n">lines</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<ul>
-<li>Add a similar dictionary in the next new line, and try to
-load the resulting file with <code>pd.read_json</code> with options: <code>orient='records',lines=True</code>. <ul>
-<li>WARNING: Use doble-quotes <code>"</code> to write the keys od the new
-dictionary</li>
-</ul>
-</li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Any Python string need to be converted first to double-quotes before to be used as JSON string.</p>
-<p><strong>Example</strong></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">numbers</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;even&quot;</span><span class="p">:</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span>   <span class="c1">#  First  key-list</span>
-         <span class="s2">&quot;odd&quot;</span> <span class="p">:</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">9</span><span class="p">]</span> <span class="p">}</span>  <span class="c1">#  Second key-list</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">numbers</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>{&#39;even&#39;: [0, 2, 4, -6, 8], &#39;odd&#39;: [1, 3, -5, 7, 9]}</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">str</span><span class="p">(</span><span class="n">numbers</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
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-<div class="jb_output_wrapper }}">
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-
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>&#34;{&#39;even&#39;: [0, 2, 4, -6, 8], &#39;odd&#39;: [1, 3, -5, 7, 9]}&#34;</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>This string can be writing in the <code>JSON</code> format by replacing the single quotes, ' , by  duoble quotes, ":</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">str</span><span class="p">(</span><span class="n">numbers</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s2">&quot;&#39;&quot;</span><span class="p">,</span><span class="s1">&#39;&quot;&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>&#39;{&#34;even&#34;: [0, 2, 4, -6, 8], &#34;odd&#34;: [1, 3, -5, 7, 9]}&#39;</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>and now can be used as an JSON input</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">read_json</span><span class="p">(</span>  <span class="nb">str</span><span class="p">(</span><span class="n">numbers</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s2">&quot;&#39;&quot;</span><span class="p">,</span><span class="s1">&#39;&quot;&#39;</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">df</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>even</th>
-      <th>odd</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>0</td>
-      <td>1</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>2</td>
-      <td>3</td>
-    </tr>
-    <tr>
-      <th>2</th>
-      <td>4</td>
-      <td>-5</td>
-    </tr>
-    <tr>
-      <th>3</th>
-      <td>-6</td>
-      <td>7</td>
-    </tr>
-    <tr>
-      <th>4</th>
-      <td>8</td>
-      <td>9</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Activity</strong>: Try to read the string as JSON without make the double-quote replacement</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Filters">Filters<a class="anchor-link" href="#Filters"> </a></h2><p>The main application of labeled data for data analysis is the possibility to make filers, or cuts, to obtain specific reduced datasets to further analysis</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>even</th>
-      <th>odd</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>0</td>
-      <td>1</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>2</td>
-      <td>3</td>
-    </tr>
-    <tr>
-      <th>2</th>
-      <td>4</td>
-      <td>-5</td>
-    </tr>
-    <tr>
-      <th>3</th>
-      <td>-6</td>
-      <td>7</td>
-    </tr>
-    <tr>
-      <th>4</th>
-      <td>8</td>
-      <td>9</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">even</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span><span class="o">&gt;</span><span class="mi">4</span><span class="p">]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>even</th>
-      <th>odd</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>3</th>
-      <td>-6</td>
-      <td>7</td>
-    </tr>
-    <tr>
-      <th>4</th>
-      <td>8</td>
-      <td>9</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#and</span>
-<span class="n">df</span><span class="p">[(</span><span class="n">df</span><span class="o">.</span><span class="n">even</span><span class="o">&gt;</span><span class="mi">0</span><span class="p">)</span> <span class="o">&amp;</span> <span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">odd</span><span class="o">&lt;</span><span class="mi">0</span><span class="p">)]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>even</th>
-      <th>odd</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>2</th>
-      <td>4</td>
-      <td>-5</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="p">[</span><span class="o">~</span><span class="p">((</span><span class="n">df</span><span class="o">.</span><span class="n">even</span><span class="o">&gt;</span><span class="mi">0</span><span class="p">)</span> <span class="o">&amp;</span> <span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">odd</span><span class="o">&lt;</span><span class="mi">0</span><span class="p">))</span> <span class="p">]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>even</th>
-      <th>odd</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>0</td>
-      <td>1</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>2</td>
-      <td>3</td>
-    </tr>
-    <tr>
-      <th>3</th>
-      <td>-6</td>
-      <td>7</td>
-    </tr>
-    <tr>
-      <th>4</th>
-      <td>8</td>
-      <td>9</td>
-    </tr>
-  </tbody>
-</table>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#or</span>
-<span class="n">df</span><span class="p">[(</span><span class="n">df</span><span class="o">.</span><span class="n">even</span><span class="o">&lt;</span><span class="mi">0</span><span class="p">)</span> <span class="o">|</span> <span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">odd</span><span class="o">&lt;</span><span class="mi">0</span><span class="p">)]</span>
-</pre></div>
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-<style scoped>
-    .dataframe tbody tr th:only-of-type {
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-    }
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-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>even</th>
-      <th>odd</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>2</th>
-      <td>4</td>
-      <td>-5</td>
-    </tr>
-    <tr>
-      <th>3</th>
-      <td>-6</td>
-      <td>7</td>
-    </tr>
-  </tbody>
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-<h2 id="The-apply-method">The <code>apply</code> method<a class="anchor-link" href="#The-apply-method"> </a></h2><p>The advantage of the spreadsheet paradigm is that the columns can be transformed with functions. All the typical functions avalaible for a spreadsheet are already implemented like the method <code>.abs()</code> used before, or the method: <code>.sum()</code></p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">even</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
-</pre></div>
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-<pre>8</pre>
-</div>
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-<p><strong>Activity</strong>: Explore the avalaible methods by using the completion system of the notebook after the last semicolon of <code>df.even.</code></p>
-
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>In addittion, for the <code>DataFrame</code> paradigm, we can easy implement any other function directly at the programming level either along the columns or along the rows</p>
-
-</div>
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-<h3 id="Column-level-apply">Column-level <code>apply</code><a class="anchor-link" href="#Column-level-apply"> </a></h3><p>We just select the column and apply the direct or implicit function:</p>
-<ul>
-<li>Pre-defined function</li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">even</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="nb">abs</span><span class="p">)</span>
-</pre></div>
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-<pre>0    0
-1    2
-2    4
-3    6
-4    8
-Name: even, dtype: int64</pre>
-</div>
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-<ul>
-<li>Implicit function</li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">even</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="nb">int</span><span class="p">))</span>
-</pre></div>
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-<pre>0    True
-1    True
-2    True
-3    True
-4    True
-Name: even, dtype: bool</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">even</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
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-<pre>0     0
-1     4
-2    16
-3    36
-4    64
-Name: even, dtype: int64</pre>
-</div>
-
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-</div>
-</div>
-</div>
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-<h3 id="Row-level-apply">Row-level apply<a class="anchor-link" href="#Row-level-apply"> </a></h3><p>The foll row is passed as dictionary to the explicit or implicit function when <code>apply</code> is used for the full <code>DataFrame</code> and the option <code>axis=1</code> is used at the end</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;even&#39;</span><span class="p">]</span><span class="o">+</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;odd&#39;</span><span class="p">]</span>
-</pre></div>
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-<pre>0     1
-1     5
-2    -1
-3     1
-4    17
-dtype: int64</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">row</span><span class="p">:</span> <span class="n">row</span><span class="p">[</span><span class="s1">&#39;even&#39;</span><span class="p">]</span><span class="o">+</span><span class="n">row</span><span class="p">[</span><span class="s1">&#39;odd&#39;</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="s1">&#39;columns&#39;</span><span class="p">)</span>
-</pre></div>
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-<pre>0     1
-1    11
-2    29
-3    43
-4    89
-dtype: int64</pre>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Chain-tools-for-data-analysis">Chain tools for data analysis<a class="anchor-link" href="#Chain-tools-for-data-analysis"> </a></h2><p>There are several chain tools for data analyis like the</p>
-<ul>
-<li>Spreadsheet based one, like Excel </li>
-<li>Relational databases with the use of more advanced SQL tabular data with some data base software like MySQL</li>
-<li>Non-relational databases with Pandas, R, or MongoDB</li>
-</ul>
-<p>Here we illustrate an example of use fo a non-relational database with Pandas</p>
-
-</div>
-</div>
-</div>
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-<h2 id="Relational-databases">Relational databases<a class="anchor-link" href="#Relational-databases"> </a></h2>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">read_excel</span><span class="p">(</span><span class="s1">&#39;http://bit.ly/spreadsheet_xlsx&#39;</span><span class="p">)</span>
-<span class="n">persona</span><span class="o">=</span><span class="n">df</span><span class="p">[[</span><span class="s1">&#39;Nombre&#39;</span><span class="p">,</span><span class="s1">&#39;Edad&#39;</span><span class="p">]]</span>
-</pre></div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">persona</span><span class="p">[</span><span class="s1">&#39;id&#39;</span><span class="p">]</span><span class="o">=</span><span class="p">[</span><span class="mi">888</span><span class="p">,</span><span class="mi">666</span><span class="p">]</span>
-<span class="n">persona</span>
-</pre></div>
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-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
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-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>Nombre</th>
-      <th>Edad</th>
-      <th>id</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>Juan Valdez</td>
-      <td>23</td>
-      <td>888</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>Álvaro Uribe Vélez</td>
-      <td>65</td>
-      <td>666</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">d1</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;id&#39;</span><span class="p">:</span><span class="mi">888</span><span class="p">,</span> <span class="s1">&#39;Inicio&#39;</span><span class="p">:</span><span class="mi">2010</span><span class="p">,</span><span class="s1">&#39;Fin:&#39;</span><span class="p">:</span><span class="kc">None</span><span class="p">,</span><span class="s1">&#39;Cargo&#39;</span><span class="p">:</span><span class="s1">&#39;Arriero&#39;</span><span class="p">,</span><span class="s1">&#39;Compañía&#39;</span><span class="p">:</span><span class="s1">&#39;Café de Colombia&#39;</span><span class="p">}</span>
-<span class="n">d2</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;id&#39;</span><span class="p">:</span><span class="mi">666</span><span class="p">,</span> <span class="s1">&#39;Inicio&#39;</span><span class="p">:</span><span class="mi">2013</span><span class="p">,</span><span class="s1">&#39;Fin:&#39;</span><span class="p">:</span><span class="mi">2020</span><span class="p">,</span><span class="s1">&#39;Cargo&#39;</span><span class="p">:</span><span class="s1">&#39;Senador&#39;</span><span class="p">,</span><span class="s1">&#39;Compañía&#39;</span><span class="p">:</span><span class="s1">&#39;Senado de la República de Colombia&#39;</span><span class="p">}</span>
-<span class="n">d3</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;id&#39;</span><span class="p">:</span><span class="mi">666</span><span class="p">,</span> <span class="s1">&#39;Inicio&#39;</span><span class="p">:</span><span class="mi">2020</span><span class="p">,</span><span class="s1">&#39;Fin:&#39;</span><span class="p">:</span><span class="kc">None</span><span class="p">,</span><span class="s1">&#39;Cargo&#39;</span><span class="p">:</span><span class="s1">&#39;Influencer&#39;</span><span class="p">,</span><span class="s1">&#39;Compañía&#39;</span><span class="p">:</span><span class="s1">&#39;Twitter&#39;</span><span class="p">}</span>
-<span class="n">trabajo</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span> <span class="p">[</span><span class="n">d1</span><span class="p">,</span><span class="n">d2</span><span class="p">,</span><span class="n">d3</span><span class="p">]</span> <span class="p">)</span>  
-<span class="n">trabajo</span>
-</pre></div>
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-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
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-        vertical-align: top;
-    }
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-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>id</th>
-      <th>Inicio</th>
-      <th>Fin:</th>
-      <th>Cargo</th>
-      <th>Compañía</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>888</td>
-      <td>2010</td>
-      <td>NaN</td>
-      <td>Arriero</td>
-      <td>Café de Colombia</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>666</td>
-      <td>2013</td>
-      <td>2020.0</td>
-      <td>Senador</td>
-      <td>Senado de la República de Colombia</td>
-    </tr>
-    <tr>
-      <th>2</th>
-      <td>666</td>
-      <td>2020</td>
-      <td>NaN</td>
-      <td>Influencer</td>
-      <td>Twitter</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p><img src="https://raw.githubusercontent.com/restrepo/ComputationalMethods/master/material/figures/relation.jpg" alt="img"></p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Actividad">Actividad<a class="anchor-link" href="#Actividad"> </a></h3><p>Obtenga el último trabajo de Álvaro Uribe Vélez</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">cc</span><span class="o">=</span><span class="n">persona</span><span class="p">[</span><span class="n">persona</span><span class="p">[</span><span class="s1">&#39;Nombre&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">str</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span><span class="o">.</span><span class="n">str</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="s1">&#39;álvaro uribe vélez&#39;</span><span class="p">)]</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;id&#39;</span><span class="p">)</span>
-<span class="n">trabajo</span><span class="p">[</span><span class="n">trabajo</span><span class="p">[</span><span class="s1">&#39;id&#39;</span><span class="p">]</span><span class="o">==</span><span class="n">cc</span><span class="p">][</span><span class="s1">&#39;Cargo&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-</pre></div>
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-<pre>&#39;Influencer&#39;</pre>
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-<h2 id="Non-relational-databases">Non-relational databases<a class="anchor-link" href="#Non-relational-databases"> </a></h2>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">persona</span><span class="o">=</span><span class="n">persona</span><span class="o">.</span><span class="n">drop</span><span class="p">(</span><span class="s1">&#39;id&#39;</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="s1">&#39;columns&#39;</span><span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">persona</span><span class="p">[</span><span class="s1">&#39;Trabajo&#39;</span><span class="p">]</span><span class="o">=</span><span class="s1">&#39;&#39;</span>
-<span class="c1">#Column need to be converted to object to assign an advance object like a list</span>
-<span class="n">persona</span><span class="p">[</span><span class="s1">&#39;Trabajo&#39;</span><span class="p">]</span><span class="o">=</span><span class="n">persona</span><span class="p">[</span><span class="s1">&#39;Trabajo&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="s1">&#39;object&#39;</span><span class="p">)</span>
-<span class="n">persona</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="s1">&#39;Trabajo&#39;</span><span class="p">]</span><span class="o">=</span><span class="p">[</span><span class="n">d1</span><span class="p">]</span>
-<span class="n">persona</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="s1">&#39;Trabajo&#39;</span><span class="p">]</span><span class="o">=</span><span class="p">[</span><span class="n">d2</span><span class="p">,</span><span class="n">d3</span><span class="p">]</span>
-</pre></div>
-
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-<style scoped>
-    .dataframe tbody tr th:only-of-type {
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-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>Nombre</th>
-      <th>Edad</th>
-      <th>Trabajo</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>Juan Valdez</td>
-      <td>23</td>
-      <td>[{'id': 888, 'Inicio': 2010, 'Fin:': None, 'Cargo': 'Arriero', 'Compañía': 'Café de Colombia'}]</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>Álvaro Uribe Vélez</td>
-      <td>65</td>
-      <td>[{'id': 666, 'Inicio': 2013, 'Fin:': 2020, 'Cargo': 'Senador', 'Compañía': 'Senado de la República de Colombia'}, {'id': 666, 'Inicio': 2020, 'Fin:': None, 'Cargo': 'Influencer', 'Compañía': 'Twit...</td>
-    </tr>
-  </tbody>
-</table>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span> <span class="n">persona</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="s1">&#39;Trabajo&#39;</span><span class="p">]</span> <span class="p">)</span>
-</pre></div>
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-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>id</th>
-      <th>Inicio</th>
-      <th>Fin:</th>
-      <th>Cargo</th>
-      <th>Compañía</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>666</td>
-      <td>2013</td>
-      <td>2020.0</td>
-      <td>Senador</td>
-      <td>Senado de la República de Colombia</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>666</td>
-      <td>2020</td>
-      <td>NaN</td>
-      <td>Influencer</td>
-      <td>Twitter</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
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-<h3 id="Actividad">Actividad<a class="anchor-link" href="#Actividad"> </a></h3><p>Obtenga el último trabajo de Álvaro Uribe Vélez</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">persona</span><span class="p">[</span><span class="n">persona</span><span class="p">[</span><span class="s1">&#39;Nombre&#39;</span><span class="p">]</span><span class="o">==</span><span class="s1">&#39;Álvaro Uribe Vélez&#39;</span>
-       <span class="p">]</span><span class="o">.</span><span class="n">Trabajo</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span>
-        <span class="k">lambda</span> <span class="n">l</span><span class="p">:</span> <span class="p">[</span><span class="n">d</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;Cargo&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">l</span> <span class="p">])</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-</pre></div>
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-<pre>1    Influencer
-Name: Trabajo, dtype: object</pre>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>We have shown that the simple two dimensional spreadsheets where each cell values is a simple type like string, integer, or float, can be represented as a dictionary of lists values or a list of dictionary column-value assignment.</p>
-<p>We can go further and allow to store in the value itself a more general data structure, like nested lists and dictionaries. This allows advanced data-analysis when the <code>apply</code> methos is used to operate inside the nested lists or dictionaries.</p>
-<p>See for example:</p>
-
-</div>
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-<h2 id="World-wide-web">World wide web<a class="anchor-link" href="#World-wide-web"> </a></h2><p>There are really three kinds of web</p>
-<ul>
-<li>The normal web, </li>
-<li>The deep web,</li>
-<li><em>The machine web</em>. The web for machine readable responses. It is served in <code>JSON</code> or <code>XML</code>  formats, which preserve programming objects.</li>
-</ul>
-
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-<h3 id="Normal-web">Normal web<a class="anchor-link" href="#Normal-web"> </a></h3>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">pd</span><span class="o">.</span><span class="n">read_html</span><span class="p">(</span><span class="s1">&#39;https://en.wikipedia.org/wiki/COVID-19_pandemic_by_country_and_territory&#39;</span><span class="p">)[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">:]</span>
-</pre></div>
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-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>COVID-19 pandemic</th>
-      <th>COVID-19 pandemic.1</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>1</th>
-      <td>Disease</td>
-      <td>COVID-19</td>
-    </tr>
-    <tr>
-      <th>2</th>
-      <td>Virus strain</td>
-      <td>SARS-CoV-2</td>
-    </tr>
-    <tr>
-      <th>3</th>
-      <td>Source</td>
-      <td>Probably bats, possibly via pangolins[1][2]</td>
-    </tr>
-    <tr>
-      <th>4</th>
-      <td>Location</td>
-      <td>Worldwide</td>
-    </tr>
-    <tr>
-      <th>5</th>
-      <td>First outbreak</td>
-      <td>Mainland China[3]</td>
-    </tr>
-    <tr>
-      <th>6</th>
-      <td>Index case</td>
-      <td>Wuhan, Hubei, China.mw-parser-output .geo-default,.mw-parser-output .geo-dms,.mw-parser-output .geo-dec{display:inline}.mw-parser-output .geo-nondefault,.mw-parser-output .geo-multi-punct{display:...</td>
-    </tr>
-    <tr>
-      <th>7</th>
-      <td>Date</td>
-      <td>1 December 2019[3] – present(1 year, 3 months, 3 weeks and 3 days)</td>
-    </tr>
-    <tr>
-      <th>8</th>
-      <td>Confirmed cases</td>
-      <td>124,971,776[4]</td>
-    </tr>
-    <tr>
-      <th>9</th>
-      <td>Active cases</td>
-      <td>51,331,455[4]</td>
-    </tr>
-    <tr>
-      <th>10</th>
-      <td>Recovered</td>
-      <td>70,893,740[4]</td>
-    </tr>
-    <tr>
-      <th>11</th>
-      <td>Deaths</td>
-      <td>2,746,581[4]</td>
-    </tr>
-    <tr>
-      <th>12</th>
-      <td>Territories</td>
-      <td>192[4]</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Real world example: microsoft academics
-<img src="https://docs.microsoft.com/en-us/academic-services/graph/media/erd/entity-relationship-diagram.png" alt="img"></p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Machine-web">Machine web<a class="anchor-link" href="#Machine-web"> </a></h3><p>For example, consider the following normal web page:</p>
-<p><a href="http://old.inspirehep.net/search?p=doi:10.1103/PhysRevLett.122.132001">http://old.inspirehep.net/search?p=doi:10.1103/PhysRevLett.122.132001</a></p>
-<p>about a Scientific paper with people from the University of Antioquia. A <em>machine web</em> version can be easily obtained in JSON just by attaching the extra parameter <code>&amp;of=recjson</code>, and direcly loaded from Pandas, which works like a <em>browser for the third web</em>:</p>
-
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">read_json</span><span class="p">(</span><span class="s1">&#39;http://old.inspirehep.net/search?p=doi:10.1103/PhysRevLett.122.132001&amp;of=recjson&#39;</span><span class="p">)</span>
-</pre></div>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span>
-</pre></div>
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-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>comment</th>
-      <th>reference</th>
-      <th>abstract</th>
-      <th>creation_date</th>
-      <th>imprint</th>
-      <th>primary_report_number</th>
-      <th>keywords</th>
-      <th>publication_info</th>
-      <th>coyright</th>
-      <th>subject</th>
-      <th>...</th>
-      <th>number_of_authors</th>
-      <th>license</th>
-      <th>accelerator_experiment</th>
-      <th>number_of_comments</th>
-      <th>url</th>
-      <th>recid</th>
-      <th>collection</th>
-      <th>thesaurus_terms</th>
-      <th>filetypes</th>
-      <th>prepublication</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>Submitted to Phys. Rev. Lett. All figures and tables can be found at http://cms-results.web.cern.ch/cms-results/public-results/publications/BPH-18-007 (CMS Public Pages). This version corrects the...</td>
-      <td>[{'title': 'Phys.Rev.Lett.,81,2432', 'year': '1998', 'doi': 'doi:10.1103/PhysRevLett.81.2432', 'order_number': '1', 'authors': 'F. Abe'}, {'title': 'Phys.Rev.,D49,5845', 'year': '1994', 'doi': 'do...</td>
-      <td>[{'number': 'arXiv', 'summary': 'Signals consistent with the B$^+_\mathrm{c}$(2S) and B$^{*+}_\mathrm{c}$(2S) states are observed in proton-proton collisions at $\sqrt{s} =$ 13 TeV, in an event sa...</td>
-      <td>2019-02-05T04:03:36</td>
-      <td>{'date': '2019-04-03'}</td>
-      <td>[arXiv:1902.00571, None, CMS-BPH-18-007, CERN-EP-2019-014]</td>
-      <td>[{'institute': 'publisher', 'term': 'Elementary Particles and Fields'}]</td>
-      <td>{'volume': '122', 'pagination': '132001', 'title': 'Phys.Rev.Lett.', 'year': '2019'}</td>
-      <td>{'date': '2019', 'holder_contact': 'publication', 'holder': 'CERN'}</td>
-      <td>[{'source': 'INSPIRE', 'term': 'Experiment-HEP'}, {'source': 'arXiv', 'term': 'hep-ex'}, {'source': 'INSPIRE', 'term': 'Experiment-HEP'}]</td>
-      <td>...</td>
-      <td>2279</td>
-      <td>[{'url': 'http://creativecommons.org/licenses/by/4.0/', 'material': 'preprint', 'license': 'CC BY 4.0'}, {'imposing': 'APS', 'url': 'https://creativecommons.org/licenses/by/4.0/', 'license': 'Crea...</td>
-      <td>{'experiment': 'CERN-LHC-CMS'}</td>
-      <td>0</td>
-      <td>[{'url': 'http://inspirehep.net/record/1718338/files/Figure_001.png', 'description': '00002 Transitions between the lightest \BC states, with solid and dashed lines indicating the emission of phot...</td>
-      <td>1718338</td>
-      <td>[{'primary': 'HEP'}, {'primary': 'Citeable'}, {'primary': 'CORE'}, {'primary': 'Fermilab'}, {'primary': 'Published'}]</td>
-      <td>[{'term': 'p p: scattering'}, {'term': 'p p: colliding beams'}, {'term': 'B/c+: hadroproduction'}, {'term': 'B/c: excited state'}, {'term': 'B/c+: hadronic decay'}, {'term': 'excited state: hadron...</td>
-      <td>[xml, pdf, pdf, pdf, png, png, png]</td>
-      <td>{'date': '2019-02-01'}</td>
-    </tr>
-  </tbody>
-</table>
-<p>1 rows × 36 columns</p>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>We can see that the column <code>authors</code> is quite nested: Is a list of dictionaries with the full information for each one of the authors of the article.</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">columns</span>
-</pre></div>
-
-    </div>
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-<pre>Index([&#39;comment&#39;, &#39;reference&#39;, &#39;abstract&#39;, &#39;creation_date&#39;, &#39;imprint&#39;,
-       &#39;primary_report_number&#39;, &#39;keywords&#39;, &#39;publication_info&#39;, &#39;coyright&#39;,
-       &#39;subject&#39;, &#39;cataloguer_info&#39;, &#39;physical_description&#39;,
-       &#39;number_of_citations&#39;, &#39;other_report_number&#39;, &#39;title&#39;,
-       &#39;persistent_identifiers_keys&#39;, &#39;corporate_name&#39;, &#39;files&#39;,
-       &#39;system_control_number&#39;, &#39;filenames&#39;, &#39;source_of_acquisition&#39;,
-       &#39;number_of_reviews&#39;, &#39;version_id&#39;, &#39;FIXME_OAI&#39;, &#39;authors&#39;, &#39;doi&#39;,
-       &#39;number_of_authors&#39;, &#39;license&#39;, &#39;accelerator_experiment&#39;,
-       &#39;number_of_comments&#39;, &#39;url&#39;, &#39;recid&#39;, &#39;collection&#39;, &#39;thesaurus_terms&#39;,
-       &#39;filetypes&#39;, &#39;prepublication&#39;],
-      dtype=&#39;object&#39;)</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">number_of_authors</span>
-</pre></div>
-
-    </div>
-</div>
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-<pre>0    2279
-Name: number_of_authors, dtype: int64</pre>
-</div>
-
-</div>
-</div>
-</div>
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-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">authors</span>
-</pre></div>
-
-    </div>
-</div>
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-<pre>0    [{&#39;INSPIRE_number&#39;: &#39;INSPIRE-00312131&#39;, &#39;affiliation&#39;: &#39;Yerevan Phys. Inst.&#39;, &#39;first_name&#39;: &#39;Albert M&#39;, &#39;last_name&#39;: &#39;Sirunyan&#39;, &#39;full_name&#39;: &#39;Sirunyan, Albert M&#39;}, {&#39;INSPIRE_number&#39;: &#39;INSPIRE-001...
-Name: authors, dtype: object</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<p><strong>Activity</strong>: Check that the lenght of the auhors list coincides with the <code>number_of_authors</code> 
-<!-- df.authors.apply(len),df.number_of_authors.values --></p>
-
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-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">authors</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="nb">len</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
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-<pre>0    2279
-Name: authors, dtype: int64</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">authors</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-</pre></div>
-
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-<pre>{&#39;INSPIRE_number&#39;: &#39;INSPIRE-00312131&#39;,
- &#39;affiliation&#39;: &#39;Yerevan Phys. Inst.&#39;,
- &#39;first_name&#39;: &#39;Albert M&#39;,
- &#39;last_name&#39;: &#39;Sirunyan&#39;,
- &#39;full_name&#39;: &#39;Sirunyan, Albert M&#39;}</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>We can use all the previous methods to extract the authors from <code>'Antioquia U.'</code>:</p>
-<p>Note: For a dictionary, <code>d</code> is safer to use <code>d.get('key')</code> instead of just <code>d['key']</code> to obtain some <code>key</code>, because not error is generated if the requested <code>key</code> does not exists at all</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">authors</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">l</span><span class="p">:</span> <span class="p">[</span><span class="n">d</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">l</span> <span class="k">if</span> <span class="n">d</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;affiliation&#39;</span><span class="p">)</span><span class="o">==</span><span class="s1">&#39;Antioquia U.&#39;</span><span class="p">])</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</pre></div>
-
-    </div>
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-<pre>[{&#39;INSPIRE_number&#39;: &#39;INSPIRE-00343518&#39;,
-  &#39;affiliation&#39;: &#39;Antioquia U.&#39;,
-  &#39;first_name&#39;: &#39;Jhovanny&#39;,
-  &#39;last_name&#39;: &#39;Mejia Guisao&#39;,
-  &#39;full_name&#39;: &#39;Mejia Guisao, Jhovanny&#39;},
- {&#39;INSPIRE_number&#39;: &#39;INSPIRE-00355050&#39;,
-  &#39;affiliation&#39;: &#39;Antioquia U.&#39;,
-  &#39;first_name&#39;: &#39;José David&#39;,
-  &#39;last_name&#39;: &#39;Ruiz Alvarez&#39;,
-  &#39;full_name&#39;: &#39;Ruiz Alvarez, José David&#39;}]</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>We also need use comprenhension list like in</p>
-
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">l</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="p">[</span><span class="n">x</span> <span class="k">if</span> <span class="n">x</span><span class="o">&gt;</span><span class="mi">1</span> <span class="k">else</span> <span class="mi">3</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">l</span> <span class="p">]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
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-<pre>[3, 2, 3]</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Then the on-fly apply function to extract the authors with affiliation 'Antioquia U.' is</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">authors</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">l</span><span class="p">:</span>  <span class="c1">#implicit function</span>
-                           <span class="p">[</span> <span class="n">d</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;full_name&#39;</span><span class="p">)</span> <span class="c1">#safer way to obtain a `key` value</span>
-                             <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">l</span>   <span class="c1">#comprehension list </span>
-                              <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">d</span><span class="p">,</span><span class="nb">dict</span><span class="p">)</span> <span class="ow">and</span> <span class="n">d</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;affiliation&#39;</span><span class="p">)</span><span class="o">==</span><span class="s1">&#39;Antioquia U.&#39;</span> <span class="c1">#condition</span>
-                           <span class="p">]</span> 
-                             <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="nb">list</span><span class="p">)</span> <span class="k">else</span> <span class="kc">None</span> <span class="c1">#  Be sure that cell have the proper format</span>
-                 <span class="p">)</span>
-</pre></div>
-
-    </div>
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-<pre>0    [Mejia Guisao, Jhovanny, Ruiz Alvarez, José David]
-Name: authors, dtype: object</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>A simpler version is possible,  but is more prompted to error when used through multiple column entries:</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">authors</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">l</span><span class="p">:</span> <span class="p">[</span> <span class="n">d</span><span class="p">[</span><span class="s1">&#39;full_name&#39;</span><span class="p">]</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">l</span>   
-                              <span class="k">if</span> <span class="n">d</span><span class="p">[</span><span class="s1">&#39;affiliation&#39;</span><span class="p">]</span><span class="o">==</span><span class="s1">&#39;Antioquia U.&#39;</span>
-                           <span class="p">]</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<pre>0    [Mejia Guisao, Jhovanny, Ruiz Alvarez, José David]
-Name: authors, dtype: object</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>For further details see: <a href="https://github.com/restrepo/inspire/blob/master/gfif.ipynb">https://github.com/restrepo/inspire/blob/master/gfif.ipynb</a></p>
-
-</div>
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-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Activity</strong>: Repeat the same activity but using directly the JSON file</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>NOTE</strong>: The same but with the new API</p>
-
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#See: https://github.com/inspirehep/rest-api-doc/issues/4#issuecomment-645218074</span>
-<span class="kn">import</span> <span class="nn">requests</span>                                                                                                                                                      
-<span class="n">response</span> <span class="o">=</span> <span class="n">requests</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;https://inspirehep.net/api/doi/10.1103/PhysRevLett.122.132001&#39;</span><span class="p">)</span>                                                                              
-<span class="n">authors</span> <span class="o">=</span> <span class="n">response</span><span class="o">.</span><span class="n">json</span><span class="p">()[</span><span class="s1">&#39;metadata&#39;</span><span class="p">][</span><span class="s1">&#39;authors&#39;</span><span class="p">]</span>                                                                                                                     
-<span class="n">names</span> <span class="o">=</span> <span class="p">[</span><span class="n">author</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;full_name&#39;</span><span class="p">)</span>
-              <span class="k">for</span> <span class="n">author</span> <span class="ow">in</span> <span class="n">authors</span> 
-               <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">aff</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;value&#39;</span><span class="p">)</span> <span class="o">==</span> <span class="s1">&#39;Antioquia U.&#39;</span> <span class="k">for</span> <span class="n">aff</span> <span class="ow">in</span> <span class="n">author</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;affiliations&#39;</span><span class="p">))]</span>
-<span class="n">names</span>
-</pre></div>
-
-    </div>
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-<pre>[&#39;Mejia Guisao, Jhovanny&#39;, &#39;Ruiz Alvarez, José David&#39;]</pre>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="ACTIVITIES">ACTIVITIES<a class="anchor-link" href="#ACTIVITIES"> </a></h2><p>See:</p>
-<ul>
-<li><a href="https://github.com/ajcr/100-pandas-puzzles">https://github.com/ajcr/100-pandas-puzzles</a></li>
-<li><a href="https://github.com/guipsamora/pandas_exercises">https://github.com/guipsamora/pandas_exercises</a></li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Final-remarks">Final remarks<a class="anchor-link" href="#Final-remarks"> </a></h2><p>With basic scripting and Pandas we already have a solid environment to analyse data. We introduce the other libraries motivated with the extending the capabilities of Pandas</p>
-
-</div>
-</div>
-</div>
-</div>
-
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-<h1 id="Algorithms-and-Convergence">Algorithms and Convergence<a class="anchor-link" href="#Algorithms-and-Convergence"> </a></h1><p>In simple terms, an <strong>algorithm</strong> can be defined as a finite sequence of unambiguous steps that must be followed in order to solve or approximate the solution to some problem. The stated procedure should be translatable into computer code through a programming language.</p>
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-<ul>
-<li><a href="#Types-of-Algorithms">Types of Algorithms</a> <ul>
-<li><a href="#Linear-algorithms">Linear algorithms</a></li>
-<li><a href="#Exponential-algorithms">Exponential algorithms</a></li>
-<li><a href="#Stable-algorithms">Stable algorithms</a></li>
-<li><a href="#Unstable-algorithms">Unstable algorithms</a></li>
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-<li><a href="#Computing Time">Computing Time</a></li>
-<li><a href="#Convergence">Convergence</a></li>
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-<p>There several ways to classify an algorithm, however those that refer to their numerical convergence and accuracy are more interesting. Among them we have:</p>
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-<h1 id="Types-of-Algorithms">Types of Algorithms<a class="anchor-link" href="#Types-of-Algorithms"> </a></h1>
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-<h2 id="Linear-algorithms">Linear algorithms<a class="anchor-link" href="#Linear-algorithms"> </a></h2>
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-<p>They are algorithms where errors scale as the number of performed iterations. This definition usually coincides with the scaling of the time computing.</p>
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-<p><strong>Example 1:</strong></p>
-<p>Consider the recursive equation:</p>
-$$ p_n = 2 p_{n-1} - p_{n-2},\ \ \ \ n = 2,3,\cdots $$<p>the solution to this equation for $p_n$ is given by</p>
-$$ p_n = A + Bn $$<p>for any constants $A$ and $B$.</p>
-<p>Setting the initial conditions as $p_0=1$ and $p_1 = 1/3$ (implying $A=1$ and $B=-2/3$), show that a <strong>float32</strong> arithmetics would lead a linear error scaling as the number of iterations $n$.</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Number of iterations</span>
-<span class="n">Niter</span> <span class="o">=</span> <span class="mi">10000</span>
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-<span class="c1">#Double precision constants (Exact solution)</span>
-<span class="n">A_d</span> <span class="o">=</span> <span class="mf">1.0</span>
-<span class="n">B_d</span> <span class="o">=</span> <span class="o">-</span><span class="mi">2</span><span class="o">/</span><span class="mf">3.</span>
-<span class="c1">#Signle precision constants</span>
-<span class="n">A_s</span> <span class="o">=</span> <span class="mf">1.0000000</span>
-<span class="n">B_s</span> <span class="o">=</span> <span class="o">-</span><span class="mf">0.666667</span>
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-<span class="c1">#Solution to n-th term</span>
-<span class="n">pn</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="n">A</span> <span class="o">+</span> <span class="n">B</span><span class="o">*</span><span class="n">n</span>
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-<span class="c1">#Arrays for storing the iterations</span>
-<span class="n">p_d</span> <span class="o">=</span> <span class="p">[]</span>
-<span class="n">p_s</span> <span class="o">=</span> <span class="p">[]</span>
-<span class="n">narray</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="n">Niter</span><span class="p">)</span>
-<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">narray</span><span class="p">:</span>
-    <span class="n">p_d</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">pn</span><span class="p">(</span> <span class="n">A_d</span><span class="p">,</span> <span class="n">B_d</span><span class="p">,</span> <span class="n">n</span> <span class="p">)</span> <span class="p">)</span>
-    <span class="n">p_s</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">pn</span><span class="p">(</span> <span class="n">A_s</span><span class="p">,</span> <span class="n">B_s</span><span class="p">,</span> <span class="n">n</span> <span class="p">)</span> <span class="p">)</span>
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-<span class="c1">#Converting to numpy arrays</span>
-<span class="n">p_d</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">p_d</span><span class="p">)</span>
-<span class="n">p_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">p_s</span><span class="p">)</span>
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-<span class="c1">#Relative error</span>
-<span class="n">error</span> <span class="o">=</span> <span class="n">p_d</span> <span class="o">-</span> <span class="n">p_s</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">narray</span><span class="p">,</span> <span class="n">error</span><span class="p">,</span> <span class="s2">&quot;-&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;blue&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Number of iterations $n$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
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-<p><strong>Example 2:</strong></p>
-<p>A linear algorithm may also refer to the computing time required for performing $n$ iterations. This example evaluates the time required for evaluating the sum of $N$ random numbers as a function of how many number are going to be added. A comparison with the built-in function of python is also performed.</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#New library!!! </span>
-<span class="c1">#Time: this library allows to access directly to the computer time. </span>
-<span class="c1">#      Useful when time calculations are required.</span>
-<span class="kn">import</span> <span class="nn">time</span> <span class="k">as</span> <span class="nn">tm</span>
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-<span class="c1">#Maximum number of iterations</span>
-<span class="n">narray</span> <span class="o">=</span> <span class="mi">10</span><span class="o">**</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mf">0.1</span><span class="p">)</span>
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-<span class="c1">#Time arrays</span>
-<span class="n">t_u</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c1">#User</span>
-<span class="n">t_p</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c1">#Python</span>
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-<span class="c1">#Iterations</span>
-<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">narray</span><span class="p">:</span>
-    <span class="c1">#Generating random numbers</span>
-    <span class="n">N</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="c1">#-------------------------------------------------------------------</span>
-    <span class="c1"># MANUAL SUMMATION</span>
-    <span class="c1">#-------------------------------------------------------------------</span>
-    <span class="c1">#Starting time counter for user</span>
-    <span class="n">start</span> <span class="o">=</span> <span class="n">tm</span><span class="o">.</span><span class="n">clock</span><span class="p">()</span>
-    <span class="c1">#Adding the numbers manually</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">xrange</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)):</span>
-        <span class="n">result</span> <span class="o">+=</span> <span class="n">N</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-    <span class="c1">#Finishing time counter for user</span>
-    <span class="n">end</span> <span class="o">=</span> <span class="n">tm</span><span class="o">.</span><span class="n">clock</span><span class="p">()</span>
-    <span class="c1">#Storing result</span>
-    <span class="n">t_u</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">end</span><span class="o">-</span><span class="n">start</span> <span class="p">)</span>
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-    <span class="c1">#-------------------------------------------------------------------</span>
-    <span class="c1"># PYTHON SUMMATION</span>
-    <span class="c1">#-------------------------------------------------------------------</span>
-    <span class="c1">#Starting time counter for user</span>
-    <span class="n">start</span> <span class="o">=</span> <span class="n">tm</span><span class="o">.</span><span class="n">clock</span><span class="p">()</span>
-    <span class="c1">#Adding the numbers using python</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span> <span class="n">N</span> <span class="p">)</span>
-    <span class="c1">#Finishing time counter for user</span>
-    <span class="n">end</span> <span class="o">=</span> <span class="n">tm</span><span class="o">.</span><span class="n">clock</span><span class="p">()</span>
-    <span class="c1">#Storing result</span>
-    <span class="n">t_p</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">end</span><span class="o">-</span><span class="n">start</span> <span class="p">)</span>
-    
-<span class="c1">#Ploting</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">semilogx</span><span class="p">(</span> <span class="n">narray</span><span class="p">,</span> <span class="n">t_u</span><span class="p">,</span> <span class="s2">&quot;-&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Manual&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">semilogx</span><span class="p">(</span> <span class="n">narray</span><span class="p">,</span> <span class="n">t_p</span><span class="p">,</span> <span class="s2">&quot;-&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;blue&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Python&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Number of taken numbers $N$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Computing time [seconds]&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
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-<p>This type of algorithms scales as an exponential factor of the number of iterations. These algorithms are usually unstable, where a small perturbation leads to a exponential growth after a few iterations.</p>
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-<p><strong>Example 3:</strong></p>
-<p>Consider the recursive equation:</p>
-$$ p_n = \frac{10}{3}p_{n-1} - p_{n-2},\ \ \ \ n = 2,3,\cdots $$<p>the solution to this equation for $p_n$ is given by</p>
-$$ p_n = A\left( \frac{1}{3} \right)^n + B\ 3^n $$<p>for any constants $A$ and $B$.</p>
-<p>Setting the initial conditions as $p_0=1$ and $p_1 = 1/3$ (implying $A=1$ and $B=0$), show that a <strong>float32</strong> arithmetics would lead an error scaling as the exponential of the number of iterations $n$.</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Number of iterations</span>
-<span class="n">Niter</span> <span class="o">=</span> <span class="mi">100</span>
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-<span class="c1">#Double precision constants (Exact solution)</span>
-<span class="n">A_d</span> <span class="o">=</span> <span class="mf">1.0</span>
-<span class="n">B_d</span> <span class="o">=</span> <span class="mf">0.</span>
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-<span class="c1">#Solution to n-th term</span>
-<span class="n">pn</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="n">A</span><span class="o">*</span><span class="p">(</span><span class="mf">3.0</span><span class="p">)</span><span class="o">**-</span><span class="n">n</span> <span class="o">+</span> <span class="n">B</span><span class="o">*</span><span class="p">(</span><span class="mf">3.0</span><span class="p">)</span><span class="o">**</span><span class="n">n</span>
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-<span class="c1">#Arrays for storing the iterations</span>
-<span class="n">p_s</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.000000</span><span class="p">,</span><span class="mf">0.333333</span><span class="p">]</span>
-<span class="n">p_d</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span><span class="mi">1</span><span class="o">/</span><span class="mf">3.</span><span class="p">]</span>
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-<span class="n">narray</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="n">Niter</span><span class="p">)</span>
-<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">narray</span><span class="p">[</span><span class="mi">2</span><span class="p">:]:</span>
-    <span class="n">p_s</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="mi">10</span><span class="o">/</span><span class="mf">3.</span><span class="o">*</span><span class="n">p_s</span><span class="p">[</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">p_s</span><span class="p">[</span><span class="n">n</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="p">)</span>
-    <span class="n">p_d</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">pn</span><span class="p">(</span> <span class="n">A_d</span><span class="p">,</span> <span class="n">B_d</span><span class="p">,</span> <span class="n">n</span> <span class="p">)</span> <span class="p">)</span>
-
-<span class="c1">#Converting to numpy arrays</span>
-<span class="n">p_d</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">p_d</span><span class="p">)</span>
-<span class="n">p_s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">p_s</span><span class="p">)</span>
-
-<span class="c1">#Relative error</span>
-<span class="n">error</span> <span class="o">=</span> <span class="n">p_d</span> <span class="o">-</span> <span class="n">p_s</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">semilogy</span><span class="p">(</span> <span class="n">narray</span><span class="p">,</span> <span class="n">error</span><span class="p">,</span> <span class="s2">&quot;-&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;blue&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Number of iterations $n$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Error $p_n-\hat</span><span class="si">{p}</span><span class="s2">_n$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
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-<p>These algorithms exhibit a small change when an initial perturbation to the initial conditions is introduced. Linear algorithms can be catalogued within this category.</p>
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-<p>Unlike stable algorithms, unstable algorithms produce a large change when iterating with respect to a small perturbation in the initial conditions. Exponential algorithms fit in this category.</p>
-<p>At first glance, unstable algorithms may seem undesirable, however there are some useful applications for them. Among the more interesting ones is the generation of random numbers in a computer. <a href="http://en.wikipedia.org/wiki/Pseudorandom_number_generator">Pseudorandom number generator</a></p>
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-<h1 id="Computing-Time">Computing Time<a class="anchor-link" href="#Computing-Time"> </a></h1>
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-<p>One of the most important skills a programmer must develop is to evaluate the computing time of certain process as well as the consumed memory. When computational resources are limited (as always happens!), estimating beforehand the computing time of an algorithm is certainly important.</p>
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-<p><strong>Example 4:</strong></p>
-<p>An interesting example that illustrates very well the issue of computing time is the N-body problem.</p>
-<p>Consider a set of $N$ masses $\{m_i\}$. The total gravitational potential energy of the $i$-th particle due to the other ones is given by</p>
-$$E_i = -G\sum_{j=1, j\neq i}^N \frac{m_i m_j}{|\vec{r}_i-\vec{r}_j|}$$<p>Now, if we want to calculate the total energy of the system, it is necessary to add each contribution, i.e.</p>
-$$E_{tot} = \sum_{i=1}^N E_i = \sum_{i=1}^N \left( -G\sum_{j=1, j\neq i}^N \frac{m_i m_j}{|\vec{r}_i-\vec{r}_j|} \right) $$<p>This implies if we want to calculate $E_{tot}$ in a computer, it is required $N(N-1)\approx N^2$ iterations, so the computing time scales as $\mathcal{O}(N^2)$. An estimation of the total time is then reached by first measuring the required time for 1 iteration, i.e., the energy of the $i$-th particle due to the $j$-th particle, and then multiplying the result by $N^2$.</p>
-<p>The large computing time of this type of algorithms (when $N$ becomes large enough) has propelled a large enterprise of more efficient algorithms, including tree-codes where the computing time is reduced to $\mathcal{O}(N \log N)$</p>
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-<span class="n">plt</span><span class="o">.</span><span class="n">loglog</span><span class="p">(</span> <span class="n">N</span><span class="p">,</span> <span class="n">N</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$N^2$&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">loglog</span><span class="p">(</span> <span class="n">N</span><span class="p">,</span> <span class="n">N</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">N</span><span class="p">),</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$N\ \log N$&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span> <span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper left&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
-</pre></div>
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-<h1 id="Convergence">Convergence<a class="anchor-link" href="#Convergence"> </a></h1>
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-<p>The last concept related to algorithms is convergence. This refers to how fast an algorithm can reach a desired result with some given precision. Some rigorous techniques can be used to quantify the convergence degree, however it is commonly a more useful approach to compare the convergence of an algorithm with other already known.</p>
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-<strong>ACTIVITY</strong></p>
-<p>In a IPython notebook and using the codes of the first task, make a figure where is compared the values obtained for the two methods for calculating $\pi$ as a function of the number of performed iterations. Which method reaches a faster convergence?
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-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/computer-arithmetics.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
-<h1 id="Computer-Arithmetics-and-Round-off-Methods">Computer Arithmetics and Round-off Methods<a class="anchor-link" href="#Computer-Arithmetics-and-Round-off-Methods"> </a></h1><p>Actividades: <a href="https://classroom.github.com/a/2v-HnNqn">https://classroom.github.com/a/2v-HnNqn</a></p>
-<p>In a symbolic computation programs, implemented in SymPy or Mathematica for example, operatios like $1/3+4/3=5/3$, $2\times 3/4 = 3/2$, $(\sqrt{2})^2 = 2$ are unambiguously defined, However, when one numerical programming languages are used to represent numbers in a computer, this is no longer true. The main reason of this is the so-called <em>finite arithmetic</em>, what is the way in which a numerical computer programs performs basic operations. Some features of <em>finite arithmetic</em> are stated below:</p>
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-<li>Only integer and rational numbers can be exactly represented.</li>
-<li>The elements of the set in which arithmetic is performed is necessarily finite.</li>
-<li>Any arithmetic operation between two or more numbers of this set should be another element of the set.</li>
-<li>Non-representable numbers like irrational numbers are approximated to the closest rational number within the defined set.</li>
-<li>Extremely large numbers produce overflows and extremely small numbers produce underflows, which are taken as null.</li>
-<li>Operations over non-representable numbers are not exact.</li>
-</ul>
-<p>In spite of this, defining adequately the set of elements in which our computer will operate, round-off methods can be systematically neglected, yielding correct results within reasonable error margins. In some pathological cases, when massive iterations are required, these errors must be taken into account more seriously.</p>
-<hr>
-<ul>
-<li><a href="#Binary-machine-numbers">Binary machine numbers</a><ul>
-<li><a href="#Single-precision-numbers">Single-precision numbers</a></li>
-<li><a href="#Double-precision-numbers">Double-precision numbers</a></li>
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-<li><a href="#Finite-Arithmetic">Finite Arithmetic</a><ul>
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-<h2 id="More-About-Float-Values">More About Float Values<a class="anchor-link" href="#More-About-Float-Values"> </a></h2><p>See also <a href="https://www.inferentialthinking.com/chapters/04/1/Numbers.html">here</a></p>
-<p>In Python a float only represents 15 or 16 significant digits for any number; the remaining precision is lost. This limited precision is enough for the vast majority of applications.</p>
-<p>In the next evaluation extra digits are discarded before any arithmetic is carried out.</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="mf">0.6666666666666666</span> <span class="o">-</span> <span class="mf">0.6666666666666666123456789</span>
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-<p>After combining float values with arithmetic, the last few digits may be incorrect. Small rounding errors are often confusing when first encountered. For example, the expression <code>2**0.5</code> computes the square root of <code>2</code>, but squaring this value does not exactly recover <code>2</code>.</p>
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-<pre>1.4142135623730951</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="mf">0.5</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
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-<pre>2.0000000000000004</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="mf">0.5</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span>
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-<p>We will obtain the general algorithm to obtain the binary r</p>
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-<p>From <a href="https://www.techcareerbooster.com/blog/binary-representation-of-an-integer">here</a>: If you recall your mathematics, then it will be easy for you to find out how we turn an integer number to its binary representation. Let's take for example the number 47. (see '//' operator <a href="https://stackoverflow.com/a/1535601">here</a>)</p>
-<ol>
-<li>We divide 47 by 2. The quotient is <code>int(47//2) → 23</code> and the remainder is <code>47%2 → 1</code></li>
-<li>We divide the quotient of the previous step by 2. The new quotient is <code>int(23//2) → 11</code> and the remainder is <code>23%2 → 1</code>.</li>
-<li>We divide the quotient of the previous step by 2. The new quotient is <code>int(11/2) → 5</code> and the remainder is <code>11%2 → 1</code>.</li>
-<li>We divide the quotient of the previous step by 2. The new quotient is <code>int(5//2) → 2</code> and the remainder is <code>5%2 → 1</code>.</li>
-<li>We divide the quotient of the previous step by 2. The new quotient is <code>int(2//2) → 1</code> and the remainder is <code>2%2 → 0</code>.</li>
-<li>We divide the quotient of the previous step by 2. The new quotient is <code>int(1//2) → 0</code> and the remainder is <code>1%2 → 1</code></li>
-<li>We stop here, when the last quotient is <code>1//2→0</code>.</li>
-</ol>
-<p>Then, the binary representation is the series of <strong>1</strong>s and <strong>0</strong>s of the remainders, taken in reverse order to the one produced, from latest to earliest: <strong>101111</strong>.
-<!-- https://www.geeksforgeeks.org/bin-in-python/ --></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Activity</strong>: Implement a function that get the binary representation of an integer.</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Activity</strong>: Return as a string with 8 charactes completed with leading zeroes. Use the string method: <code>.rjust(8,"0")</code></p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Activity</strong> Write your function in the following format</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="ch">#!/usr/bin/env python3</span>
-<span class="k">def</span> <span class="nf">mybin</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;__name__ = </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="vm">__name__</span><span class="p">))</span>
-    <span class="c1">#Write your code here and change the next line as required</span>
-    <span class="k">return</span> <span class="nb">bin</span><span class="p">(</span><span class="n">x</span><span class="p">)[</span><span class="mi">2</span><span class="p">:]</span><span class="o">.</span><span class="n">rjust</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="s2">&quot;0&quot;</span><span class="p">)</span>
-<span class="k">if</span> <span class="vm">__name__</span><span class="o">==</span><span class="s1">&#39;__main__&#39;</span><span class="p">:</span>
-    <span class="n">n</span><span class="o">=</span><span class="nb">input</span><span class="p">(</span><span class="s1">&#39;Escriba un entero:</span><span class="se">\n</span><span class="s1">&#39;</span><span class="p">)</span>
-    <span class="n">b</span><span class="o">=</span><span class="n">mybin</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-    <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Su representación binaria es: </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>Overwriting mybin.py
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>The advantage of this standard format is that can be used directly as a python module, because in that case the variable <code>__name__</code> evaluates to the name
-of the function. To check this, we first use a cell of the notebook just as a editor with the Jupyter magic function <code>%%writefile</code></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%%writefile</span> mybin.py
-<span class="ch">#!/usr/bin/env python3</span>
-<span class="k">def</span> <span class="nf">mybin</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;__name__ = </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="vm">__name__</span><span class="p">))</span>
-    <span class="k">return</span> <span class="nb">bin</span><span class="p">(</span><span class="n">x</span><span class="p">)[</span><span class="mi">2</span><span class="p">:]</span><span class="o">.</span><span class="n">rjust</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="s2">&quot;0&quot;</span><span class="p">)</span>
-<span class="k">if</span> <span class="vm">__name__</span><span class="o">==</span><span class="s1">&#39;__main__&#39;</span><span class="p">:</span>
-    <span class="n">n</span><span class="o">=</span><span class="nb">input</span><span class="p">(</span><span class="s1">&#39;Escriba un entero;</span><span class="se">\n</span><span class="s1">&#39;</span><span class="p">)</span>
-    <span class="n">b</span><span class="o">=</span><span class="n">mybin</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-    <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Su representación binaria es: </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>Overwriting mybin.py
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>We now can use the full contents of the file as an usual python module, it it is
-in the working directory</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">mybin</span> <span class="k">as</span> <span class="nn">my</span>
-<span class="n">my</span><span class="o">.</span><span class="n">mybin</span><span class="p">(</span><span class="mi">45</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>__name__ = mybin
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>&#39;00101101&#39;</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>while here, or in a terminal:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="vm">__name__</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>&#39;__main__&#39;</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Check with <code>bin</code></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>The binary representation of a float in 32 bits done by the following steps</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">bin</span><span class="p">(</span><span class="mi">47</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>&#39;0b101111&#39;</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>The format is not the standard one. We need drop the first '0b'</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">bin</span><span class="p">(</span><span class="mi">47</span><span class="p">)[</span><span class="mi">2</span><span class="p">:]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>&#39;101111&#39;</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>We must enforce the 8 bits representation by padding any additional leading zeroes</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">bin</span><span class="p">(</span><span class="mi">47</span><span class="p">)[</span><span class="mi">2</span><span class="p">:]</span><span class="o">.</span><span class="n">rjust</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="s1">&#39;0&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>&#39;00101111&#39;</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>To convert a binary into the integer we can use the <code>int</code> function with <code>base=0</code> upont the proper Python formatted string, starting with: <font color='red'>`0b`</font></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">int</span><span class="p">(</span><span class="s1">&#39;0b101111&#39;</span><span class="p">,</span><span class="n">base</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>47</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>or with any number of padding zeroes in between</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">int</span><span class="p">(</span><span class="s1">&#39;0b0000000101111&#39;</span><span class="p">,</span><span class="n">base</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>47</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Floats">Floats<a class="anchor-link" href="#Floats"> </a></h3><p>Floats in 32 bits are representated trough 4 8-bit integers.</p>
-<p>The processes of obtain the binary representation for a float can be complicated because the design used to really  store the number  in one specific programming language like Python. In fact, from <a href="https://stackoverflow.com/a/16444778/2268280">https://stackoverflow.com/a/16444778/2268280</a>:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<ul>
-<li>To obtain the four integers associated to a real number we need first to ask for the packed structure : <code>'!'</code>, in 32 bits: <code>'f'</code>, with the full string <code>'!f'</code> as</li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">struct</span>
-<span class="n">packed</span><span class="o">=</span><span class="n">struct</span><span class="o">.</span><span class="n">pack</span><span class="p">(</span><span class="s1">&#39;!f&#39;</span><span class="p">,</span><span class="mf">0.15625</span><span class="p">)</span>
-<span class="n">packed</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>b&#39;&gt; \x00\x00&#39;</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">type</span><span class="p">(</span><span class="n">packed</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>bytes</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>This is a packed Python list with the four integers inside</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="p">[</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">packed</span><span class="p">]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>[62, 32, 0, 0]</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Using the previous function to convert into a list of 8-bit strings</p>
-
-</div>
-</div>
-</div>
-</div>
-
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">lb</span><span class="o">=</span><span class="p">[</span> <span class="n">mybin</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">packed</span><span class="p">]</span>
-<span class="n">lb</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>__name__ = __main__
-__name__ = __main__
-__name__ = __main__
-__name__ = __main__
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
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-<pre>[&#39;00111110&#39;, &#39;00100000&#39;, &#39;00000000&#39;, &#39;00000000&#39;]</pre>
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-<p>The float representation is just the string formed with the four binaries</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="s1">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">lb</span><span class="p">)</span>
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-<pre>&#39;00111110001000000000000000000000&#39;</pre>
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-<p><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Float_example.svg/590px-Float_example.svg.png" alt="32-bits"></p>
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-<h4 id="Full-implementation">Full implementation<a class="anchor-link" href="#Full-implementation"> </a></h4>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">struct</span>
-
-<span class="k">def</span> <span class="nf">binary</span><span class="p">(</span><span class="n">num</span><span class="p">):</span>
-<span class="c1">#num=3</span>
-<span class="c1">#if True:</span>
-    <span class="c1"># Struct can provide us with the float packed into bytes. The &#39;!&#39; ensures that</span>
-    <span class="c1"># it&#39;s in network byte order (big-endian) and the &#39;f&#39; says that it should be</span>
-    <span class="c1"># packed as a float: 32 bites. Alternatively, for double-precision, you could use &#39;d&#39;.</span>
-    <span class="n">packed</span> <span class="o">=</span> <span class="n">struct</span><span class="o">.</span><span class="n">pack</span><span class="p">(</span><span class="s1">&#39;!f&#39;</span><span class="p">,</span> <span class="n">num</span><span class="p">)</span>
-    <span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;Packed: </span><span class="si">%s</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="nb">repr</span><span class="p">(</span><span class="n">packed</span><span class="p">))</span>
-
-    <span class="c1"># For each character in the returned string, we&#39;ll turn it into its corresponding</span>
-    <span class="c1"># integer code point</span>
-    <span class="c1"># </span>
-    <span class="n">integers</span> <span class="o">=</span> <span class="p">[</span><span class="n">c</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">packed</span><span class="p">]</span>
-    <span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;Integers: </span><span class="si">%s</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">integers</span><span class="p">)</span>
-
-    <span class="c1"># For each integer, we&#39;ll convert it to its binary representation.</span>
-    <span class="n">binaries</span> <span class="o">=</span> <span class="p">[</span><span class="nb">bin</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">integers</span><span class="p">]</span>
-    <span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;Binaries: </span><span class="si">%s</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">binaries</span><span class="p">)</span>
-
-    <span class="c1"># Now strip off the &#39;0b&#39; from each of these</span>
-    <span class="n">stripped_binaries</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s1">&#39;0b&#39;</span><span class="p">,</span> <span class="s1">&#39;&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">binaries</span><span class="p">]</span>
-    <span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;Stripped: </span><span class="si">%s</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">stripped_binaries</span><span class="p">)</span>
-
-    <span class="c1"># Pad each byte&#39;s binary representation&#39;s with 0&#39;s to make sure it has all 8 bits:</span>
-    <span class="c1">#</span>
-    <span class="c1"># [&#39;00111110&#39;, &#39;10100011&#39;, &#39;11010111&#39;, &#39;00001010&#39;]</span>
-    <span class="n">padded</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span><span class="o">.</span><span class="n">rjust</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="s1">&#39;0&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">stripped_binaries</span><span class="p">]</span>
-    <span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;Padded: </span><span class="si">%s</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">padded</span><span class="p">)</span>
-
-    <span class="c1"># At this point, we have each of the bytes for the network byte ordered float</span>
-    <span class="c1"># in an array as binary strings. Now we just concatenate them to get the total</span>
-    <span class="c1"># representation of the float:</span>
-    <span class="k">return</span> <span class="s1">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">padded</span><span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">BIN</span><span class="o">=</span><span class="n">binary</span><span class="p">(</span><span class="mf">0.15625</span><span class="p">)</span>
-<span class="n">BIN</span>
-</pre></div>
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-<pre>Packed: b&#39;&gt; \x00\x00&#39;
-Integers: [62, 32, 0, 0]
-Binaries: [&#39;0b111110&#39;, &#39;0b100000&#39;, &#39;0b0&#39;, &#39;0b0&#39;]
-Stripped: [&#39;111110&#39;, &#39;100000&#39;, &#39;0&#39;, &#39;0&#39;]
-Padded: [&#39;00111110&#39;, &#39;00100000&#39;, &#39;00000000&#39;, &#39;00000000&#39;]
-</pre>
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-<pre>&#39;00111110001000000000000000000000&#39;</pre>
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-<p><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Float_example.svg/590px-Float_example.svg.png" alt="32-bits"></p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="s1">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span> <span class="nb">list</span><span class="p">(</span><span class="n">BIN</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="p">)</span> <span class="c1">#inverted list joined into a string</span>
-</pre></div>
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-<pre>&#39;00000000000000000000010001111100&#39;</pre>
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-<h2 id="Binary-machine-numbers">Binary machine numbers<a class="anchor-link" href="#Binary-machine-numbers"> </a></h2>
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-<p>As everyone knows, the base of the modern computation is the binary numbers. The binary base or base-2 numeral system is the simplest one among the existing numeral bases. As every electronic devices are based on logic circuits (circuits operating with <a href="#LogicGates">logic gates</a>), the implementation of a binary base is straightforward, besides, any other numeral system can be reduced to a binary representation.</p>
-<p><img src="http://www.ee.surrey.ac.uk/Projects/CAL/digital-logic/gatesfunc/graphics/symtab.gif" alt="LogicGates"></p>
-<p>According to the standard <a href="http://en.wikipedia.org/wiki/IEEE_floating_point">IEEE 754-2008</a>, representation of real numbers can be done in several ways, <a href="http://en.wikipedia.org/wiki/Single-precision_floating-point_format">single-precision</a> and <a href="http://en.wikipedia.org/wiki/Double-precision_floating-point_format">double precision</a> are the most used ones.</p>
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-<h3 id="Single-precision-numbers">Single-precision numbers<a class="anchor-link" href="#Single-precision-numbers"> </a></h3>
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-<p>Single-precision numbers are used when one does not need very accurate results and/or need to save memory. These numbers are represented by a <strong>32-bits</strong> (Binary digIT) lenght binary number, where the real number is stored following the next rules:</p>
-<p><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Float_example.svg/590px-Float_example.svg.png" alt="32-bits"></p>
-<ol>
-<li>The fist digit (called <em>s</em>) indicates the sign of the number (s=0 means a positive number, s=1 a negative one).</li>
-<li>The next 8 bits represent the exponent of the number.</li>
-<li>The last 23 bits represent the fractional part of the number.</li>
-</ol>
-<p>The formula for recovering the real number is then given by:</p>
-$$r = \frac{(-1)^s}{2^{127-e}} \left( 1 + \sum_{i=0}^{22}\frac{b_{i}}{2^{23-i}} \right)$$<p>where $s$ is the sign, $b_{23-i}$ the fraction bits and $e$ is given by:</p>
-$$e = \sum_{i=0}^7 b_{23+i}2^i$$<p>Next, it is shown a little routine for calculating the value of the represented 32-bits number</p>
-
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-<h3 id="ACTIVITY">ACTIVITY<a class="anchor-link" href="#ACTIVITY"> </a></h3><p>With the binary representation please try to implement the formula to recover the number.</p>
-<p><strong>Hint</strong>: Use as input the binary representation as a string and invert its order</p>
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-<p><strong>SOLUTION</strong>:</p>
-
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-<p>The goal of this course is try to implement all the algorithms in terms of array operations, instead of use the loops of Pyhton like the <code>for</code> loop.</p>
-<p>In this way, we need to interpret the formula in terms of  arrays operations. If we manage to achieve that, then NumPy will be take care of the internal loops, which are much faster than the Python loops. As a bonus, the code is more compact and readable.</p>
-<p>For example:</p>
-\begin{align}
-\sum_{i=0}^{22}\frac{b_{i}}{2^{23-i}}=&amp;\frac{b[0]}{2^{23}}+\frac{b[1]}{2^{22}}
- +\cdots +\frac{b[21]}{2^2}+\frac{b[22]}{2^1}\nonumber\\
-=&amp;\frac{b}{2^{[23,22,...,1]}}.\operatorname{sum()}\nonumber\\
-=&amp;\frac{b}{2^{[1,2,...,23][::-1]}}.\operatorname{sum()}
-\end{align}<p>To implement the full formula:
-$$r = \frac{(-1)^s}{2^{127-e}} \left( 1 + \sum_{i=0}^{22}\frac{b_{i}}{2^{23-i}} \right)$$
-with 
-$$e = \sum_{i=0}^7 b_{23+i}2^i$$</p>
-$$e_{\text{max}} = \sum_{i=0}^7 2^i$$<p>Note that 
-we have</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="k">def</span> <span class="nf">number32</span><span class="p">(</span><span class="n">BIN</span><span class="p">):</span>
-<span class="c1">#if True:</span>
-    <span class="c1">#BIN=&#39;00111110001000000000000000000000&#39;</span>
-    <span class="c1">#Inverting binary string</span>
-    <span class="n">b_inverted</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">BIN</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-    
-    <span class="n">s</span><span class="o">=</span><span class="n">b_inverted</span><span class="p">[</span><span class="mi">31</span><span class="p">]</span>
-    <span class="c1">#Exponent part</span>
-    <span class="n">be</span><span class="o">=</span><span class="n">b_inverted</span><span class="p">[</span><span class="mi">23</span><span class="p">:</span><span class="mi">31</span><span class="p">]</span>
-    <span class="n">i</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">be</span><span class="o">.</span><span class="n">size</span><span class="p">)</span>
-    <span class="n">e</span><span class="o">=</span><span class="p">(</span><span class="n">be</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">i</span><span class="p">))</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
-    
-    <span class="n">bf</span><span class="o">=</span><span class="n">b_inverted</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">23</span><span class="p">]</span>
-    <span class="n">i_inverted</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">bf</span><span class="o">.</span><span class="n">size</span><span class="o">+</span><span class="mi">1</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-
-    <span class="n">numero</span><span class="o">=</span><span class="p">(</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">s</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">127</span><span class="o">-</span><span class="n">e</span><span class="p">)</span>  <span class="p">)</span><span class="o">*</span><span class="p">(</span> <span class="mi">1</span> <span class="o">+</span> <span class="p">(</span><span class="n">bf</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="n">i_inverted</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span> <span class="p">)</span> 
-    <span class="k">return</span> <span class="n">numero</span>
-</pre></div>
-
-    </div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">number32</span><span class="p">(</span><span class="s1">&#39;00111110001000000000000000000000&#39;</span><span class="p">))</span>
-</pre></div>
-
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-<pre>0.15625
-</pre>
-</div>
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-<p>Check aginst the implementation with <code>for</code>'s</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># %load numtobin.py</span>
-<span class="k">def</span> <span class="nf">num32</span><span class="p">(</span> <span class="n">binary</span> <span class="p">):</span>
-    <span class="c1">#Inverting binary string</span>
-    <span class="n">binary</span> <span class="o">=</span> <span class="n">binary</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">s</span><span class="o">=</span><span class="n">binary</span><span class="p">[</span><span class="mi">31</span><span class="p">]</span>
-    <span class="c1">#Decimal part</span>
-    <span class="n">dec</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">24</span><span class="p">):</span>
-        <span class="n">dec</span> <span class="o">+=</span> <span class="nb">int</span><span class="p">(</span><span class="n">binary</span><span class="p">[</span><span class="mi">23</span><span class="o">-</span><span class="n">i</span><span class="p">])</span><span class="o">*</span><span class="mi">2</span><span class="o">**-</span><span class="n">i</span>
-    <span class="c1">#Exponent part</span>
-    <span class="n">exp</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">8</span><span class="p">):</span>
-        <span class="n">exp</span> <span class="o">+=</span> <span class="nb">int</span><span class="p">(</span><span class="n">binary</span><span class="p">[</span><span class="mi">23</span><span class="o">+</span><span class="n">i</span><span class="p">])</span><span class="o">*</span><span class="mi">2</span><span class="o">**</span><span class="n">i</span>
-    <span class="c1">#Total number</span>
-    <span class="n">number</span> <span class="o">=</span><span class="p">(</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="nb">int</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">127</span><span class="o">-</span><span class="n">exp</span><span class="p">)</span> <span class="p">)</span><span class="o">*</span><span class="n">dec</span>
-    <span class="k">return</span> <span class="n">number</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">num32</span><span class="p">(</span><span class="s1">&#39;00111110001000000000000000000000&#39;</span><span class="p">))</span>
-</pre></div>
-
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-<pre>0.15625
-</pre>
-</div>
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-</div>
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-<p>%load sln.py</p>
-
-</div>
-</div>
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-<p>Single-precision system can represent real numbers within the interval $\pm 10^{-38} \cdots 10^{38}$, with $7-8$ decimal digits.</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Decimal digits </span>
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span> <span class="s2">&quot;Decimal digits contributions for single precision number&quot;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span> <span class="mi">2</span><span class="o">**-</span><span class="mf">23.</span><span class="p">,</span> <span class="mi">2</span><span class="o">**-</span><span class="mf">15.</span><span class="p">,</span> <span class="mi">2</span><span class="o">**-</span><span class="mf">5.</span> <span class="p">,</span> <span class="s2">&quot;</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
-
-<span class="c1">#Largest and smallest exponent  </span>
-<span class="n">emax</span> <span class="o">=</span> <span class="mi">0</span> 
-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">8</span><span class="p">):</span>
-    <span class="n">emax</span> <span class="o">+=</span> <span class="mi">2</span><span class="o">**</span><span class="n">i</span>
-<span class="nb">print</span><span class="p">(</span> <span class="s2">&quot;Largest and smallest exponent for single precision number&quot;</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">emax</span><span class="o">-</span><span class="mf">127.</span><span class="p">),</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mf">127.</span><span class="p">),</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">&quot;</span> <span class="p">)</span>
-</pre></div>
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-<pre>
-
-Decimal digits contributions for single precision number
-1.1920928955078125e-07 3.0517578125e-05 0.03125 
-
-Largest and smallest exponent for single precision number
-3.402823669209385e+38 5.877471754111438e-39 
-
-</pre>
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Double-precision-numbers">Double-precision numbers<a class="anchor-link" href="#Double-precision-numbers"> </a></h3>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Double-precision numbers are used when high accuracy is required. These numbers are represented by a <strong>64-bits</strong> (Binary digIT) lenght binary number, where the real number is stored following the next rules:</p>
-<p><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/a/a9/IEEE_754_Double_Floating_Point_Format.svg/618px-IEEE_754_Double_Floating_Point_Format.svg.png" alt="64-bits"></p>
-<ol>
-<li>The fist digit (called <em>s</em>) indicates the sign of the number (s=0 means a positive number, s=1 a negative one).</li>
-<li>The next 11 bits represent the exponent of the number.</li>
-<li>The last  bits represent the fractional part of the number.</li>
-</ol>
-<p>The formula for recovering the real number is then given by:</p>
-$$r = (-1)^s\times \left( 1 + \sum_{i=1}^{52}b_{52-i}2^{-i} \right)\times 2^{e-1023}$$<p>where $s$ is the sign, $b_{23-i}$ the fraction bits and $e$ is given by:</p>
-$$e = \sum_{i=0}^{10} b_{52+i}2^i$$<p>Double-precision system can represent real numbers within the interval $\pm 10^{-308} \cdots 10^{308}$, with $16-17$ decimal digits.</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="mf">1e307</span> <span class="o">*</span> <span class="mi">10</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-
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-
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-<pre>1e+308</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="mf">1e307</span> <span class="o">*</span> <span class="mi">20</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<div class="output_text output_subarea output_execute_result">
-<pre>inf</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="mf">1e-323</span>
-</pre></div>
-
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-</div>
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-<pre>1e-323</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="mf">1e-324</span>
-</pre></div>
-
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-</div>
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-<pre>0.0</pre>
-</div>
-
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-</div>
-</div>
-</div>
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-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="ACTIVITY"><strong>ACTIVITY</strong><a class="anchor-link" href="#ACTIVITY"> </a></h3><p><strong>1.</strong> Write a python script that calculates the double precision number represented by a 64-bits binary.</p>
-<p><strong>2.</strong> What is the number represented by:</p>
-<p>0 10000000011 1011100100001111111111111111111111111111111111111111</p>
-<p><font color='white'>
-    <strong>ANSWER:</strong>  27.56640625</p>
-
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-<h2 id="Finite-Arithmetic">Finite Arithmetic<a class="anchor-link" href="#Finite-Arithmetic"> </a></h2>
-</div>
-</div>
-</div>
-</div>
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-<p>The most basic arithmetic operations are addition and multiplication. Further operations such as subtraction, division and power are secondary as they can be reached by iteratively use the latter ones.</p>
-
-</div>
-</div>
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-</div>
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-<h3 id="Addition">Addition<a class="anchor-link" href="#Addition"> </a></h3>
-</div>
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-</div>
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-<!-- As mentioned before, arithmetic operations are not exact in a computer due to the inherent limitations in number representing. Even when adding two already approximate numbers, say a single-precision couple of numbers, the result may not be a representable number, being necessary to apply approximation rules.
-```python
-N = 9
-xe=0; x = 0; 
-for i in range(N):
-    xe += np.float(1.0/N)
-    x += np.float16(1.0/N)
-print('expected: {}; obtained: {}'.format(xe,x))
-xe
-```
-Note that the sucessive application of rounded-off numbers produces a final result less precise.
-```python
-print( "5/7".format( np.float32(5/7.) ) )
-print( "1/3", np.float32(1/3.) )
-print( np.float32(5/7.+1/3.), 22/21. )
-```
--->
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Error:&quot;</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">(</span><span class="mi">5</span><span class="o">/</span><span class="mf">7.</span><span class="o">+</span><span class="mi">1</span><span class="o">/</span><span class="mf">3.</span><span class="p">)</span><span class="o">-</span><span class="mi">22</span><span class="o">/</span><span class="mf">21.</span><span class="p">)</span>
-</pre></div>
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-    </div>
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-</div>
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-<pre>Error: 5.676632830464712e-08
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Therefore, at the numerical computation level we must be aware that an expected 
-zero result is really
-$$0\approx 10^{-16}$$</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Multiplication">Multiplication<a class="anchor-link" href="#Multiplication"> </a></h3>
-</div>
-</div>
-</div>
-</div>
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-<p>For multiplication it is applied the same round-off rules as the addition, however, be aware that multiplicative errors propagate more quickly than additive errors.</p>
-
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">N</span> <span class="o">=</span> <span class="mi">20</span>
-<span class="n">xe</span><span class="o">=</span><span class="mi">1</span><span class="p">;</span> <span class="n">x</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
-    <span class="n">xe</span> <span class="o">*=</span> <span class="nb">float</span><span class="p">(</span><span class="mf">2.0</span><span class="o">**</span><span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">N</span><span class="p">))</span>
-    <span class="n">x</span> <span class="o">*=</span> <span class="n">np</span><span class="o">.</span><span class="n">float16</span><span class="p">(</span><span class="mf">2.0</span><span class="o">**</span><span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">N</span><span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;expected: </span><span class="si">{}</span><span class="s1">; obtained: </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">xe</span><span class="p">,</span><span class="n">x</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<div class="output_area">
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-<div class="output_subarea output_stream output_stdout output_text">
-<pre>expected: 2.000000000000003; obtained: 1.9958053041750938
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">N</span> <span class="o">=</span> <span class="mi">20</span>
-<span class="n">xe</span><span class="o">=</span><span class="mi">1</span><span class="p">;</span> <span class="n">x</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
-    <span class="n">xe</span> <span class="o">*=</span> <span class="mf">2.0</span><span class="o">**</span><span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">N</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">float16</span><span class="p">(</span><span class="mf">3.1415926535897932384626433832795028841971</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>3.14</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">(</span><span class="mf">3.1415926535897932384626433832795028841971</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>3.1415927</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
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-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">float</span><span class="p">(</span><span class="mf">3.1415926535897932384626433832795028841971</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
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-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>3.141592653589793</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
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-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">(</span><span class="mf">3.1415926535897932384626433832795028841971</span><span class="p">)</span>
-</pre></div>
-
-    </div>
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-<pre>3.141592653589793</pre>
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">float128</span><span class="p">(</span><span class="mf">3.1415926535897932384626433832795028841971</span><span class="p">)</span>
-</pre></div>
-
-    </div>
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-<pre>3.141592653589793116</pre>
-</div>
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-<p>The final result has an error at the third decimal digit, one more than the case of addition.</p>
-
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-<p><strong>ACTIVITY</strong></p>
-<p>Find the error associated to the finite representation in the next operations</p>
-$$
-x-u, \frac{x-u}{w}, (x-u)*v, u+v 
-$$<p>considering the values</p>
-$$
-x = \frac{5}{7}, y = \frac{1}{3}, u = 0.71425
-$$$$
-v = 0.98765\times 10^5, w = 0.111111\times 10^{-4}
-$$
-</div>
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-    <div id="page-info"><div id="page-title">Diferential equations</div>
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-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/differential-equations.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<h1 id="Differential-Equations">Differential Equations<a class="anchor-link" href="#Differential-Equations"> </a></h1>
-</div>
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-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/differential-equations.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
-<p>Differential equations is without doubt one of the more useful branch of mathematics in science. They are used to model problems involving the change of some variable with respect to another. Differential equations cover a wide range of different applications, ranging from ordinary differential equations (ODE) until boundary-value problems involving many variables. For the sake of simplicity, throughout this section we shall cover only ODE systems as they are more elemental and equally useful. First, we shall cover first order methods, then second order methods and finally, system of differential equations.</p>
-<p>See:</p>
-<ul>
-<li><a href="http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture21.pdf">http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture21.pdf</a></li>
-<li><a href="http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture22.pdf">http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture22.pdf</a></li>
-<li>Infectious Disease Modelling:<ul>
-<li><a href="https://towardsdatascience.com/infectious-disease-modelling-part-i-understanding-sir-28d60e29fdfc">Understanding the models that are used to model Coronavirus</a>: Explaining the background and deriving the formulas of the SIR model from scratch. Coding and visualizing the model in Python.</li>
-<li><a href="https://towardsdatascience.com/infectious-disease-modelling-beyond-the-basic-sir-model-216369c584c4">Beyond the Basic SIR Model</a></li>
-</ul>
-</li>
-</ul>
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-<ul>
-<li><a href="#First-Order-Methods">First Order Methods</a> <ul>
-<li><a href="#Euler&#39;s-method">Euler's method</a></li>
-<li><a href="#Example-1">Example 1</a></li>
-</ul>
-</li>
-<li><a href="#High-Order-Methods">High Order Methods</a><ul>
-<li><a href="#Second-order-Runge-Kutta-methods">Second-order Runge-Kutta methods</a></li>
-<li><a href="#Example-2">Example 2</a></li>
-<li><a href="#Fourth-order-Runge-Kutta-method">Fourth-order Runge-Kutta method</a></li>
-</ul>
-</li>
-<li><a href="#Two-Point-Boundary-Value-Problems">Two-Point Boundary Value Problems</a><ul>
-<li><a href="#Example-3">Example 3</a></li>
-</ul>
-</li>
-<li><a href="./material/">SIR Model</a></li>
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-<span class="c1"># JSAnimation import available at https://github.com/jakevdp/JSAnimation</span>
-<span class="c1">#from JSAnimation import IPython_display</span>
-<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">animation</span>
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-<h2 id="Physical-motivation">Physical motivation<a class="anchor-link" href="#Physical-motivation"> </a></h2><h3 id="Momentum">Momentum<a class="anchor-link" href="#Momentum"> </a></h3><p>In absence a forces a body moves freely to a constant velocity (See figure)</p>
-<p>The quantity which can be associated with the change of speed of a body is the instantaneous <em>momentum</em> of a particle of mass $m$ moving with instantaneous velocity $\boldsymbol{v}$, given by
-\begin{align}
-   \boldsymbol{p}=&amp;\gamma m \mathbf{v}\,, &amp;\text{where: } \gamma=&amp;\frac{1}{\sqrt{1-{|\boldsymbol{v}|^2}/{c^2}}}\,,
-\end{align}
-and $c\approx 3\times 10^8\ $m/s is the speed of light.</p>
-<p>If $\gamma\approx1$, or equivalent, sif $|\boldsymbol{v}|\ll c$:
-\begin{align}
-  \boldsymbol{p}\approx &amp;m \boldsymbol{v}
-\end{align}</p>
-<h3 id="Equation-of-motion">Equation of motion<a class="anchor-link" href="#Equation-of-motion"> </a></h3><p>Any change of the momentum can be atribuited to some force, $\boldsymbol{F}$. In fact, the Newton's second law can be defined in a general way as</p>
-<blockquote><p>The change of the momentum of particle is equal to the net force acting upon the system times the duration of the interaction</p>
-</blockquote>
-<p>For one instaneous duration $\Delta t$, this law can be written as
-$$
-\Delta \boldsymbol{p} = \boldsymbol{F}\Delta t,\qquad \Delta t \to 0 
-$$
-or as the <em>equation of motion</em>, which is the <em>differential equation</em>
-\begin{align}
-\boldsymbol{F}=&amp;\lim_{t\to 0} \frac{\Delta \boldsymbol{p}}{\Delta t} \\
-\boldsymbol{F}=&amp;\frac{\operatorname{d}\boldsymbol{p}}{\operatorname{d} t} \,.
-\end{align}
-This equation of motion is of general validity and can be applied numerically to solve any problem.</p>
-<h3 id="Constant-mass">Constant mass<a class="anchor-link" href="#Constant-mass"> </a></h3><p>In the special case of constant mass
-\begin{align}
-\boldsymbol{F}=&amp;m \boldsymbol{a}
-\end{align}</p>
-
-</div>
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-<h3 id="Example:-Drag-force">Example: Drag force<a class="anchor-link" href="#Example:-Drag-force"> </a></h3><p>For a body falling in the air, in addition to the gravitiy force $\boldsymbol{W}$, there is a dragging force in the opposite direction given by <a href="https://en.wikipedia.org/wiki/Drag_(physics">[see here for details]</a>)<img src="https://raw.githubusercontent.com/restrepo/ComputationalMethods/master/material/figures/friction.svg" alt="img">
-$$
-F_D=\frac{1}{2}{\rho A C_d} v^2(t)=\frac{1}{2m^2}{\rho A C_d} p^2(t)\,,
-$$
-where</p>
-<ul>
-<li>$A$: is the frontal area on a plane perpendicular to the direction of motion—e.g. for objects with a simple shape, such as a sphere</li>
-<li>$\rho$ is the density of the air</li>
-<li>$C_d$: <em>Drag coefficient</em>. $C_d=0.346$ for a <a href="http://baseball.physics.illinois.edu/DragTPTMay2014.pdf">baseball</a></li>
-<li>$v(t)$: speed of the baseball</li>
-<li>$p(t)$: momentum of baseball</li>
-</ul>
-<p>The equation of motion is
-\begin{align}
-W-F_D=&amp; \frac{d p}{dt}\\
-\frac{d p}{dt}=&amp;m g - \frac{\rho A C_d}{2m^2} p^2(t). 
-\end{align}</p>
-<p>We see then that, in general
-$$
-\frac{d p}{dt}=f(t,p)\,.
-$$</p>
-
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-<h2 id="Mathematical-background">Mathematical background<a class="anchor-link" href="#Mathematical-background"> </a></h2><p>Ordinary differential equations normally implies the solution of an initial-value problem, i.e., the solution has to satisfy the differential equation together with some initial condition. Real-life problems usually imply very complicated problems and even non-soluble ones, making any analytical approximation unfeasible. Fortunately, there are two ways to handle this. First, for almost every situation, it is generally posible to simplify the original problem and obtain a simpler one that can be easily solved. Then, using perturbation theory, we can perturbate this solution in order to approximate the real one. This approach is useful, however, it depends very much on the specific problem and a systematic study is rather complicated.</p>
-<p>The second approximation, and the one used here, consists of a complete numerical reduction of the problem, solving it exactly within the accuracy allowed by implicit errors of the methods. For this part, we are going to assume well-defined problems, where solutions are expected to be well-behaved.</p>
-
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-<h2 id="Euler's-method">Euler's method<a class="anchor-link" href="#Euler's-method"> </a></h2><h3 id="First-order-differential-equations">First order differential equations<a class="anchor-link" href="#First-order-differential-equations"> </a></h3>
-</div>
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-<p>This method is the most basic of all methods, however, it is useful to understand basic concepts and definitions of differential equations problems.</p>
-<p>Suppose we have a well-posed initial-value problem given by:</p>
-$$ \frac{dy}{dt}=f(t,y),\ \ \ a\leq t\leq b, \ \ \ \ y(a) = \alpha $$<p>Now, let's define a <strong>mesh points</strong> as a set of values $\{t_i\}$ where we are going to approximate the solution $y(t)$. These points can be equal-spaced such that</p>
-$$ t_i = a+ i \Delta t,\ \ \ \ \mbox{with}\ \ i=1,2,3,\cdots,N \ \ \mbox{and}\ \Delta t = \frac{b-a}{N}. $$<p>Here, $h$ is called the <strong>step size</strong> of the <strong>mesh points</strong>.</p>
-<p>Now, using the first-order approximation of the derivative that we saw in <a href="http://nbviewer.ipython.org/github/sbustamante/ComputationalMethods/blob/master/material/numerical-calculus.ipynb#Numerical-Differentiation">Numerical Differentiation</a>, we obtain</p>
-$$ \frac{dy}{dt}\approx \frac{y(t+\Delta t)-y(t)}{\Delta t} $$<p>or</p>
-$$ \left.\frac{dy}{dt}\right|_{t=t_i}\approx \frac{y(t_{i+1})-y(t_i)}{\Delta t} $$<p>The original problem can be rewritten as</p>
-$$ y_{i+1} = y_i + f(t_i, y_i)\Delta t \pm \frac{\Delta t^2}{2}y^{''}(\xi_i) $$<p>where the notation $y(t_i)\equiv y_i$ has been introduced and the last term (error term) can be obtained taking a second-order approximation of the derivative. $\xi_i$ is the value that give the maximum in absolute value for $y''$</p>
-<p>This equation can be solved recursively as we know the initial value $y_0 = y(a) = \alpha$.</p>
-
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-<h3 id="Second-order-differential-equations">Second order differential equations<a class="anchor-link" href="#Second-order-differential-equations"> </a></h3>
-</div>
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-<p>The formalism of the Euler's method can be applied for any system of the form:</p>
-$$ \frac{dy}{dt}=f(t,y),\ \ \ a\leq t\leq b, \ \ \ \ y(a) = \alpha $$<p>However, it is possible to extend this to second-order systems, i.e., systems involving second derivatives. Let's suppose a general system of the form:</p>
-$$ \frac{d^2y}{dt^2}+ g(t,y)\frac{dy}{dt}=f(t,y),\ \ \ a\leq t\leq b, \ \ \ \ y(a) = \alpha\ \ \mbox{and}\ y'(a) = \beta$$<p>For this system we have to provide both, the initial value $y(a)$ and the initial derivative $y'(a)$.</p>
-<p>Now, let's define a new variable $w(t) = y'(t)$, the previous problem can be then written as</p>
-\begin{align}
-\frac{dy}{dt}=&amp;w(t) \\  
-\frac{dw}{dt}=&amp;f(t,y)-g(t,y)w\, ,\ \ \ a\leq t\leq b\,, \ \ \ \ y(a) = \alpha\ \ \mbox{and}\ w(a) = \beta
-\end{align}<p>Each of this problem has now the form required by the Euler's algorithm, and the solution can be computed as:</p>
-\begin{align}
-w_{i+1}=&amp; w_{i} + [f(t_i,y_i)-g(t_i,y_i)w_i]\Delta t \\  
-y_{i+1}=&amp;y_i + w_i \Delta t\, ,\ \ \ a\leq t\leq b\,, \ \ \ \ y(a) = \alpha\ \ \mbox{and}\ w(a) = \beta
-\end{align}
-</div>
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-<h3 id="Euler-method-for-second-order-differential-equations">Euler method for second order differential equations<a class="anchor-link" href="#Euler-method-for-second-order-differential-equations"> </a></h3><p>We can define the column vectors
-\begin{align}
-\boldsymbol{U}=\begin{bmatrix}
-U_1\\
-U_2\\
-\end{bmatrix}=&amp;\begin{bmatrix}
-y(t)\\
-w(t)\\
-\end{bmatrix}\,,&amp; 
-\boldsymbol{V}(\boldsymbol{U},t)=\begin{bmatrix}
-w(t)\\
-f(t,y)-g(t,y)w(t)\\
-\end{bmatrix}=&amp;\begin{bmatrix}
-U_2\\
-f(t,U_1)-g(t,U_1)U_2\\
-\end{bmatrix}
-\end{align}
-such that</p>
-
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-<h2 id="Newton's-third-law-of-motion">Newton's third law of motion<a class="anchor-link" href="#Newton's-third-law-of-motion"> </a></h2><p>In fact, the equation of motion can be seen as the system of equations
-\begin{align}
-\frac{\boldsymbol{p}}{m}=&amp;\frac{\operatorname{d}\boldsymbol{x}}{\operatorname{d}t}\\
-\boldsymbol{p}=&amp; \gamma m \boldsymbol{v}\\
-\boldsymbol{F}=&amp;\frac{\operatorname{d}\boldsymbol{p}}{\operatorname{d} t} 
-\end{align}</p>
-
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-<h3 id="Example:">Example:<a class="anchor-link" href="#Example:"> </a></h3><!-- Para ilustrar la integración numérica considere el siguiente programa en `vpython` de la solución al problema de una partícula de masa $m=0.5\ $Kg, con velocidad inicial $\mathbf{v}_i=(5,0,0)\ $m/s, moviendose bajo la influencia de una fuerza neta gravitacional dada por
-\begin{align}
-\textbf{F}_{\text{neta}}=&(0,-mg,0)\nonumber\\
-=&(0,-4.9,0)\ \text{N}\,,
-\end{align}
-La implementación con un paso de integración dado por un $\Delta t=0.01\ \text{s}$ se ilustra en el siguiente programa:
--->
-
-<p>An object of 0.5 Kg is launched from the top of a building of 50 m with an horizontal speed of 5 m/s. Study the evolution of the movement</p>
-<ul>
-<li>Neglecting the air friction</li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-<span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">()</span>
-<span class="n">m</span><span class="o">=.</span><span class="mi">5</span>
-<span class="n">g</span><span class="o">=</span><span class="mf">9.8</span>
-<span class="c1"># intial conditions</span>
-<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">50</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-<span class="n">v</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-<span class="n">p</span><span class="o">=</span><span class="n">m</span><span class="o">*</span><span class="n">v</span>
-<span class="n">deltat</span><span class="o">=</span><span class="mf">0.01</span>
-<span class="c1"># Analysis for the first 3 s</span>
-<span class="n">df</span><span class="o">=</span><span class="n">df</span><span class="o">.</span><span class="n">append</span><span class="p">({</span><span class="s1">&#39;x&#39;</span><span class="p">:</span><span class="n">x</span><span class="p">,</span><span class="s1">&#39;p&#39;</span><span class="p">:</span><span class="n">p</span><span class="p">},</span><span class="n">ignore_index</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-
-<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span> <span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="n">deltat</span><span class="p">):</span>
-    <span class="n">Fg</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="n">m</span><span class="o">*</span><span class="n">g</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-    <span class="n">p</span><span class="o">=</span><span class="n">p</span><span class="o">+</span><span class="n">Fg</span><span class="o">*</span><span class="n">deltat</span>
-    <span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="o">+</span><span class="p">(</span><span class="n">p</span><span class="o">/</span><span class="n">m</span><span class="p">)</span><span class="o">*</span><span class="n">deltat</span>
-    <span class="n">df</span><span class="o">=</span><span class="n">df</span><span class="o">.</span><span class="n">append</span><span class="p">({</span><span class="s1">&#39;x&#39;</span><span class="p">:</span><span class="n">x</span><span class="p">,</span><span class="s1">&#39;p&#39;</span><span class="p">:</span><span class="n">p</span><span class="p">},</span><span class="n">ignore_index</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="c1">#[:3]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
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-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>p</th>
-      <th>x</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>[2.5, 0.0, 0.0]</td>
-      <td>[0, 50, 0]</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>[2.5, -0.049, 0.0]</td>
-      <td>[0.05, 49.99902, 0.0]</td>
-    </tr>
-    <tr>
-      <th>2</th>
-      <td>[2.5, -0.098, 0.0]</td>
-      <td>[0.1, 49.997060000000005, 0.0]</td>
-    </tr>
-    <tr>
-      <th>3</th>
-      <td>[2.5, -0.14700000000000002, 0.0]</td>
-      <td>[0.15000000000000002, 49.99412, 0.0]</td>
-    </tr>
-    <tr>
-      <th>4</th>
-      <td>[2.5, -0.196, 0.0]</td>
-      <td>[0.2, 49.9902, 0.0]</td>
-    </tr>
-    <tr>
-      <th>5</th>
-      <td>[2.5, -0.245, 0.0]</td>
-      <td>[0.25, 49.9853, 0.0]</td>
-    </tr>
-    <tr>
-      <th>6</th>
-      <td>[2.5, -0.294, 0.0]</td>
-      <td>[0.3, 49.979420000000005, 0.0]</td>
-    </tr>
-    <tr>
-      <th>7</th>
-      <td>[2.5, -0.34299999999999997, 0.0]</td>
-      <td>[0.35, 49.97256, 0.0]</td>
-    </tr>
-    <tr>
-      <th>8</th>
-      <td>[2.5, -0.39199999999999996, 0.0]</td>
-      <td>[0.39999999999999997, 49.96472, 0.0]</td>
-    </tr>
-    <tr>
-      <th>9</th>
-      <td>[2.5, -0.44099999999999995, 0.0]</td>
-      <td>[0.44999999999999996, 49.9559, 0.0]</td>
-    </tr>
-    <tr>
-      <th>10</th>
-      <td>[2.5, -0.48999999999999994, 0.0]</td>
-      <td>[0.49999999999999994, 49.9461, 0.0]</td>
-    </tr>
-    <tr>
-      <th>11</th>
-      <td>[2.5, -0.5389999999999999, 0.0]</td>
-      <td>[0.5499999999999999, 49.935320000000004, 0.0]</td>
-    </tr>
-    <tr>
-      <th>12</th>
-      <td>[2.5, -0.588, 0.0]</td>
-      <td>[0.6, 49.92356, 0.0]</td>
-    </tr>
-    <tr>
-      <th>13</th>
-      <td>[2.5, -0.637, 0.0]</td>
-      <td>[0.65, 49.91082, 0.0]</td>
-    </tr>
-    <tr>
-      <th>14</th>
-      <td>[2.5, -0.686, 0.0]</td>
-      <td>[0.7000000000000001, 49.8971, 0.0]</td>
-    </tr>
-    <tr>
-      <th>15</th>
-      <td>[2.5, -0.7350000000000001, 0.0]</td>
-      <td>[0.7500000000000001, 49.882400000000004, 0.0]</td>
-    </tr>
-    <tr>
-      <th>16</th>
-      <td>[2.5, -0.7840000000000001, 0.0]</td>
-      <td>[0.8000000000000002, 49.86672, 0.0]</td>
-    </tr>
-    <tr>
-      <th>17</th>
-      <td>[2.5, -0.8330000000000002, 0.0]</td>
-      <td>[0.8500000000000002, 49.85006, 0.0]</td>
-    </tr>
-    <tr>
-      <th>18</th>
-      <td>[2.5, -0.8820000000000002, 0.0]</td>
-      <td>[0.9000000000000002, 49.83242, 0.0]</td>
-    </tr>
-    <tr>
-      <th>19</th>
-      <td>[2.5, -0.9310000000000003, 0.0]</td>
-      <td>[0.9500000000000003, 49.8138, 0.0]</td>
-    </tr>
-    <tr>
-      <th>20</th>
-      <td>[2.5, -0.9800000000000003, 0.0]</td>
-      <td>[1.0000000000000002, 49.794200000000004, 0.0]</td>
-    </tr>
-    <tr>
-      <th>21</th>
-      <td>[2.5, -1.0290000000000004, 0.0]</td>
-      <td>[1.0500000000000003, 49.77362, 0.0]</td>
-    </tr>
-    <tr>
-      <th>22</th>
-      <td>[2.5, -1.0780000000000003, 0.0]</td>
-      <td>[1.1000000000000003, 49.75206, 0.0]</td>
-    </tr>
-    <tr>
-      <th>23</th>
-      <td>[2.5, -1.1270000000000002, 0.0]</td>
-      <td>[1.1500000000000004, 49.72952, 0.0]</td>
-    </tr>
-    <tr>
-      <th>24</th>
-      <td>[2.5, -1.1760000000000002, 0.0]</td>
-      <td>[1.2000000000000004, 49.706, 0.0]</td>
-    </tr>
-    <tr>
-      <th>25</th>
-      <td>[2.5, -1.225, 0.0]</td>
-      <td>[1.2500000000000004, 49.6815, 0.0]</td>
-    </tr>
-    <tr>
-      <th>26</th>
-      <td>[2.5, -1.274, 0.0]</td>
-      <td>[1.3000000000000005, 49.65602, 0.0]</td>
-    </tr>
-    <tr>
-      <th>27</th>
-      <td>[2.5, -1.323, 0.0]</td>
-      <td>[1.3500000000000005, 49.62956, 0.0]</td>
-    </tr>
-    <tr>
-      <th>28</th>
-      <td>[2.5, -1.3719999999999999, 0.0]</td>
-      <td>[1.4000000000000006, 49.60212, 0.0]</td>
-    </tr>
-    <tr>
-      <th>29</th>
-      <td>[2.5, -1.4209999999999998, 0.0]</td>
-      <td>[1.4500000000000006, 49.5737, 0.0]</td>
-    </tr>
-    <tr>
-      <th>...</th>
-      <td>...</td>
-      <td>...</td>
-    </tr>
-    <tr>
-      <th>271</th>
-      <td>[2.5, -13.278999999999971, 0.0]</td>
-      <td>[13.550000000000058, 13.88112000000001, 0.0]</td>
-    </tr>
-    <tr>
-      <th>272</th>
-      <td>[2.5, -13.327999999999971, 0.0]</td>
-      <td>[13.600000000000058, 13.61456000000001, 0.0]</td>
-    </tr>
-    <tr>
-      <th>273</th>
-      <td>[2.5, -13.37699999999997, 0.0]</td>
-      <td>[13.650000000000059, 13.347020000000011, 0.0]</td>
-    </tr>
-    <tr>
-      <th>274</th>
-      <td>[2.5, -13.42599999999997, 0.0]</td>
-      <td>[13.70000000000006, 13.078500000000012, 0.0]</td>
-    </tr>
-    <tr>
-      <th>275</th>
-      <td>[2.5, -13.47499999999997, 0.0]</td>
-      <td>[13.75000000000006, 12.809000000000013, 0.0]</td>
-    </tr>
-    <tr>
-      <th>276</th>
-      <td>[2.5, -13.523999999999969, 0.0]</td>
-      <td>[13.800000000000061, 12.538520000000014, 0.0]</td>
-    </tr>
-    <tr>
-      <th>277</th>
-      <td>[2.5, -13.572999999999968, 0.0]</td>
-      <td>[13.850000000000062, 12.267060000000015, 0.0]</td>
-    </tr>
-    <tr>
-      <th>278</th>
-      <td>[2.5, -13.621999999999968, 0.0]</td>
-      <td>[13.900000000000063, 11.994620000000015, 0.0]</td>
-    </tr>
-    <tr>
-      <th>279</th>
-      <td>[2.5, -13.670999999999967, 0.0]</td>
-      <td>[13.950000000000063, 11.721200000000016, 0.0]</td>
-    </tr>
-    <tr>
-      <th>280</th>
-      <td>[2.5, -13.719999999999967, 0.0]</td>
-      <td>[14.000000000000064, 11.446800000000016, 0.0]</td>
-    </tr>
-    <tr>
-      <th>281</th>
-      <td>[2.5, -13.768999999999966, 0.0]</td>
-      <td>[14.050000000000065, 11.171420000000015, 0.0]</td>
-    </tr>
-    <tr>
-      <th>282</th>
-      <td>[2.5, -13.817999999999966, 0.0]</td>
-      <td>[14.100000000000065, 10.895060000000017, 0.0]</td>
-    </tr>
-    <tr>
-      <th>283</th>
-      <td>[2.5, -13.866999999999965, 0.0]</td>
-      <td>[14.150000000000066, 10.617720000000018, 0.0]</td>
-    </tr>
-    <tr>
-      <th>284</th>
-      <td>[2.5, -13.915999999999965, 0.0]</td>
-      <td>[14.200000000000067, 10.339400000000019, 0.0]</td>
-    </tr>
-    <tr>
-      <th>285</th>
-      <td>[2.5, -13.964999999999964, 0.0]</td>
-      <td>[14.250000000000068, 10.06010000000002, 0.0]</td>
-    </tr>
-    <tr>
-      <th>286</th>
-      <td>[2.5, -14.013999999999964, 0.0]</td>
-      <td>[14.300000000000068, 9.77982000000002, 0.0]</td>
-    </tr>
-    <tr>
-      <th>287</th>
-      <td>[2.5, -14.062999999999963, 0.0]</td>
-      <td>[14.350000000000069, 9.49856000000002, 0.0]</td>
-    </tr>
-    <tr>
-      <th>288</th>
-      <td>[2.5, -14.111999999999963, 0.0]</td>
-      <td>[14.40000000000007, 9.216320000000021, 0.0]</td>
-    </tr>
-    <tr>
-      <th>289</th>
-      <td>[2.5, -14.160999999999962, 0.0]</td>
-      <td>[14.45000000000007, 8.933100000000021, 0.0]</td>
-    </tr>
-    <tr>
-      <th>290</th>
-      <td>[2.5, -14.209999999999962, 0.0]</td>
-      <td>[14.500000000000071, 8.648900000000022, 0.0]</td>
-    </tr>
-    <tr>
-      <th>291</th>
-      <td>[2.5, -14.258999999999961, 0.0]</td>
-      <td>[14.550000000000072, 8.363720000000024, 0.0]</td>
-    </tr>
-    <tr>
-      <th>292</th>
-      <td>[2.5, -14.30799999999996, 0.0]</td>
-      <td>[14.600000000000072, 8.077560000000025, 0.0]</td>
-    </tr>
-    <tr>
-      <th>293</th>
-      <td>[2.5, -14.35699999999996, 0.0]</td>
-      <td>[14.650000000000073, 7.790420000000026, 0.0]</td>
-    </tr>
-    <tr>
-      <th>294</th>
-      <td>[2.5, -14.40599999999996, 0.0]</td>
-      <td>[14.700000000000074, 7.502300000000027, 0.0]</td>
-    </tr>
-    <tr>
-      <th>295</th>
-      <td>[2.5, -14.45499999999996, 0.0]</td>
-      <td>[14.750000000000075, 7.213200000000027, 0.0]</td>
-    </tr>
-    <tr>
-      <th>296</th>
-      <td>[2.5, -14.503999999999959, 0.0]</td>
-      <td>[14.800000000000075, 6.923120000000028, 0.0]</td>
-    </tr>
-    <tr>
-      <th>297</th>
-      <td>[2.5, -14.552999999999958, 0.0]</td>
-      <td>[14.850000000000076, 6.632060000000029, 0.0]</td>
-    </tr>
-    <tr>
-      <th>298</th>
-      <td>[2.5, -14.601999999999958, 0.0]</td>
-      <td>[14.900000000000077, 6.34002000000003, 0.0]</td>
-    </tr>
-    <tr>
-      <th>299</th>
-      <td>[2.5, -14.650999999999957, 0.0]</td>
-      <td>[14.950000000000077, 6.047000000000031, 0.0]</td>
-    </tr>
-    <tr>
-      <th>300</th>
-      <td>[2.5, -14.699999999999957, 0.0]</td>
-      <td>[15.000000000000078, 5.753000000000032, 0.0]</td>
-    </tr>
-  </tbody>
-</table>
-<p>301 rows × 2 columns</p>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">to_json</span><span class="p">(</span><span class="s1">&#39;mvto.json&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#pd.read_json(&#39;mvto.json&#39;).x.str[1]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#py=df.p.str[1]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#               x       y</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">df</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;$x$ [m]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;$y$ [m]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-</pre></div>
-
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-</div>
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-<div class="output">
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-<div class="jb_output_wrapper }}">
-<div class="output_area">
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-<div class="output_png output_subarea ">
-<img src="../images/features/differential-equations_23_0.png"
->
-</div>
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-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Consider now the case the movement inside a fluid with a draggin force proportinal to the velocity
-$$
-\boldsymbol{F}_D=c \boldsymbol{v} = c \frac{\boldsymbol{p}}{m}\,,
-$$
-where $c$ can take the values 0 (not dragging force), 0.1 y  0.5</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-<span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">()</span>
-<span class="n">m</span><span class="o">=.</span><span class="mi">5</span>
-<span class="n">g</span><span class="o">=</span><span class="mf">9.8</span>
-<span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.5</span><span class="p">]:</span>
-    <span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">50</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-    <span class="n">v</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-    <span class="n">p</span><span class="o">=</span><span class="n">m</span><span class="o">*</span><span class="n">v</span>
-    <span class="n">deltat</span><span class="o">=</span><span class="mf">0.01</span>
-    <span class="n">t</span><span class="o">=</span><span class="mi">0</span>
-    <span class="n">df</span><span class="o">=</span><span class="n">df</span><span class="o">.</span><span class="n">append</span><span class="p">({</span><span class="s1">&#39;x&#39;</span><span class="p">:</span><span class="n">x</span><span class="p">,</span><span class="s1">&#39;p&#39;</span><span class="p">:</span><span class="n">p</span><span class="p">,</span><span class="s1">&#39;t&#39;</span><span class="p">:</span><span class="n">t</span><span class="p">,</span><span class="s1">&#39;c&#39;</span><span class="p">:</span><span class="n">c</span><span class="p">},</span><span class="n">ignore_index</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-    
-    <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span> <span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="n">deltat</span><span class="p">):</span>
-        <span class="n">Fg</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="n">m</span><span class="o">*</span><span class="n">g</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span><span class="o">-</span><span class="n">c</span><span class="o">*</span><span class="p">(</span><span class="n">p</span><span class="o">/</span><span class="n">m</span><span class="p">)</span>
-        <span class="n">p</span><span class="o">=</span><span class="n">p</span><span class="o">+</span><span class="n">Fg</span><span class="o">*</span><span class="n">deltat</span>
-        <span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="o">+</span><span class="p">(</span><span class="n">p</span><span class="o">/</span><span class="n">m</span><span class="p">)</span><span class="o">*</span><span class="n">deltat</span>
-        <span class="n">t</span><span class="o">=</span><span class="n">t</span><span class="o">+</span><span class="n">deltat</span>
-        <span class="n">df</span><span class="o">=</span><span class="n">df</span><span class="o">.</span><span class="n">append</span><span class="p">({</span><span class="s1">&#39;x&#39;</span><span class="p">:</span><span class="n">x</span><span class="p">,</span><span class="s1">&#39;p&#39;</span><span class="p">:</span><span class="n">p</span><span class="p">,</span><span class="s1">&#39;t&#39;</span><span class="p">:</span><span class="n">t</span><span class="p">,</span><span class="s1">&#39;c&#39;</span><span class="p">:</span><span class="n">c</span><span class="p">},</span><span class="n">ignore_index</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">.</span><span class="n">unique</span><span class="p">()</span>
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-<pre>array([0. , 0.1, 0.5])</pre>
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-<p>Apply a mask  upon the DataFrame:</p>
-<ul>
-<li>Example</li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">logical_or</span><span class="p">(</span> <span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="mi">0</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="mf">0.5</span>  <span class="p">)]</span><span class="o">.</span><span class="n">c</span><span class="o">.</span><span class="n">unique</span><span class="p">()</span>
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-<pre>array([0. , 0.5])</pre>
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-<ul>
-<li>Filter <code>c==0</code></li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="mi">0</span><span class="p">][:</span><span class="mi">3</span><span class="p">]</span>
-</pre></div>
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-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>c</th>
-      <th>p</th>
-      <th>t</th>
-      <th>x</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>0.0</td>
-      <td>[2.5, 0.0, 0.0]</td>
-      <td>0.00</td>
-      <td>[0, 50, 0]</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>0.0</td>
-      <td>[2.5, -0.049, 0.0]</td>
-      <td>0.01</td>
-      <td>[0.05, 49.99902, 0.0]</td>
-    </tr>
-    <tr>
-      <th>2</th>
-      <td>0.0</td>
-      <td>[2.5, -0.098, 0.0]</td>
-      <td>0.02</td>
-      <td>[0.1, 49.997060000000005, 0.0]</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
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-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="mf">0.5</span><span class="p">][:</span><span class="mi">3</span><span class="p">]</span>
-</pre></div>
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-</div>
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-<div class="output_wrapper">
-<div class="output">
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-<div class="jb_output_wrapper }}">
-<div class="output_area">
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-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>c</th>
-      <th>p</th>
-      <th>t</th>
-      <th>x</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>602</th>
-      <td>0.5</td>
-      <td>[2.5, 0.0, 0.0]</td>
-      <td>0.00</td>
-      <td>[0, 50, 0]</td>
-    </tr>
-    <tr>
-      <th>603</th>
-      <td>0.5</td>
-      <td>[2.475, -0.049, 0.0]</td>
-      <td>0.01</td>
-      <td>[0.0495, 49.99902, 0.0]</td>
-    </tr>
-    <tr>
-      <th>604</th>
-      <td>0.5</td>
-      <td>[2.45025, -0.09751, 0.0]</td>
-      <td>0.02</td>
-      <td>[0.09850500000000001, 49.9970698, 0.0]</td>
-    </tr>
-  </tbody>
-</table>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">mpl_toolkits.mplot3d</span> <span class="kn">import</span> <span class="n">Axes3D</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
-<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">gca</span><span class="p">(</span><span class="n">projection</span><span class="o">=</span><span class="s1">&#39;3d&#39;</span><span class="p">)</span>
-<span class="n">c</span><span class="o">=</span><span class="mi">0</span>
-<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="p">,</span> <span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="p">,</span> <span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="p">)</span>
-<span class="n">c</span><span class="o">=</span><span class="mf">0.1</span>
-<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="p">,</span> <span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="p">,</span> <span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="p">)</span>
-<span class="n">c</span><span class="o">=</span><span class="mf">0.5</span>
-<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="p">,</span> <span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="p">,</span> <span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="p">)</span>
-
-<span class="n">ax</span><span class="o">.</span><span class="n">view_init</span><span class="p">(</span><span class="mi">80</span><span class="p">,</span> <span class="mi">250</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;$x$ [m]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;$y$ [m]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
-</pre></div>
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-<pre>Text(0.5, 0, &#39;$y$ [m]&#39;)</pre>
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->
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">c</span><span class="o">=</span><span class="mi">0</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-<span class="n">c</span><span class="o">=</span><span class="mf">0.1</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-<span class="n">c</span><span class="o">=</span><span class="mf">0.5</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">c</span><span class="o">==</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">str</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;$x$ [m]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;$y$ [m]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-</pre></div>
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-<p><strong>Activity</strong>: Find the time to reach the position <code>y=0</code> in each case. By using an algorithm to find roots</p>
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-<p><strong>Activity</strong>: Repeat the previous analysis for a baseball in the air with a dragging force proportinal to the square of the velocity. The diameter of the baseball is 7.5cm</p>
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-<h3 id="Example.-Spring">Example. Spring<a class="anchor-link" href="#Example.-Spring"> </a></h3><p>In order to apply this, let's assume a simple mass-spring.</p>
-<p><img src="https://raw.githubusercontent.com/sbustamante/ComputationalMethods/master/material/figures/mass_spring.png" alt=""></p>
-<p>The equation of motion according to Newton's second law is</p>
-$$ F = -kx $$<p>Using the previous results, we can rewrite this as:</p>
-$$ \frac{d p}{dt} = -k x $$$$ \frac{dx}{dt} = \frac{p}{m}$$<p>And the equivalent Euler system is
-\begin{align}
-v_{i+1}=&amp; v_{i} - \Delta t\frac{k}{m}x_i \\  
-x_{i+1}=&amp;x_i +  v_i \Delta t\, ,\ \ \ x(0) = x_0\ \ \mbox{and}\ v(0) = v_0
-\end{align}</p>
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-<h2 id="-----**Activity**-"><font color="red">     **Activity** </font><a class="anchor-link" href="#-----**Activity**-"> </a></h2>
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-<p><font color='red'>    
-**1.** Using the initial conditions $x(0) = 0$ and $v(0) = 3$, solve the previous system. Plot the solutions $x(t)$ and $y(t)$ and compare with real solutions. Furthermore, calculate the total energy of the system. What can you conclude about the behaviour of the energy? Does it make any sense?
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-<p><font color='red'>    
-**2.** Using the same reasoning, derive the equations for a simple pendulum. Compare the solution for small oscillations with the analytic one. What happens when increasing the initial amplitude of the movement?
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-<h2 id="Implementation-in-Scipy">Implementation in Scipy<a class="anchor-link" href="#Implementation-in-Scipy"> </a></h2>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">scipy.integrate</span> <span class="k">as</span> <span class="nn">integrate</span>
-</pre></div>
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-<h3 id="First-order-ordinary-differential-equations">First order ordinary differential equations<a class="anchor-link" href="#First-order-ordinary-differential-equations"> </a></h3><p><code>integrate.odeint</code>:
-Integrate a system of ordinary differential equations</p>
-<p>Uso básico</p>
-<div class="highlight"><pre><span></span><span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">func</span><span class="p">,</span><span class="n">y0</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
-</pre></div>
-<p>Solves the initial value problem for stiff or non-stiff systems
-of first order ordinary differential equations:
-$$
-\frac{dy}{dt}=f(y,t)
-$$
-where $y$ can be a vector.</p>
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-<h3 id="Example">Example<a class="anchor-link" href="#Example"> </a></h3><p>Consider the following differential equation
-$$ \frac{dy(t)}{dt}=-k y(t),$$
-where $k$ is constant</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">dy_dt</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">t</span><span class="p">):</span>
-    <span class="n">k</span> <span class="o">=</span> <span class="mf">0.3</span>
-    <span class="n">dydt</span> <span class="o">=</span> <span class="o">-</span><span class="n">k</span> <span class="o">*</span> <span class="n">y</span>
-    <span class="k">return</span> <span class="n">dydt</span>
-</pre></div>
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-<p>For a first evolution after one step of $\Delta t=0.1$: con condición inicial $y(0)=y_0=5$</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># initial condition</span>
-<span class="n">y0</span> <span class="o">=</span> <span class="mi">5</span>
-<span class="n">t</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.1</span><span class="p">]</span>
-<span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">dy_dt</span><span class="p">,</span><span class="n">y0</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
-</pre></div>
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-<pre>array([[5.        ],
-       [4.85222774]])</pre>
-</div>
-
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-<p>while the evolution  from 0 to 20 s is</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># time points</span>
-<span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">20</span><span class="p">)</span>
-
-<span class="c1"># solve ODE</span>
-<span class="n">y</span> <span class="o">=</span> <span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">dy_dt</span><span class="p">,</span><span class="n">y0</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
-
-<span class="c1"># plot results</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;t&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;y(t)&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-</pre></div>
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-<p>Compare with the analytical solution $$y(t)=y_0\operatorname{e}^{-kt}$$</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># time points</span>
-<span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">20</span><span class="p">)</span>
-
-<span class="c1">#k = 0.3 solve ODE</span>
-<span class="n">y0</span><span class="o">=</span><span class="mi">5</span>
-<span class="n">y</span> <span class="o">=</span> <span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">dy_dt</span><span class="p">,</span><span class="n">y0</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
-
-<span class="n">k</span><span class="o">=</span><span class="mf">0.3</span>
-<span class="c1"># plot results</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="s1">&#39;c-&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;t&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">y0</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">k</span><span class="o">*</span><span class="n">t</span><span class="p">),</span><span class="s1">&#39;k:&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;t&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;y(t)&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-</pre></div>
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-<h3 id="Second-order-differential-equations">Second order differential equations<a class="anchor-link" href="#Second-order-differential-equations"> </a></h3>
-</div>
-</div>
-</div>
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-<p>Although first-order schemes like Euler's method are illustrative and allow a good understanding of the numerical problem, real applications cannot be dealt with them, instead more precise and accurate high-order methods must be invoked. In this section we shall cover a well-known family of numerical integrators, the Runge-Kutta methods.</p>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">scipy.integrate</span> <span class="k">as</span> <span class="nn">integrate</span>
-</pre></div>
-
-    </div>
-</div>
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-<h3 id="Some-examples">Some examples<a class="anchor-link" href="#Some-examples"> </a></h3><ol>
-<li><a href="http://www.southampton.ac.uk/~fangohr/teaching/python/book/Python-for-Computational-Science-and-Engineering.pdf">http://www.southampton.ac.uk/~fangohr/teaching/python/book/Python-for-Computational-Science-and-Engineering.pdf</a><ul>
-<li><a href="http://www.southampton.ac.uk/~fangohr/teaching/python/book/ipynb/">http://www.southampton.ac.uk/~fangohr/teaching/python/book/ipynb/</a></li>
-</ul>
-</li>
-<li><a href="http://sam-dolan.staff.shef.ac.uk/mas212/">http://sam-dolan.staff.shef.ac.uk/mas212/</a><ul>
-<li><a href="http://sam-dolan.staff.shef.ac.uk/mas212/notebooks/ODE_Example.ipynb">http://sam-dolan.staff.shef.ac.uk/mas212/notebooks/ODE_Example.ipynb</a></li>
-</ul>
-</li>
-<li><a href="https://apmonitor.com/pdc/index.php/Main/SolveDifferentialEquations">https://apmonitor.com/pdc/index.php/Main/SolveDifferentialEquations</a></li>
-<li><a href="http://csc.ucdavis.edu/~cmg/Group/readings/pythonissue_3of4.pdf">http://csc.ucdavis.edu/~cmg/Group/readings/pythonissue_3of4.pdf</a></li>
-<li><a href="http://www.tau.ac.il/~kineret/amit/scipy_tutorial/#x1-210004.4">http://www.tau.ac.il/~kineret/amit/scipy_tutorial/#x1-210004.4</a></li>
-<li><a href="https://www.udacity.com/course/differential-equations-in-action--cs222">https://www.udacity.com/course/differential-equations-in-action--cs222</a></li>
-<li><a href="https://github.com/robclewley/pydstool">https://github.com/robclewley/pydstool</a><ul>
-<li><a href="http://www2.gsu.edu/~matrhc/PyDSTool.htm">http://www2.gsu.edu/~matrhc/PyDSTool.htm</a></li>
-</ul>
-</li>
-<li><a href="https://github.com/JuliaDiffEq">https://github.com/JuliaDiffEq</a>   </li>
-</ol>
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-<p><strong>Example</strong>: From <a href="http://sam-dolan.staff.shef.ac.uk/mas212/notebooks/ODE_Example.ipynb">http://sam-dolan.staff.shef.ac.uk/mas212/</a></p>
-<p>As explained before, we need to write a second order ordinary differential equations in terms of first order matricial ordinary differential equation. In terms of a parameter, $x$, this implay to have a column vector 
-$$
-U=\begin{bmatrix}
-y\\
-z\\
-\end{bmatrix}
-$$4
-$$
-\frac{dU}{dt}=V(U,t).
-$$</p>
-
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-<p>Suppose we have a second-order ODE such as a damped simple harmonic motion equation,
-$$
-\quad y'' + 2 y' + 2 y = \cos(2x), \quad \quad y(0) = 0, \; y'(0) = 0
-$$
-We can turn this into two first-order equations by defining a new depedent variable. For example,
-$$
-\quad z \equiv y' \quad \Rightarrow \quad z' + 2 z + 2y = \cos(2x), \quad z(0)=y(0) = 0.
-$$
-So that
-$$
-y'=z \quad \Rightarrow \quad z'  =- 2 z - 2y + \cos(2x), \quad z(0)=y(0) = 0.
-$$
-where
-$$
-\frac{dU}{dx}=\begin{bmatrix}y'\\z'\end{bmatrix}
-$$
-We can solve this system of ODEs using "odeint" with lists, as follows:</p>
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-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Let 
-$$
-U=\begin{bmatrix}
-U_0\\
-U_1
-\end{bmatrix}=\begin{bmatrix}
-y\\
-z
-\end{bmatrix}.
-$$
-Therefore
-\begin{align}
-\frac{\operatorname{d}}{\operatorname{d} x} 
-\begin{bmatrix} 
-y\\
-z\\
-\end{bmatrix}=&amp;
-\begin{bmatrix}
-z\\
--2z-2y+\cos(2x)\\
-\end{bmatrix}\\
-\frac{\operatorname{d}}{\operatorname{d} x} 
-U=&amp;
-\begin{bmatrix}
-U_1\\
--2U_1-2U_0+\cos(2x)\\
-\end{bmatrix}
-\end{align}</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Implementation by using only $U$</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">dU_dx</span><span class="p">(</span><span class="n">U</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Here U is a vector such that y=U[0] and z=U[1]. </span>
-<span class="sd">    This function should return [y&#39;, z&#39;]</span>
-<span class="sd">    &#39;&#39;&#39;</span>    
-    <span class="k">return</span> <span class="p">[</span>      <span class="n">U</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> 
-            <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">U</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">U</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">U0</span> <span class="c1"># → x=0</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>[0, 0]</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>x→0.1</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">dU_dx</span><span class="p">,</span> <span class="n">U0</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.1</span><span class="p">])</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([[0.        , 0.        ],
-       [0.00465902, 0.08970031]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">array</span><span class="p">(</span>
-        <span class="p">[[</span><span class="mf">0.</span>        <span class="p">,</span> <span class="mf">0.</span>         <span class="p">],</span><span class="c1">#→ U(0)</span>
-         <span class="p">[</span><span class="mf">0.00465902</span><span class="p">,</span> <span class="mf">0.08970031</span><span class="p">]]</span> <span class="c1">#→ U(0.1)=[y(0.1),y&#39;(0.1)]</span>
-     <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([[0.        , 0.        ],
-       [0.00465902, 0.08970031]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Us</span><span class="p">[:</span><span class="mi">3</span><span class="p">]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([[0.        , 0.        ],
-       [0.02601336, 0.1841925 ],
-       [0.08083294, 0.229454  ]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Note that <code>y=Us[:,0]</code></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">U0</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> 
-      <span class="mi">0</span><span class="p">]</span>
-<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">200</span><span class="p">)</span>
-<span class="n">Us</span> <span class="o">=</span> <span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">dU_dx</span><span class="p">,</span> <span class="n">U0</span><span class="p">,</span> <span class="n">xs</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Us</span><span class="p">[:</span><span class="mi">3</span><span class="p">]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([[0.        , 0.        ],
-       [0.02601336, 0.1841925 ],
-       [0.08083294, 0.229454  ]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Us</span><span class="p">[:,</span><span class="mi">0</span><span class="p">][:</span><span class="mi">3</span><span class="p">]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([0.        , 0.02601336, 0.08083294])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ys</span> <span class="o">=</span> <span class="n">Us</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="c1">#First column is extracted</span>
-
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;y&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Damped harmonic oscillator&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span><span class="n">ys</span><span class="p">);</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/differential-equations_71_0.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">zs</span> <span class="o">=</span> <span class="n">Us</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span>
-
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;z&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Damped harmonic oscillator&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span><span class="n">zs</span><span class="p">);</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/differential-equations_72_0.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Implementation by using explicitly $y(x)$ and $z(x)$
-\begin{align}
-\frac{\operatorname{d}}{\operatorname{d} x} 
-\begin{bmatrix} 
-y\\
-z\\
-\end{bmatrix}=&amp;
-\begin{bmatrix}
-z\\
--2z-2y+\cos(2x)\\
-\end{bmatrix}
-\end{align}</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">dU_dx</span><span class="p">(</span><span class="n">U</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Here U is a vector such that y=U[0] and z=U[1]. </span>
-<span class="sd">    This function should return [y&#39;, z&#39;]</span>
-<span class="sd">    &#39;&#39;&#39;</span>    
-    <span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="o">=</span><span class="n">U</span> <span class="c1"># y → U[0]; z → U[1] </span>
-    <span class="k">return</span> <span class="p">[</span>        <span class="n">z</span><span class="p">,</span> 
-            <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">z</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)]</span>
-
-<span class="n">U0</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">200</span><span class="p">)</span>
-<span class="n">Us</span> <span class="o">=</span> <span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">dU_dx</span><span class="p">,</span> <span class="n">U0</span><span class="p">,</span> <span class="n">xs</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ys</span> <span class="o">=</span> <span class="n">Us</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span>
-
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;y&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Damped harmonic oscillator&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span><span class="n">ys</span><span class="p">);</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/differential-equations_75_0.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Activity</strong>: Apply the previous example to the problem of parabolic motion with air friction:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-\begin{align}
-\frac{\boldsymbol{p}}{m}=&amp;\frac{\operatorname{d}\boldsymbol{x}}{\operatorname{d}t}\\
-\boldsymbol{F}=&amp;\frac{\operatorname{d}\boldsymbol{p}}{\operatorname{d} t} 
-\end{align}\begin{align}
-\frac{\operatorname{d}}{\operatorname{d} t} 
-\begin{bmatrix} 
- \boldsymbol{x}\\
- \boldsymbol{p}\\
-\end{bmatrix}=&amp;
-\begin{bmatrix}
-\boldsymbol{p}/m\\
--m\boldsymbol{g}-c \boldsymbol{p}/m \\
-\end{bmatrix}\,,
-\end{align}<p>such that in two dimensions</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-$$
-\boldsymbol{U}=\begin{bmatrix}
-{U}_0\\
-{U}_1\\
-{U}_2\\
-{U}_3\\
-\end{bmatrix}=\begin{bmatrix}
-{x}\\
-y\\
-p_x\\
-p_y\\
-\end{bmatrix}
-$$\begin{align}
-\frac{\operatorname{d}}{\operatorname{d} t} \boldsymbol{U}=&amp;
-\begin{bmatrix}
-{U}_2/m\\
-{U}_3/m\\
--c{U}_2/m\\
--m{g}-c{U}_3/m\\
-\end{bmatrix}
-\end{align}
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">dU_dt</span><span class="p">(</span><span class="n">U</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span><span class="n">c</span><span class="o">=</span><span class="mf">0.</span><span class="p">,</span><span class="n">m</span><span class="o">=.</span><span class="mi">5</span><span class="p">,</span><span class="n">g</span><span class="o">=</span><span class="mf">9.8</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Here U is a vector such that y=U[0] and z=U[1]. </span>
-<span class="sd">    This function should return [y&#39;, z&#39;]</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">U</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">/</span><span class="n">m</span><span class="p">,</span>
-            <span class="n">U</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span><span class="o">/</span><span class="n">m</span><span class="p">,</span>
-            <span class="o">-</span><span class="n">c</span><span class="o">*</span><span class="n">U</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">/</span><span class="n">m</span><span class="p">,</span> 
-            <span class="o">-</span><span class="n">m</span><span class="o">*</span><span class="n">g</span><span class="o">-</span><span class="n">c</span><span class="o">*</span><span class="n">U</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span><span class="o">/</span><span class="n">m</span><span class="p">]</span>
-
-<span class="n">m</span><span class="o">=</span><span class="mf">0.5</span>
-<span class="c1"># p_x(0)=5*m → v(0)=5 m/s</span>
-<span class="n">U0</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span><span class="n">m</span><span class="o">*</span><span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
-<span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">200</span><span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">isinstance</span><span class="p">((</span><span class="mf">0.1</span><span class="p">),</span><span class="nb">float</span><span class="p">)</span>
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-<pre>True</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">isinstance</span><span class="p">((</span><span class="mf">0.1</span><span class="p">,),</span><span class="nb">tuple</span><span class="p">)</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Us</span> <span class="o">=</span> <span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">dU_dt</span><span class="p">,</span> <span class="n">U0</span><span class="p">,</span> <span class="n">t</span> <span class="p">)</span>  
-<span class="n">xs</span> <span class="o">=</span> <span class="n">Us</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span>
-<span class="n">ys</span> <span class="o">=</span> <span class="n">Us</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span><span class="n">ys</span><span class="p">)</span>
-<span class="n">Us</span> <span class="o">=</span> <span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">dU_dt</span><span class="p">,</span> <span class="n">U0</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span><span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,)</span> <span class="p">)</span>  
-<span class="n">xs</span> <span class="o">=</span> <span class="n">Us</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span>
-<span class="n">ys</span> <span class="o">=</span> <span class="n">Us</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span><span class="n">ys</span><span class="p">)</span>
-<span class="n">Us</span> <span class="o">=</span> <span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">dU_dt</span><span class="p">,</span> <span class="n">U0</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span><span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,)</span> <span class="p">)</span>  
-<span class="n">xs</span> <span class="o">=</span> <span class="n">Us</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span>
-<span class="n">ys</span> <span class="o">=</span> <span class="n">Us</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span><span class="n">ys</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;$x$ [m]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;$y$ [m]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-</pre></div>
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-<pre>Text(0, 0.5, &#39;$y$ [m]&#39;)</pre>
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-<p><font color="red">Activity</font>: Implemet directly:
-\begin{align}
-\frac{\operatorname{d}}{\operatorname{d} t} 
-\begin{bmatrix} 
- \boldsymbol{x}\\
- \boldsymbol{p}\\
-\end{bmatrix}=&amp;
-\begin{bmatrix}
-\boldsymbol{p}/m\\
--m\boldsymbol{g}-c \boldsymbol{p}/m \\
-\end{bmatrix}\,,
-\end{align}
-as
-\begin{align}
-\frac{\operatorname{d}}{\operatorname{d} t} 
-\begin{bmatrix} 
- x\\
- y\\
- p_x\\
- p_y
-\end{bmatrix}=&amp;
-\begin{bmatrix}
-p_x/m\\
-p_y/m\\
--c p_x/m \\
--mg-c p_y/m \\
-\end{bmatrix}\,,
-\end{align}</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">dU_dt</span><span class="p">(</span><span class="n">U</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span><span class="n">c</span><span class="o">=</span><span class="mf">0.</span><span class="p">,</span><span class="n">m</span><span class="o">=.</span><span class="mi">5</span><span class="p">,</span><span class="n">g</span><span class="o">=</span><span class="mf">9.8</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Here U is a vector such that y=U[0] and z=U[1]. </span>
-<span class="sd">    This function should return [y&#39;, z&#39;]</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">px</span><span class="p">,</span><span class="n">py</span><span class="o">=</span><span class="n">U</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">px</span><span class="o">/</span><span class="n">m</span><span class="p">,</span>
-            <span class="n">py</span><span class="o">/</span><span class="n">m</span><span class="p">,</span>
-            <span class="o">-</span><span class="n">c</span><span class="o">*</span><span class="n">px</span><span class="o">/</span><span class="n">m</span><span class="p">,</span> 
-            <span class="o">-</span><span class="n">m</span><span class="o">*</span><span class="n">g</span><span class="o">-</span><span class="n">c</span><span class="o">*</span><span class="n">py</span><span class="o">/</span><span class="n">m</span><span class="p">]</span>
-
-<span class="n">m</span><span class="o">=</span><span class="mf">0.5</span>
-<span class="n">U0</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span><span class="mi">5</span><span class="o">*</span><span class="n">m</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
-<span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">200</span><span class="p">)</span>
-
-<span class="n">Us</span> <span class="o">=</span> <span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">dU_dt</span><span class="p">,</span> <span class="n">U0</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span><span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,)</span> <span class="p">)</span>  
-<span class="n">xs</span> <span class="o">=</span> <span class="n">Us</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span>
-<span class="n">ys</span> <span class="o">=</span> <span class="n">Us</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span><span class="n">ys</span><span class="p">)</span>
-</pre></div>
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-<pre>[&lt;matplotlib.lines.Line2D at 0x7f60f36e3128&gt;]</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%</span><span class="k">pylab</span> inline
-</pre></div>
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-<pre>Populating the interactive namespace from numpy and matplotlib
-</pre>
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-<h2 id="Example-2">Example 2<a class="anchor-link" href="#Example-2"> </a></h2>
-</div>
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-<p>In this example we are going to use the Scipy implementation for mapping the phase space of a pendulum.</p>
-<p>The equations of the pendulum are given by:</p>
-$$ \frac{d\theta}{dt} = \omega $$$$ \frac{d\omega}{dt} = -\frac{g}{l}\sin \theta $$\begin{align}
-\frac{d}{dt} U=\begin{pmatrix}\omega\\
- -\dfrac{g}{l}\sin\theta \end{pmatrix},
-\end{align}<p>where
-\begin{align}
- U=\begin{pmatrix}\theta\\
-\omega \end{pmatrix}.
-\end{align}</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">integrate</span>
-<span class="k">def</span> <span class="nf">dU_dt</span><span class="p">(</span><span class="n">U</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span><span class="n">l</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">g</span><span class="o">=</span><span class="mf">9.8</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Here U is a vector such that   θ=U[0] and ω=U[1]. </span>
-<span class="sd">    This function should return [θ&#39;, ω&#39;]</span>
-<span class="sd">    &#39;&#39;&#39;</span>    
-    <span class="n">θ</span><span class="p">,</span><span class="n">ω</span><span class="o">=</span><span class="n">U</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">ω</span><span class="p">,</span> <span class="o">-</span><span class="n">g</span><span class="o">/</span><span class="n">l</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span> <span class="n">θ</span> <span class="p">)</span> <span class="p">]</span>
-</pre></div>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">tmax</span> <span class="o">=</span> <span class="mi">6</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span>
-<span class="n">omega_max</span> <span class="o">=</span> <span class="mi">8</span> <span class="c1">#rad/s</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Nic</span> <span class="o">=</span> <span class="mi">1000</span>
-<span class="c1">#Maxim angular velocity</span>
-
-<span class="n">theta0s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span><span class="mi">4</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span><span class="n">Nic</span><span class="p">)</span>
-<span class="n">omega0s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="n">omega_max</span><span class="p">,</span><span class="n">omega_max</span><span class="p">,</span><span class="n">Nic</span><span class="p">)</span>
-</pre></div>
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-<p>TAREA: Generar un rango aleatorio uniforme de varios ordenes de magnitud.
-En particular, 1000 números aleatorios entre  $3\times 10^{-6}$ hasta $5\times 10^{4}$</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">U0</span><span class="o">=</span><span class="p">[</span><span class="n">theta0s</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">omega0s</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-<span class="n">U0</span>
-</pre></div>
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-<pre>[-1.094232903981899, -2.8722875660380076]</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">t</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">tmax</span><span class="p">,</span><span class="mi">400</span><span class="p">)</span>
-<span class="n">Us</span><span class="o">=</span><span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">dU_dt</span><span class="p">,</span><span class="n">U0</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Us</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span><span class="n">Us</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span><span class="n">lw</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">color</span> <span class="o">=</span> <span class="s2">&quot;black&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$\theta$ [rad]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$\omega$ [rad/s]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
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-<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span> <span class="p">)</span>
-<span class="k">for</span> <span class="n">theta0</span><span class="p">,</span> <span class="n">omega0</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">theta0s</span><span class="p">,</span> <span class="n">omega0s</span><span class="p">):</span>
-    <span class="n">t</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">tmax</span><span class="p">,</span><span class="mi">400</span><span class="p">)</span>
-    <span class="n">U0</span><span class="o">=</span><span class="p">[</span><span class="n">theta0</span><span class="p">,</span><span class="n">omega0</span><span class="p">]</span>
-    <span class="n">Us</span><span class="o">=</span><span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">dU_dt</span><span class="p">,</span><span class="n">U0</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Us</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span><span class="n">Us</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span><span class="n">lw</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">,</span> <span class="n">color</span> <span class="o">=</span> <span class="s2">&quot;black&quot;</span> <span class="p">)</span>
-    <span class="k">if</span> <span class="n">j</span><span class="o">==</span><span class="n">Nic</span><span class="p">:</span> <span class="c1">#secutity stop</span>
-        <span class="k">break</span>
-    <span class="n">j</span><span class="o">=</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span>
-    
-<span class="c1">#Format of figure</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span> <span class="s2">&quot;$</span><span class="se">\\</span><span class="s2">theta/\pi$&quot;</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">18</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span> <span class="s2">&quot;$\omega$ [rad/s]&quot;</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">18</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span> <span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span> <span class="p">(</span><span class="o">-</span><span class="n">omega_max</span><span class="p">,</span> <span class="n">omega_max</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span> <span class="s2">&quot;Phase space of a pendulum&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>    
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-<p>The nearly closed curves around (0,0) represent the regular small swings of the pendulum near its
-rest position.  The oscillatory curves up and down of the closed curves represent the movement when  the pendulum start at $\theta=0$ but with high enough angular speed such that the pendulum goes all the way around. Of course, its
-angular speed will slow down on the way up but then it will speed up on the way down again.
-In the absence of friction, it just keeps spinning around indefinitely. The counterclockwise
-motions of the pendulum of this kind are shown in the graph by the wavy lines
-at the top with positive angular speed, while the curves on the bottom
-which go from right to left represent clockwise rotations. The phase space have a periodicity of $2\pi$.</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">t</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">tmax</span><span class="p">,</span><span class="mi">400</span><span class="p">)</span>
-<span class="n">Us</span><span class="o">=</span><span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">dU_dt</span><span class="p">,[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span><span class="n">t</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Us</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span><span class="n">Us</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span><span class="n">lw</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">color</span> <span class="o">=</span> <span class="s2">&quot;black&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">t</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">tmax</span><span class="p">,</span><span class="mi">400</span><span class="p">)</span>
-<span class="n">Us</span><span class="o">=</span><span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">dU_dt</span><span class="p">,[</span><span class="mi">0</span><span class="p">,</span><span class="mi">7</span><span class="p">],</span><span class="n">t</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Us</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span><span class="n">Us</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span><span class="n">lw</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">color</span> <span class="o">=</span> <span class="s2">&quot;black&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span>
-</pre></div>
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-<span class="n">Us</span><span class="o">=</span><span class="n">integrate</span><span class="o">.</span><span class="n">odeint</span><span class="p">(</span><span class="n">dU_dt</span><span class="p">,[</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">7</span><span class="p">],</span><span class="n">t</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Us</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span><span class="n">Us</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span><span class="n">lw</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">color</span> <span class="o">=</span> <span class="s2">&quot;black&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Us</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span><span class="n">Us</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span><span class="n">lw</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">color</span> <span class="o">=</span> <span class="s2">&quot;black&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span>
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-<p><strong>Activity</strong>: Check the anlytical expression for the period of a pendulum</p>
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-<h1 id="Appendix">Appendix<a class="anchor-link" href="#Appendix"> </a></h1>
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-<h2 id="-----**Activity**-"><font color="red">     **Activity** </font><a class="anchor-link" href="#-----**Activity**-"> </a></h2>
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-<p><font color='red'>    
-Using the previous example, explore the phase space of a simple oscillator and a damped pendulum.
-</font></p>
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-<h2 id="Fourth-order-Runge-Kutta-method">Fourth-order Runge-Kutta method<a class="anchor-link" href="#Fourth-order-Runge-Kutta-method"> </a></h2>
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-<p>Finally, the most used general purpose method is the fourth-order Runge-Kutta scheme. Its derivation follows the same previous reasoning, however the procedure is rather long and it makes no sense to reprouce it here. Instead, we will give the direct algorithm:</p>
-<p>Let's assume again a problem of the form:</p>
-$$ \frac{dy}{dt}=y'=f(t,y),\ \ \ a\leq t\leq b, \ \ \ \ y(a) = \alpha $$<p>The Runge-Kutta-4 (RK4) method allows us to predict the solution at the time $t+h$ as:</p>
-$$ y(t+h) = y(t) + \frac{1}{6}( \mathbf{K}_0 + 2\mathbf{K}_1 + 2\mathbf{K}_2 + \mathbf{K}_3  ) $$<p>where:</p>
-$$ \mathbf{K}_0 = hf(t,y)$$$$ \mathbf{K}_1 = hf\left( t + \frac{h}{2},y + \frac{\mathbf{K}_0}{2}\right)$$$$ \mathbf{K}_2 = hf\left( t + \frac{h}{2},y + \frac{\mathbf{K}_1}{2}\right)$$$$ \mathbf{K}_3 = hf\left( t + h,y + \mathbf{K}_2\right)$$
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-<h2 id="-----**Activity**-"><font color="red">     **Activity** </font><a class="anchor-link" href="#-----**Activity**-"> </a></h2>
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-The Lorenz attractor is a common set of differential equations of some models of terrestrial atmosphere studies. It is historically known as one of the first system to exhibit deterministic caos. The equations are:</p>
-$$ \frac{dx}{dt} = a(y-x) $$$$ \frac{dy}{dt} = x(b-z)-y $$$$ \frac{dz}{dt} = xy-cz $$<p>with $a = 10$, $b=28$ and $c = 8/3$ the solution shows periodic orbits.
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-Write a routine that gives a step of RK4 and integrate the previous system. Plot the resulting 3D solution $(x,y,z)$.
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-<h2 id="Second-order-Runge-Kutta-methods">Second-order Runge-Kutta methods<a class="anchor-link" href="#Second-order-Runge-Kutta-methods"> </a></h2><p>For this method, let's assume a problem of the form:</p>
-$$ \frac{dy}{dt}=y'=f(t,y),\ \ \ a\leq t\leq b, \ \ \ \ y(a) = \alpha $$<p>Now, we want to know the solution in the next timestep, i.e. $y(t+h)$. For this, we propose the next solution:</p>
-$$ y(t+h) = y(t) + c_0 f(t,y)h + c_1f[  t+ph, y+qhf(t,y) ]h $$<p>determining the coefficients $c_0, c_1, q$ and $p$, we will have the complete approximated solution of the problem.</p>
-<p>One way to determine them is by comparing with the taylor expansion around $t$</p>
-$$ y(t+h) = y(t) + f(t,y)h + \frac{1}{2}\left( \frac{\partial f}{\partial t} + \frac{\partial f}{\partial y}  \right)h^2 + \mathcal{O}(h^3) $$<p>Now, we can also expand the function $f[  t+ph, y+qhf(t,y) ]$ around the point $(t,y)$, yielding:</p>
-$$ f[  t+ph, y+qhf(t,y) ] = f(t,y) + \frac{\partial f}{\partial t}ph + \frac{\partial f}{\partial y}qh + \mathcal{O}(h^2)  $$<p>Replacing this into the original expression:</p>
-$$ y(t+h) = y(t) + c_0 f(t,y)h + c_1\left[ f(t,y) + \frac{\partial f}{\partial t}ph + \frac{\partial f}{\partial y}qh \right]h + \mathcal{O}(h^3)$$<p>ordering the terms we obtain:</p>
-$$ y(t+h) = y(t) + (c_0+c_1) f(t,y)h + c_1\left[ \frac{\partial f}{\partial t}p + \frac{\partial f}{\partial y}q \right]h^2 + \mathcal{O}(h^3)$$<p>Equalling the next conditions are obtained:</p>
-$$ c_0 + c_1 =1 \ \ \ c_1p=\frac{1}{2}\ \ \ c_1q = \frac{1}{2} $$<p>This set of equations are undetermined so there are several solutions, each one yielding a different method:</p>
-$$ \matrix{ 
-c_0 = 0 &amp; c_1=1 &amp; p = 1/2 &amp; q = 1/2 &amp; \mbox{Modified Euler's Method} \\
-c_0 = 1/2 &amp; c_1=1/2 &amp; p = 1 &amp; q = 1 &amp; \mbox{Heun's Method} \\
-c_0 = 1/3 &amp; c_1=2/3 &amp; p = 3/4 &amp; q = 3/4 &amp; \mbox{Ralston's Method}
-} $$<p>The algorithm is then:</p>
-$$ y(t+h) = y(t) + (c_1+c_2)\mathbf{K}_1 $$<p>with</p>
-$$ \mathbf{K}_1 = hf( t+pt, y+q\mathbf{K}_0 ) $$$$ \mathbf{K}_0 = hf(t,y) $$<p>All these methods constitute the second-order Runge-Kutta methods.</p>
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-<h1 id="Two-Point-Boundary-Value-Problems">Two-Point Boundary Value Problems<a class="anchor-link" href="#Two-Point-Boundary-Value-Problems"> </a></h1>
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-<p>Up to now we have solved initial value problems of the form:</p>
-$$ \frac{dy}{dt}=y'=f(t,y),\ \ \ a\leq t\leq b, \ \ \ \ y(a) = \alpha $$<p>Second order equations can be similarly planted as</p>
-$$ \frac{d^2y}{dt^2}=y''=f(t,y,y'),\ \ \ a\leq t\leq b, \ \ \ \ y(a) = \alpha \ \ \ y'(a) = u $$<p>This type of systems can be readily solved by defining the auxiliar variable $w = y'$, turning it into a first order system of equations.</p>
-<p>Now, we shall solve two-point boundary problem, where we have two conditions on the solution $y(t)$ instead of having the function and its derivative at some initial point, i.e.</p>
-$$ \frac{d^2y}{dt^2}=y''=f(t,y,y'),\ \ \ a\leq t\leq b, \ \ \ \ y(a) = \alpha \ \ \ y(b) = \beta $$<p>In spite of its similar form to the initial value problem, two-point boundary problems pose a increased difficulty for numerical methods. The main reason of this is the iterative procedure performed by numerical approaches, where from an initial point, further points are found. Trying to fit two different values at different points implies then a continuous readjusting of the solution.</p>
-<p>A common way to solve these problems is by turning them into a initial-value problem</p>
-$$ \frac{d^2y}{dt^2}=y''=f(t,y,y'),\ \ \ a\leq t\leq b, \ \ \ \ y(a) = \alpha \ \ \ y'(a) = u $$<p>Let's suppose some choice of $u$, integrating by using some of the previous methods, we obtain the final boundary condition $y(b)=\theta$. If the produced value is not the one we wanted from our initial problem, we try another value $u$. This can be repeated until we get a reasonable approach to $y(b)=\beta$. This method works fine, but it is so expensive and terribly inefficient.</p>
-<p>Note when we change $u$, the final boundary value also change, so we can assume $y(b) = \theta$. The solution to the problem can be thought then as a root-finding problem:</p>
-$$ y(b) = \theta(u) = \beta $$<p>or</p>
-<p>$ r(u) \equiv \theta(u) - \beta = 0 $</p>
-<p>where $r(u)$ is the residual function. This problem can be thus solved using some of the methods previously seen for the root-finding problem.</p>
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-<h2 id="Example-3">Example 3<a class="anchor-link" href="#Example-3"> </a></h2>
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-<p>A very simplified model of interior solid planets consists of a set of spherically symmetric radial layers, where the next properties are computed: density $\rho(r)$, enclosed mass $m(r)$, gravity $g(r)$ and pressure $P(r)$. Each of these properties are assumed to be only radially dependent. The set of equations that rules the planetary interior is:</p>
-<p><strong>Hydrostatic equilibrium equation</strong></p>
-$$\frac{dP}{dr} = -\rho(r)g(r)$$<p><strong>Adams-Williamson equation</strong></p>
-$$\frac{dg}{dr} = 4\pi G\rho(r) - \frac{2Gm(r)}{r^3}$$<p><strong>Continuity equation</strong></p>
-$$\frac{dm}{dr} = 4\pi r^2 \rho(r)$$<p><strong>Equation of state</strong></p>
-$$\frac{d\rho}{dr} = -\frac{\rho(r)^2g(r)}{K_s}$$<p>For accurate results the term $K_s$, called the adiabatic bulk modulus, is temperature and radii dependent. However, for the sake of simplicity we shall assume a constant value.</p>
-<p>Solving simultaneously the previous set of equations, we can find the complete internal structure of a planet.</p>
-<p>We have four functions to be determined and four equations, so the problem is solvable. It is only necessary to provide a set of boundary conditions of the form:</p>
-$$ \rho(R) = \rho_{surf},\ \ \ m(R) = M_p, \ \ \ g(R) = g_{surf},\ \ \ P(R) = P_{atm} $$<p>where $R$ is the planetary radius, $\rho_{surf}$ the surface density, $M_p$ the mass of the planet, $g_{surf}$ the surface gravity and $P_{atm}$ the atmospheric pressure. However, there is a problem, we do not know the planetary radius $R$, so an extra condition is required to determine this value. This is reached through the physical condition $m(0) = 0$, this is, the enclosed mass at a radius $r = 0$ (center of the planet) must be 0.</p>
-<p>The two-value boundary nature of this problem lies then in fitting the mass function at $m(R) = M_p$ and at $m(0) = 0$. To do so, let's call the residual mass $m(0) = M_r$. This value should depend on the chosen radius $R$, so a large value would imply a mass defect $M_r(R)&lt;0$, and a small value a mass excess $M_r(0)&gt;0$. The problem is then solving the radius $R$ for which $m(0) = M_r(R) = 0$. This can be done by using the bisection method.</p>
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-<p>For this problem, we are going to assume an one-layer planet made of perovskite, so $K_s \approx 200\ GPa$. A planet mass equal to earth, so $M_p = 5.97219 \times 10^{24}\ kg$, a surface density $\rho_{surf} = 3500\ kg/m^3$ and a atmospheric pressure of $P_{atm} = 1\ atm = 1\times 10^5\ Pa$.</p>
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-<h2 id="Implementation-of-the-pendulum-phase-space">Implementation of the pendulum phase space<a class="anchor-link" href="#Implementation-of-the-pendulum-phase-space"> </a></h2>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#RK2 integrator</span>
-<span class="k">def</span> <span class="nf">RK2_step</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">h</span> <span class="p">):</span>
-    <span class="c1">#Creating solutions</span>
-    <span class="n">K0</span> <span class="o">=</span> <span class="n">h</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-    <span class="n">K1</span> <span class="o">=</span> <span class="n">h</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">t</span> <span class="o">+</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">h</span><span class="p">,</span> <span class="n">y</span> <span class="o">+</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">K0</span><span class="p">)</span>
-    <span class="n">y1</span> <span class="o">=</span> <span class="n">y</span> <span class="o">+</span> <span class="n">K1</span>
-    <span class="c1">#Returning solution</span>
-    <span class="k">return</span> <span class="n">y1</span>
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-<p>The phase space of a dynamical system is a space in which all the possible states of that system are univocally represented. For the case of the pendulum, a complete state of the system is given by the set $(\theta, \omega)$, so its phase space is two-dimensional. In order to explore all the possible states, we are going to generate a set of initial conditions and integrate them.</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#========================================================</span>
-<span class="c1">#Defining parameters</span>
-<span class="c1">#========================================================</span>
-<span class="c1">#Gravity</span>
-<span class="n">g</span> <span class="o">=</span> <span class="mf">9.8</span>
-<span class="c1">#Pendulum&#39;s lenght</span>
-<span class="n">l</span> <span class="o">=</span> <span class="mf">1.0</span>
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-<span class="c1">#Number of initial conditions</span>
-<span class="n">Nic</span> <span class="o">=</span> <span class="mi">1000</span>
-<span class="c1">#Maxim angular velocity</span>
-<span class="n">omega_max</span> <span class="o">=</span> <span class="mi">8</span>
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-<span class="c1">#Maxim time of integration</span>
-<span class="n">tmax</span> <span class="o">=</span> <span class="mi">6</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span>
-<span class="c1">#Timestep</span>
-<span class="n">h</span> <span class="o">=</span> <span class="mf">0.01</span>
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-<span class="c1">#========================================================</span>
-<span class="c1">#Dynamical function of the system</span>
-<span class="c1">#========================================================</span>
-<span class="k">def</span> <span class="nf">function</span><span class="p">(</span> <span class="n">t</span><span class="p">,</span> <span class="n">y</span> <span class="p">):</span>
-    <span class="c1">#Using the convention y = [theta, omega]</span>
-    <span class="n">theta</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">omega</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="c1">#Derivatives</span>
-    <span class="n">dtheta</span> <span class="o">=</span> <span class="n">omega</span>
-    <span class="n">domega</span> <span class="o">=</span> <span class="o">-</span><span class="n">g</span><span class="o">/</span><span class="n">l</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">dtheta</span><span class="p">,</span> <span class="n">domega</span><span class="p">])</span>
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-<span class="c1">#Generating set of initial conditions</span>
-<span class="c1">#========================================================</span>
-<span class="n">theta0s</span> <span class="o">=</span> <span class="o">-</span><span class="mi">4</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="n">Nic</span><span class="p">)</span><span class="o">*</span><span class="mi">8</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span>
-<span class="n">omega0s</span> <span class="o">=</span> <span class="o">-</span><span class="n">omega_max</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="n">Nic</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span><span class="o">*</span><span class="n">omega_max</span>
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-<span class="c1">#Integrating and plotting the solution for each IC</span>
-<span class="c1">#========================================================</span>
-<span class="c1">#Setting figure</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">jj</span><span class="o">=</span><span class="mi">0</span>
-<span class="k">for</span> <span class="n">theta0</span><span class="p">,</span> <span class="n">omega0</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">theta0s</span><span class="p">,</span> <span class="n">omega0s</span><span class="p">):</span>
-    <span class="c1">#Arrays for solutions</span>
-    <span class="n">time</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,]</span>
-    <span class="n">theta</span> <span class="o">=</span> <span class="p">[</span><span class="n">theta0</span><span class="p">,]</span>
-    <span class="n">omega</span> <span class="o">=</span> <span class="p">[</span><span class="n">omega0</span><span class="p">,]</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">tmax</span><span class="o">/</span><span class="n">h</span><span class="p">)),</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span> <span class="mi">0</span><span class="p">,</span> <span class="n">tmax</span><span class="p">,</span> <span class="n">h</span> <span class="p">)):</span>
-        <span class="c1">#Building current condition</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="n">theta</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">omega</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span>
-        <span class="c1">#Integrating the system</span>
-        <span class="n">thetai</span><span class="p">,</span> <span class="n">omegai</span> <span class="o">=</span> <span class="n">RK2_step</span><span class="p">(</span> <span class="n">function</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">h</span> <span class="p">)</span>
-        <span class="c1">#Appending new components</span>
-        <span class="n">theta</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">thetai</span> <span class="p">)</span>
-        <span class="n">omega</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">omegai</span> <span class="p">)</span>
-        <span class="n">time</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">t</span> <span class="p">)</span>
-        <span class="c1">#if i==10:</span>
-        <span class="c1">#    break</span>
-    <span class="c1">#Plotting solution</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">theta</span><span class="p">,</span> <span class="n">omega</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">color</span> <span class="o">=</span> <span class="s2">&quot;blue&quot;</span> <span class="p">)</span>
-    <span class="k">if</span> <span class="n">jj</span><span class="o">==</span><span class="mi">50</span><span class="p">:</span>
-        <span class="k">break</span>
-    <span class="n">jj</span><span class="o">=</span><span class="n">jj</span><span class="o">+</span><span class="mi">1</span>
-
-<span class="c1">#Format of figure</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span> <span class="s2">&quot;$</span><span class="se">\\</span><span class="s2">theta$&quot;</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">18</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span> <span class="s2">&quot;$\omega$&quot;</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">18</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span> <span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span> <span class="p">(</span><span class="o">-</span><span class="n">omega_max</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="n">omega_max</span><span class="o">+</span><span class="mi">3</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span> <span class="s2">&quot;Phase space of a pendulum&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-</pre></div>
-
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-<p>Due to the discrete, and sometimes sparse, nature of experiments and observations, data taking procedures will always produce discrete data as well. Even, as we have seen before, information only can be discretely presented into a computer due to the binary representation. However, when we are dealing with physical models, continuous and smooth properties are of course preferred. Interpolation techniques allow then to recover a continuous field from sparse datasets. Throughout this section we shall cover some of these interpolation methods.</p>
-<hr>
-<p><!-- __Homework__: https://classroom.github.com/a/SB94rDVJ --></p>
-<h2 id="Bibliography:">Bibliography:<a class="anchor-link" href="#Bibliography:"> </a></h2><p>[1a] Gonzalo Galiano Casas, Esperanza García Gonzalo <a href="https://www.unioviedo.es/compnum/expositive/handbook/metnum.pdf">Numerical Computation</a> - <a href="https://drive.google.com/file/d/1gwHYLx5aBf3oXRhmGeG-n1toTtAUhvfX/view?usp=sharing">GD</a> - <a href="https://www.unioviedo.es/compnum/index.php/english-menu">Web page with notebooks</a><br/>
-[1g] Mo Mu, <a href="https://www.math.ust.hk/~mamu/courses/230/course.htm">MATH 3311: Introduction to Numerical Methods</a> <a href="https://drive.google.com/drive/folders/1vv38SS7Zpw8mOHG7Kq_3lZr6o9H4xOYP?usp=sharing">GD</a> - <a href="https://www.math.ust.hk/~mamu/courses/231/Slides/CH03_1B.pdf">Demostration Error in Lagrange Polynomials</a><br/>
-[1h] Zhiliang Xu, <a href="https://www3.nd.edu/~zxu2/ACMS40390-F16.html">ACMS 40390: Fall 2016 - Numerical Analysis</a> <a href="https://drive.google.com/drive/folders/1Z9Fws1v34PuPdGAp_mBCWmTvZM_Qd6Xk?usp=sharing">GD</a>&lt;/br&gt;</p>
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-<li><a href="#NumPy-Polynomials">NumPy polynomials</a></li>
-<li><a href="#Linear-Interpolation">Linear Interpolation</a><ul>
-<li><a href="#Steps-LI">Steps</a></li>
-<li><a href="#Example-1">Example 1</a></li>
-</ul>
-</li>
-<li><a href="#Lagrange-Polynomial">Lagrange Polynomial</a><ul>
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-<li><a href="#Activity-LP">Activity</a></li>
-</ul>
-</li>
-<li><a href="#Divided-Differences">Divided Differences</a><ul>
-<li><a href="#Example-2">Example 2</a></li>
-</ul>
-</li>
-<li><a href="#Hermite-Interpolation">Hermite Interpolation</a><ul>
-<li><a href="#Derivation-in-terms-of-divided-differences">Derivation in terms of divided differences</a></li>
-<li><a href="#Example-3">Example 3</a></li>
-</ul>
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-</ul>
-<hr>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%</span><span class="k">pylab</span> inline
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-<pre>Populating the interactive namespace from numpy and matplotlib
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span><span class="p">,</span> <span class="n">Markdown</span><span class="p">,</span> <span class="n">Latex</span><span class="p">,</span> <span class="n">Image</span> 
-<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">interpolate</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="c1"># JSAnimation import available at https://github.com/jakevdp/JSAnimation</span>
-<span class="c1">#from JSAnimation import IPython_display</span>
-<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">animation</span>
-<span class="kn">from</span> <span class="nn">IPython.core.display</span> <span class="kn">import</span> <span class="n">Image</span> 
-</pre></div>
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-<p>Pretty print inside colaboratory</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">IPython</span>
-
-<span class="k">def</span> <span class="nf">setup_typeset</span><span class="p">():</span>
-  <span class="sd">&quot;&quot;&quot;MathJax initialization for the current cell.</span>
-<span class="sd">  </span>
-<span class="sd">  This installs and configures MathJax for the current output.</span>
-<span class="sd">  &quot;&quot;&quot;</span>
-  <span class="n">IPython</span><span class="o">.</span><span class="n">display</span><span class="o">.</span><span class="n">display</span><span class="p">(</span><span class="n">IPython</span><span class="o">.</span><span class="n">display</span><span class="o">.</span><span class="n">HTML</span><span class="p">(</span><span class="s1">&#39;&#39;&#39;</span>
-<span class="s1">      &lt;script src=&quot;https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js?config=TeX-AMS_HTML-full,Safe&amp;delayStartupUntil=configured&quot;&gt;&lt;/script&gt;</span>
-<span class="s1">      &lt;script&gt;</span>
-<span class="s1">        (() =&gt; {</span>
-<span class="s1">          const mathjax = window.MathJax;</span>
-<span class="s1">          mathjax.Hub.Config({</span>
-<span class="s1">          &#39;tex2jax&#39;: {</span>
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-<span class="s1">            &#39;displayMath&#39;: [[&#39;$$&#39;, &#39;$$&#39;], [&#39;</span><span class="se">\\</span><span class="s1">[&#39;, &#39;</span><span class="se">\\</span><span class="s1">]&#39;]],</span>
-<span class="s1">            &#39;processEscapes&#39;: true,</span>
-<span class="s1">            &#39;processEnvironments&#39;: true,</span>
-<span class="s1">            &#39;skipTags&#39;: [&#39;script&#39;, &#39;noscript&#39;, &#39;style&#39;, &#39;textarea&#39;, &#39;code&#39;],</span>
-<span class="s1">            &#39;displayAlign&#39;: &#39;center&#39;,</span>
-<span class="s1">          },</span>
-<span class="s1">          &#39;HTML-CSS&#39;: {</span>
-<span class="s1">            &#39;styles&#39;: {&#39;.MathJax_Display&#39;: {&#39;margin&#39;: 0}},</span>
-<span class="s1">            &#39;linebreaks&#39;: {&#39;automatic&#39;: true},</span>
-<span class="s1">            // Disable to prevent OTF font loading, which aren&#39;t part of our</span>
-<span class="s1">            // distribution.</span>
-<span class="s1">            &#39;imageFont&#39;: null,</span>
-<span class="s1">          },</span>
-<span class="s1">          &#39;messageStyle&#39;: &#39;none&#39;</span>
-<span class="s1">        });</span>
-<span class="s1">        mathjax.Hub.Configured();</span>
-<span class="s1">      })();</span>
-<span class="s1">      &lt;/script&gt;</span>
-<span class="s1">      &#39;&#39;&#39;</span><span class="p">))</span>
-    
-<span class="k">def</span> <span class="nf">Polynomial_to_LaTeX</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Small function to print nicely the polynomial p as we write it in maths, in LaTeX code.&quot;&quot;&quot;</span>
-    <span class="n">coefs</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">coef</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>  <span class="c1"># List of coefficient, sorted by increasing degrees</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="s2">&quot;&quot;</span>  <span class="c1"># The resulting string</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">coefs</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>  <span class="c1"># Remove the trailing .0</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>  <span class="c1"># First coefficient, no need for X</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">res</span> <span class="o">+=</span> <span class="s2">&quot;</span><span class="si">{a}</span><span class="s2"> + &quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">a</span><span class="o">=</span><span class="n">a</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">a</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>  <span class="c1"># Negative a is printed like (a)</span>
-                <span class="n">res</span> <span class="o">+=</span> <span class="s2">&quot;(</span><span class="si">{a}</span><span class="s2">) + &quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">a</span><span class="o">=</span><span class="n">a</span><span class="p">)</span>
-            <span class="c1"># a = 0 is not displayed </span>
-        <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>  <span class="c1"># Second coefficient, only X and not X**i</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>  <span class="c1"># a = 1 does not need to be displayed</span>
-                <span class="n">res</span> <span class="o">+=</span> <span class="s2">&quot;x + &quot;</span>
-            <span class="k">elif</span> <span class="n">a</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">res</span> <span class="o">+=</span> <span class="s2">&quot;</span><span class="si">{a}</span><span class="s2"> \;x + &quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">a</span><span class="o">=</span><span class="n">a</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">a</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">res</span> <span class="o">+=</span> <span class="s2">&quot;(</span><span class="si">{a}</span><span class="s2">) \;x + &quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">a</span><span class="o">=</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="c1"># A special care needs to be addressed to put the exponent in {..} in LaTeX</span>
-                <span class="n">res</span> <span class="o">+=</span> <span class="s2">&quot;x^</span><span class="si">{i}</span><span class="s2"> + &quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="o">=</span><span class="s2">&quot;{</span><span class="si">%d</span><span class="s2">}&quot;</span> <span class="o">%</span> <span class="n">i</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">a</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">res</span> <span class="o">+=</span> <span class="s2">&quot;</span><span class="si">{a}</span><span class="s2"> \;x^</span><span class="si">{i}</span><span class="s2"> + &quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">a</span><span class="o">=</span><span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="o">=</span><span class="s2">&quot;{</span><span class="si">%d</span><span class="s2">}&quot;</span> <span class="o">%</span> <span class="n">i</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">a</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">res</span> <span class="o">+=</span> <span class="s2">&quot;(</span><span class="si">{a}</span><span class="s2">) \;x^</span><span class="si">{i}</span><span class="s2"> + &quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">a</span><span class="o">=</span><span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="o">=</span><span class="s2">&quot;{</span><span class="si">%d</span><span class="s2">}&quot;</span> <span class="o">%</span> <span class="n">i</span><span class="p">)</span>
-    <span class="k">return</span> <span class="s2">&quot;$&quot;</span> <span class="o">+</span> <span class="n">res</span><span class="p">[:</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span> <span class="o">+</span> <span class="s2">&quot;$&quot;</span> <span class="k">if</span> <span class="n">res</span> <span class="k">else</span> <span class="s2">&quot;&quot;</span>    
-</pre></div>
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-<h1 id="NumPy-Polynomials">NumPy Polynomials<a class="anchor-link" href="#NumPy-Polynomials"> </a></h1>
-</div>
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-<p>In Numpy there is an implementation of Polynomials. The object is initialized giving the polynomial coefficients:</p>
-<p>More information about this</p>
-<ul>
-<li><a href="https://perso.crans.org/besson/publis/notebooks/Demonstration%20of%20numpy.polynomial.Polynomial%20and%20nice%20display%20with%20LaTeX%20and%20MathJax%20(python3">LaTeX print</a>.html): </li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">])</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
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-<pre>   2
-1 x + 2 x - 3
-</pre>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;</span><span class="se">\t</span><span class="s1">hola</span><span class="se">\n</span><span class="s1">mundo</span><span class="se">\\</span><span class="s1">&#39;</span><span class="p">)</span>
-</pre></div>
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-</div>
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-<pre>	hola
-mundo\
-</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Polynomial_to_LaTeX</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-</pre></div>
-
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-<pre>&#39;$(-3) + 2 \\;x + x^{2}$&#39;</pre>
-</div>
-
-</div>
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-<p>Copy an Paster in a MarkDown cell: $(-3) + 2 \;x + x^{2}$</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Latex</span><span class="p">(</span><span class="n">Polynomial_to_LaTeX</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-</pre></div>
-
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-$(-3) + 2 \;x + x^{2}$
-</div>
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-<p>The numpy polynomial is automatically a function of its variable $x$</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p</span><span class="p">(</span><span class="mf">1.3</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<div class="output_wrapper">
-<div class="output">
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-<div class="output_text output_subarea output_execute_result">
-<pre>1.29</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>By default, the assigned the attribute <code>variable</code> is <code>x</code>:</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p</span><span class="o">.</span><span class="n">variable</span>
-</pre></div>
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-<pre>&#39;x&#39;</pre>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>which can be assigned at initialization</p>
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">],</span><span class="n">variable</span><span class="o">=</span><span class="s1">&#39;t&#39;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>   2
-1 t + 2 t - 3
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Change of variable</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">])</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>   2
-1 x + 2 x - 3
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p</span><span class="o">.</span><span class="n">coef</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([ 1,  2, -3])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">pp</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">coef</span><span class="p">,</span><span class="n">variable</span><span class="o">=</span><span class="s1">&#39;t&#39;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">pp</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>   2
-1 t + 2 t - 3
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Polynomial can be added but not multiplied, simplified or expanded</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p1</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;p1(x)=</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">p1</span><span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;*&#39;</span><span class="o">*</span><span class="mi">20</span><span class="p">)</span>
-<span class="n">p2</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;p2(x)=</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">p2</span><span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;*&#39;</span><span class="o">*</span><span class="mi">20</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;p1(x)+p2(x)=</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">p1</span><span class="o">+</span><span class="n">p2</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>p1(x)= 
-1 x + 1
-********************
-p2(x)= 
--1 x + 1
-********************
-p1(x)+p2(x)= 
-2
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p1</span><span class="o">+</span><span class="n">p2</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>poly1d([2])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">p1</span><span class="o">+</span><span class="n">p2</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre> 
-2
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>The object have in particular methods for <br/>
-<strong>Integration</strong>:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>   2
-1 x + 2 x - 3
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">])</span>
-<span class="nb">print</span><span class="p">(</span> <span class="n">p</span><span class="o">.</span><span class="n">integ</span><span class="p">()</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>        3     2
-0.3333 x + 1 x - 3 x
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p</span><span class="o">.</span><span class="n">integ</span><span class="p">()</span><span class="o">.</span><span class="n">coef</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([ 0.33333333,  1.        , -3.        ,  0.        ])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Derivatives</strong></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span> <span class="n">p</span><span class="o">.</span><span class="n">deriv</span><span class="p">()</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre> 
-2 x + 2
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>roots</strong>:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">setup_typeset</span><span class="p">()</span> <span class="c1">#active colab pretty print</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">roots</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea ">
-
-<script src="https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js?config=TeX-AMS_HTML-full,Safe&delayStartupUntil=configured"></script>
-<script>
-  (() => {
-    const mathjax = window.MathJax;
-    mathjax.Hub.Config({
-    'tex2jax': {
-      'inlineMath': [['$', '$'], ['\(', '\)']],
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-      'processEnvironments': true,
-      'skipTags': ['script', 'noscript', 'style', 'textarea', 'code'],
-      'displayAlign': 'center',
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-    'HTML-CSS': {
-      'styles': {'.MathJax_Display': {'margin': 0}},
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-      'imageFont': null,
-    },
-    'messageStyle': 'none'
-  });
-  mathjax.Hub.Configured();
-})();
-</script>
-
-</div>
-
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>[-3.  1.]
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Latex</span><span class="p">(</span> <span class="s1">&#39;$p(</span><span class="si">{}</span><span class="s1">)$=</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">roots</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="mi">1</span><span class="p">),</span>
-                                      <span class="n">p</span><span class="p">((</span><span class="n">p</span><span class="o">.</span><span class="n">roots</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="p">)</span>  <span class="p">)</span> <span class="p">)</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_latex output_subarea output_execute_result">
-$p(-3.0)$=1.7763568394002505e-15
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Latex</span><span class="p">(</span> <span class="sa">f</span><span class="s1">&#39;$p(</span><span class="si">{</span><span class="nb">round</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">roots</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="mi">0</span><span class="p">)</span><span class="si">}</span><span class="s1">)$=</span><span class="si">{</span><span class="n">p</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">roots</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="si">}</span><span class="s1">&#39;</span> <span class="p">)</span>
-                                    
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_latex output_subarea output_execute_result">
-$p(-3.0)$=1.7763568394002505e-15
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>It is possible to define polynomial by given the list of roots and</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">([</span><span class="o">-</span><span class="mf">246.2</span><span class="p">,</span><span class="o">-</span><span class="mi">40</span><span class="p">,</span><span class="mi">40</span><span class="p">,</span><span class="mf">246.2</span><span class="p">],</span><span class="n">r</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>   4             2
-1 x - 6.221e+04 x + 9.698e+07
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p</span><span class="p">(</span><span class="o">-</span><span class="mi">40</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>1.4901161193847656e-08</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>For further details check the official help:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span>np.poly1d<span class="o">?</span>
-</pre></div>
-
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-</div>
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-</div>
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Activity</strong>:  Movement with uniform acceleration</p>
-<ol>
-<li>Define a polynomial for the movement with uniform acceleration:
-\begin{align}
-x(t)=x_0+v_0 (t-t_0)+\tfrac{1}{2} a (t-t_0)^2 \,,
-\end{align}</li>
-<li>Use the previous formula expressed as polynomial of degree 2, to solve the following problem with <code>np.poly1d</code>: <ul>
-<li>A car departs from rest with a constant acceleration of $6~\text{m}\cdot\text{s}^{-2}$ and travels through a flat and straight road. 10 seconds later a second pass for the same starting point and in the same direction with an initial speed of $10~\text{m}\cdot\text{s}^{-1}$ and a constant acelleration of $10~\text{m}\cdot\text{s}^{-2}$. Find the time and distance at which the two cars meet.</li>
-</ul>
-</li>
-</ol>
-<p><em>Hint</em>. 
-\begin{align}
-x(t)=x_0-v_0t_0+\frac{1}{2}at_0^2 +(v_0-at_0)t+\tfrac{1}{2} a t^2 
-\end{align}</p>
-
-</div>
-</div>
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x0</span><span class="o">=</span><span class="mi">0</span>
-<span class="n">t0</span><span class="o">=</span><span class="mi">0</span>
-<span class="n">v0</span><span class="o">=</span><span class="mi">0</span>
-<span class="n">a</span><span class="o">=</span><span class="mi">6</span> <span class="c1">#m/s^2</span>
-<span class="n">x_1</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">([</span><span class="mf">0.5</span><span class="o">*</span><span class="n">a</span><span class="p">,</span><span class="n">v0</span><span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="n">t0</span><span class="p">,</span><span class="n">x0</span><span class="o">-</span><span class="n">v0</span><span class="o">*</span><span class="n">t0</span><span class="o">+</span><span class="mf">0.5</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">t0</span><span class="o">**</span><span class="mi">2</span><span class="p">],</span><span class="n">variable</span><span class="o">=</span><span class="s1">&#39;t&#39;</span><span class="p">)</span>
-<span class="n">x0</span><span class="o">=</span><span class="mi">0</span>
-<span class="n">t0</span><span class="o">=</span><span class="mi">10</span> <span class="c1">#s</span>
-<span class="n">v0</span><span class="o">=</span><span class="mi">10</span> <span class="c1">#m/s</span>
-<span class="n">a</span><span class="o">=</span><span class="mi">10</span> <span class="c1">#m/s^2</span>
-<span class="n">x_2</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">([</span><span class="mf">0.5</span><span class="o">*</span><span class="n">a</span><span class="p">,</span><span class="n">v0</span><span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="n">t0</span><span class="p">,</span><span class="n">x0</span><span class="o">-</span><span class="n">v0</span><span class="o">*</span><span class="n">t0</span><span class="o">+</span><span class="mf">0.5</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">t0</span><span class="o">**</span><span class="mi">2</span><span class="p">],</span><span class="n">variable</span><span class="o">=</span><span class="s1">&#39;t&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-</div>
-</div>
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-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">x_1</span><span class="p">)</span>
-</pre></div>
-
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-<pre>   2
-3 t
-</pre>
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">x_2</span><span class="p">)</span>
-</pre></div>
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-<pre>   2
-5 t - 90 t + 400
-</pre>
-</div>
-</div>
-</div>
-</div>
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-</div>
-</div>
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-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<!-- ### <font color=red>Solution</font>
-setup_typeset() #active colab pretty print
-def x(x0,t0,v0,a):
-    return np.poly1d( [0.5*a,v0-a*t0,x0-v0*t0+0.5*a*t0**2],
-                     variable='t'  )
-x1=x(0,0,0,6)
-x2=x(0,10,10,10)
-display(Latex('$x_1(t)=$'))
-print(x1) 
-display(Latex('$x_2(t)=$'))
-print(x2)
-t=np.linspace(0,45,100)
-plt.plot(t,x2(t)-x1(t))
-plt.grid()
-#plt.plot(t,(x2-x1)(t))
-#plt.grid()
-#Physica solution is the one after 10s
-Latex(r'meeting time $t_{\rm end}=$ %g s; meeting point $x_{\rm end}=$ %g m' 
-          %(  (x2-x1).r[0]  ,  x2(  (x2-x1).r[0]  )   )) 
-#plt.plot(t,x2(t),'ro')
-#plt.plot(t,x2(t),'k-')
-#plt.xlim(10,15)
-#plt.ylim(0,200)
-          -->
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Sea el tiempo de encuentro definido $t_e$
-$$x_1(t_e)=x_2(t_e)\,,$$
-que se interpreta como
-$$x_1(t_e)-x_2(t_e)=0\,,$$
-Puedo definir un nuevo polinomio
-$$
-x(t)=x_1(t)-x_2(t)
-$$
-Entonces $t_e$, es la raíz del polinomio $x(t)$</p>
-
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-</div>
-</div>
-</div>
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span><span class="o">=</span><span class="n">x_1</span><span class="o">-</span><span class="n">x_2</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<pre>    2
--2 x + 90 x - 400
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span><span class="o">.</span><span class="n">roots</span>
-</pre></div>
-
-    </div>
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-<pre>array([40.,  5.])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>La raíz 5 no es física porque el tiempo incial para el segundo carro es 10</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x_1</span><span class="p">[</span><span class="mi">40</span><span class="p">]</span>
-</pre></div>
-
-    </div>
-</div>
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-<pre>0</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x_2</span><span class="p">[</span><span class="mi">40</span><span class="p">]</span>
-</pre></div>
-
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-<pre>0</pre>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">setup_typeset</span><span class="p">()</span> <span class="c1">#active colab pretty print</span>
-<span class="k">def</span> <span class="nf">x</span><span class="p">(</span><span class="n">x0</span><span class="p">,</span><span class="n">t0</span><span class="p">,</span><span class="n">v0</span><span class="p">,</span><span class="n">a</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span> <span class="p">[</span><span class="mf">0.5</span><span class="o">*</span><span class="n">a</span><span class="p">,</span><span class="n">v0</span><span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="n">t0</span><span class="p">,</span><span class="n">x0</span><span class="o">-</span><span class="n">v0</span><span class="o">*</span><span class="n">t0</span><span class="o">+</span><span class="mf">0.5</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">t0</span><span class="o">**</span><span class="mi">2</span><span class="p">],</span>
-                     <span class="n">variable</span><span class="o">=</span><span class="s1">&#39;t&#39;</span>  <span class="p">)</span>
-<span class="n">x1</span><span class="o">=</span><span class="n">x</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span>
-<span class="n">x2</span><span class="o">=</span><span class="n">x</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span>
-<span class="n">display</span><span class="p">(</span><span class="n">Latex</span><span class="p">(</span><span class="s1">&#39;$x_1(t)=$&#39;</span><span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span> 
-<span class="n">display</span><span class="p">(</span><span class="n">Latex</span><span class="p">(</span><span class="s1">&#39;$x_2(t)=$&#39;</span><span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">x2</span><span class="p">)</span>
-<span class="n">t</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">45</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">x2</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">-</span><span class="n">x1</span><span class="p">(</span><span class="n">t</span><span class="p">))</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;$t$ [s]&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;$x_2(t)-x_1(t)$ [m]&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="c1">#plt.plot(t,(x2-x1)(t))</span>
-<span class="c1">#plt.grid()</span>
-<span class="c1">#Physica solution is the one after 10s</span>
-<span class="n">Latex</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;meeting time $t_{\rm end}=$ </span><span class="si">%g</span><span class="s1"> s; meeting point $x_{\rm end}=$ </span><span class="si">%g</span><span class="s1"> m&#39;</span> 
-          <span class="o">%</span><span class="p">(</span>  <span class="p">(</span><span class="n">x2</span><span class="o">-</span><span class="n">x1</span><span class="p">)</span><span class="o">.</span><span class="n">r</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>  <span class="p">,</span>  <span class="n">x2</span><span class="p">(</span>  <span class="p">(</span><span class="n">x2</span><span class="o">-</span><span class="n">x1</span><span class="p">)</span><span class="o">.</span><span class="n">r</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>  <span class="p">)</span>   <span class="p">))</span> 
-<span class="c1">#plt.plot(t,x2(t),&#39;ro&#39;)</span>
-<span class="c1">#plt.plot(t,x2(t),&#39;k-&#39;)</span>
-<span class="c1">#plt.xlim(10,15)</span>
-<span class="c1">#plt.ylim(0,200)          </span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<div class="output">
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-<div class="output_area">
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-<div class="output_html rendered_html output_subarea ">
-
-<script src="https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js?config=TeX-AMS_HTML-full,Safe&delayStartupUntil=configured"></script>
-<script>
-  (() => {
-    const mathjax = window.MathJax;
-    mathjax.Hub.Config({
-    'tex2jax': {
-      'inlineMath': [['$', '$'], ['\(', '\)']],
-      'displayMath': [['$$', '$$'], ['\[', '\]']],
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-      'styles': {'.MathJax_Display': {'margin': 0}},
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-      // Disable to prevent OTF font loading, which aren't part of our
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-      'imageFont': null,
-    },
-    'messageStyle': 'none'
-  });
-  mathjax.Hub.Configured();
-})();
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-$x_1(t)=$
-</div>
-
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>   2
-3 t
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_latex output_subarea ">
-$x_2(t)=$
-</div>
-
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>   2
-5 t - 90 t + 400
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_latex output_subarea output_execute_result">
-meeting time $t_{\rm end}=$ 40 s; meeting point $x_{\rm end}=$ 4800 m
-</div>
-
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/interpolation_58_6.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">v</span><span class="o">=</span><span class="n">x_2</span><span class="o">.</span><span class="n">deriv</span><span class="p">()</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
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-<div class="jb_output_wrapper }}">
-<div class="output_area">
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-<pre> 
-10 x - 90
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">deriv</span><span class="p">())</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<div class="output_wrapper">
-<div class="output">
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-<div class="jb_output_wrapper }}">
-<div class="output_area">
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-<pre> 
-10
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h1 id="Linear-Interpolation">Linear Interpolation<a class="anchor-link" href="#Linear-Interpolation"> </a></h1>
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>When we have a set of discrete points of the form $(x_i, y_i)$ for $1\leq i \leq N$, the most natural way to obtain (approximate) any intermediate value is assuming points connected by lines. Let's assume a set of points $(x_i, y_i)$ such that $y_i = f(x_i)$ for an unknown function $f(x)$, if we want to approximate the value $f(x)$ for $x_i\leq x \leq x_{i+1}$, we construct an equation of a line passing through $(x_i,y_i)$ and $(x_{i+1},y_{i+1})$.</p>
-<p>The linear equation is</p>
-$$y=mx+b$$<p>where</p>
-$$m=\frac{y_{i+1}-y_i}{x_{i+1}-x_i} $$<p>and $b$ is obtained by evaluating with either $(x_i,y_i)$ or $(x_{i+1},y_{i+1})$</p>
-$$y=\frac{y_{i+1}-y_i}{x_{i+1}-x_i}x+b$$$$b=y_i-\frac{y_{i+1}-y_i}{x_{i+1}-x_i}x_i$$\begin{align}
-%$$  
-f(x)\approx y = &amp;\frac{y_{i+1}-y_i}{x_{i+1}-x_i}(x-x_i) + y_i \\
-=&amp;\frac{y_{i+1}-y_i}{x_{i+1}-x_i}x+\left[y_i-\frac{y_{i+1}-y_i}{x_{i+1}-x_i}x_i\right] \\
-%$$
-\end{align}<p>and this can be applied for any $x$ such that $x_0\leq x \leq x_N$ and where it has been assumed an ordered set $\left\{x_i\right\}_i$.</p>
-
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-<h2 id="Steps-LI">Steps LI<a class="anchor-link" href="#Steps-LI"> </a></h2>
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Once defined the mathematical basis behind linear interpolation, we proceed to establish the algorithmic steps for an implementation.</p>
-<ol>
-<li>Establish the dataset you want to interpolate, i.e. you must provide a set of the form $(x_i,y_i)$.</li>
-<li>Give the value $x$ where you want to approximate the value $f(x)$.</li>
-<li>Find the interval $[x_i, x_{i+1}]$ in which $x$ is embedded.</li>
-<li>Use the above expression in order to find $y=f(x)$.</li>
-</ol>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>To make the linear interpolation we will use</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">scipy</span> <span class="k">as</span> <span class="nn">sp</span>
-sp.interpolate.interp1d<span class="o">?</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>The option <code>kind</code> specifies the kind of interpolation:</p>
-<ul>
-<li><code>'linear', 'nearest', 'zero', 'slinear', 'quadratic', 'cubic'</code>,
-  where <code>'zero', 'slinear', 'quadratic' and 'cubic'</code> refer to a spline
-  interpolation of zeroth, first, second or third order) </li>
-<li>or as an
-  integer specifying the order of the spline interpolator to use.</li>
-</ul>
-<p>Default is <code>'linear'</code> correponding to integer 1.</p>
-
-</div>
-</div>
-</div>
-</div>
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-<h2 id="Example-1">Example 1<a class="anchor-link" href="#Example-1"> </a></h2>
-</div>
-</div>
-</div>
-</div>
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-<p>Sample the function $f(x) = \sin(x)$ between $0$ and $10$ using $N=10$ points (9 intervals). Plot both, the interpolation and the original function.</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">scipy</span> <span class="k">as</span> <span class="nn">sp</span>
-<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">interpolate</span> 
-</pre></div>
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-    </div>
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-</div>
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-<p><code>sp</code> reemplaza completamente a numpy con <code>np</code></p>
-
-</div>
-</div>
-</div>
-</div>
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-<p>Interpolation with 9 equal intervals</p>
-
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">n_points</span> <span class="o">=</span> <span class="mi">10</span>
-<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">n_points</span><span class="p">)</span>
-<span class="n">f</span><span class="o">=</span><span class="n">interpolate</span><span class="o">.</span><span class="n">interp1d</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">kind</span><span class="o">=</span><span class="s1">&#39;linear&#39;</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Plotting the results adding the real function with enough points</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Ninter</span> <span class="o">=</span> <span class="mi">100</span>
-<span class="n">X</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">Ninter</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="s1">&#39;ro&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;Data&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">),</span><span class="s1">&#39;k-&#39;</span><span class="p">,</span><span class="n">lw</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;linear interpolation&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">X</span><span class="p">),</span><span class="s1">&#39;c-&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;real function&#39;</span><span class="p">)</span>
-
-<span class="c1">#Formatting</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="mf">2.35</span><span class="p">,</span><span class="n">f</span><span class="p">(</span><span class="mf">2.35</span><span class="p">),</span><span class="s1">&#39;y*&#39;</span><span class="p">,</span><span class="n">markersize</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;sample point&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span> <span class="s2">&quot;Linear interpolation of $\sin(x)$&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span> <span class="s2">&quot;$x$&quot;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span> <span class="s2">&quot;$y$&quot;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-</pre></div>
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-<p>Other <code>kind</code> values: 0,1,2</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%</span><span class="k">pylab</span> inline
-</pre></div>
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-<pre>Populating the interactive namespace from numpy and matplotlib
-</pre>
-</div>
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-</div>
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-<div class="output_area">
-
-<div class="output_subarea output_stream output_stderr output_text">
-<pre>/usr/local/lib/python3.5/dist-packages/IPython/core/magics/pylab.py:160: UserWarning: pylab import has clobbered these variables: [&#39;f&#39;]
-`%matplotlib` prevents importing * from pylab and numpy
-  &#34;\n`%matplotlib` prevents importing * from pylab and numpy&#34;
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
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-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">n_points</span> <span class="o">=</span> <span class="mi">10</span>
-<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">n_points</span><span class="p">)</span>
-<span class="n">Ninter</span> <span class="o">=</span> <span class="mi">100</span>
-<span class="n">X</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">Ninter</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="s1">&#39;ro&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;Data&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">X</span><span class="p">),</span><span class="s1">&#39;c-&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;real function&#39;</span><span class="p">)</span>
-
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">interpolate</span><span class="o">.</span><span class="n">interp1d</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">kind</span><span class="o">=</span><span class="mi">0</span><span class="p">)(</span><span class="n">X</span><span class="p">),</span>
-         <span class="s1">&#39;g--&#39;</span><span class="p">,</span><span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;interpolation with horizontal lines&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">interpolate</span><span class="o">.</span><span class="n">interp1d</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">kind</span><span class="o">=</span><span class="mi">1</span><span class="p">)(</span><span class="n">X</span><span class="p">),</span>
-         <span class="s1">&#39;k-&#39;</span><span class="p">,</span><span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;linear interpolation&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">interpolate</span><span class="o">.</span><span class="n">interp1d</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">kind</span><span class="o">=</span><span class="mi">2</span><span class="p">)(</span><span class="n">X</span><span class="p">),</span>
-         <span class="s1">&#39;r:&#39;</span><span class="p">,</span><span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;quadratic interpolation&#39;</span><span class="p">)</span>
-<span class="c1">#plt.plot(X,interpolate.interp1d( x,np.sin(x),kind=3)(X),</span>
-<span class="c1">#         &#39;k:&#39;,lw=1,label=&#39;quadratic interpolation&#39;)</span>
-
-<span class="c1">#Formatting</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="mf">2.35</span><span class="p">,</span><span class="n">f</span><span class="p">(</span><span class="mf">2.35</span><span class="p">),</span><span class="s1">&#39;y*&#39;</span><span class="p">,</span><span class="n">markersize</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;sample point&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span> <span class="s2">&quot;Linear interpolation of $\sin(x)$&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span> <span class="s2">&quot;$x$&quot;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span> <span class="s2">&quot;$y$&quot;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span> <span class="p">)</span>
-
-<span class="c1">#Formatting</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span> <span class="s2">&quot;Linear interpolation of $\sin(x)$&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span> <span class="s2">&quot;$x$&quot;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span> <span class="s2">&quot;$y$&quot;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-</pre></div>
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-<p>We can see that the data poinst are just joined by straight lines:</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f</span><span class="p">(</span><span class="mf">2.35</span><span class="p">),</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mf">2.35</span><span class="p">)</span>
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-<pre>(array(0.67198441), 0.7114733527908443)</pre>
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-<p>However, the object <code>f</code> behaves like a function. For example. We can evaluate both the real and the interpolated function in $x_0=3$</p>
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-<p><strong>Activity</strong></p>
-<p>Use the previous code and explore the behaviour of the Linear Interpolation algorithm when varying the number of data used.</p>
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-<!--
-N=20
-x=np.linspace(0,6,N)
-f=interpolate.interp1d( x,np.sin(x) )
-X=np.linspace(0,6,100)
-plt.plot(x,f(x),'ro')
-plt.plot(x,f(x),'k-',lw=3)
-plt.plot(X,np.sin(X),'c-')
-plt.plot(2.35,f(2.35),'y*',markersize=10)
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-<h2 id="Example:">Example:<a class="anchor-link" href="#Example:"> </a></h2><p>Generate three points that do not lie upon a stright line, and try to make a manual interpolation with a polynomial of degree two.</p>
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-<span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">({</span> <span class="s1">&#39;X&#39;</span><span class="p">:[</span><span class="o">-</span><span class="mf">2.4</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mf">21.3</span><span class="p">],</span>
-                  <span class="s1">&#39;Y&#39;</span><span class="p">:[</span><span class="o">-</span><span class="mf">10.</span><span class="p">,</span><span class="mf">8.</span><span class="p">,</span><span class="mf">3.</span><span class="p">]</span>
-                 <span class="p">}</span>  
-                <span class="p">)</span>
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-</style>
-<table border="1" class="dataframe">
-  <thead>
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-      <th></th>
-      <th>X</th>
-      <th>Y</th>
-    </tr>
-  </thead>
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-      <th>0</th>
-      <td>-2.4</td>
-      <td>-10.0</td>
-    </tr>
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-      <th>1</th>
-      <td>3.0</td>
-      <td>8.0</td>
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-      <th>2</th>
-      <td>21.3</td>
-      <td>3.0</td>
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-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-</pre></div>
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-</pre></div>
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-</pre></div>
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-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">f</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">),</span><span class="s1">&#39;b-&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="n">f</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span><span class="s1">&#39;y*&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-</pre></div>
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-<h3 id="Polynomial-object-in-numpy">Polynomial object in <code>numpy</code><a class="anchor-link" href="#Polynomial-object-in-numpy"> </a></h3><p>In <code>numpy</code> it is possible to define polynomials friom either its coefficients o its roots with <code>np.poly1d</code></p>
-
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-<p>Define a two degree polynomial from its roots:</p>
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-</pre></div>
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-<p>We try to make a fit by using an inverted parabola passing trough the tree points and using the roots as a guess.
-In fact, we can try with a polynomial of degree two with roots at 1 and 22.</p>
-<p>With $k$ we can flip the curve and reduce the maximum without change the roots</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">k</span><span class="o">=-</span><span class="mf">0.1</span>
-<span class="n">P</span><span class="o">=</span><span class="n">k</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">23</span><span class="p">],</span><span class="n">r</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">P</span><span class="p">)</span>
-</pre></div>
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-<pre>      2
--0.1 x + 2.3 x - 0
-</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">P</span><span class="o">.</span><span class="n">roots</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">30</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">(</span> <span class="n">x</span><span class="p">),</span><span class="s1">&#39;b-&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">30</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">15</span><span class="p">,</span><span class="mi">20</span><span class="p">)</span>
-</pre></div>
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-<pre>(-15.0, 20.0)</pre>
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-<p><strong>Activity</strong>: Scipy interpolation.</p>
-<p>Define an interplation function which passes throgh the three points by using <code>interp1D</code> of Scipy with several linear and quadratic curves between the points.</p>
-
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-<!-- 
-plt.plot(df.X,df.Y,'ro')
-x=sp.linspace(df.X.min(),df.X.max() )
-plt.plot(x, sp.interpolate.interp1d(df.X,df.Y,kind=1)(x),label='linear')
-plt.plot(x, sp.interpolate.interp1d(df.X,df.Y,kind=2)(x),label='qudratic')
-plt.legend()
-plt.grid()
-plt.xlabel( "$x$",size=15 )
-plt.ylabel( "$y$",size=15 )
-plt.xlim(-8,30)
-plt.ylim(-15,20)
--->
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-<h3 id="Interpolation-with-numpy">Interpolation with <code>numpy</code><a class="anchor-link" href="#Interpolation-with-numpy"> </a></h3><p><code>numpy</code> already include an interpolation function with polynomials called <code>np.polyfit</code></p>
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">polyfit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">deg</span><span class="p">,</span> <span class="n">rcond</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">full</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">w</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">cov</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
-</pre></div>
-<p>Least squares polynomial fit.</p>
-<p>Fit a polynomial <code>p(x) = p[0] * x**deg + ... + p[deg]</code> of degree <code>deg</code>
-to points <code>(x, y)</code>. Returns a vector of coefficients <code>p</code> that minimises
-the squared error.</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>To fit a set of <code>x</code> and <code>y</code> we need to specify the degree of the polynial to make the fit with the madatory argument: <code>deg</code></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Example</strong>: fit the points of the previous DataFrame with a polynomial of degree 3</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">coeffs</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">polyfit</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="n">deg</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">coeffs</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>array([-0.15217542,  3.42463858, -0.904337  ])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span><span class="o">.</span><span class="n">roots</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>array([22.2373041 ,  0.26724135])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">coeffs</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>         2
--0.1522 x + 3.425 x - 0.9043
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">coeffs</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">polyfit</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="n">deg</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-<span class="n">P</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">30</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">(</span> <span class="n">x</span><span class="p">),</span><span class="s1">&#39;b-&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">20</span><span class="p">,</span><span class="mi">20</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
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-<div class="jb_output_wrapper }}">
-<div class="output_area">
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-<pre>(-20.0, 20.0)</pre>
-</div>
-
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/interpolation_113_1.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Example:-Least-action">Example: Least action<a class="anchor-link" href="#Example:-Least-action"> </a></h2><p>see Least action Notebook: <a href="./least_action.ipynb">[local]</a> <a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/least_action.ipynb">[GitHub]</a></p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#np.poly1d?</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h1 id="Lagrange-Polynomial">Lagrange Polynomial<a class="anchor-link" href="#Lagrange-Polynomial"> </a></h1>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Algebraic polynomials are very special functions as they have properties like differentiability (unlike linear interpolation) and continuity that make them useful for approximations like interpolation. A Polynomial is defined as a function given by the general expression:</p>
-$$P_n(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$$<p>where $n$ is the polynomial degree.</p>
-<p>Another important property of polynomials is given by the <a href="http://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem">Weierstrass Approximation Theorem</a>, which states given a continuous function $f$ defined on a interval $[a,b]$, for all $\epsilon &gt;0$, there exits a polynomial $P(x)$ such that</p>
-$$|f(x) - P(x)|&lt;\epsilon\ \ \ \ \  \mbox{for all }\ x\ \mbox{ in }\ [a,b].$$<p>This theorem guarantees the existence of such a polynomial, however it is necessary to propose a scheme to build it.</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-<span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">({</span> <span class="s1">&#39;X&#39;</span><span class="p">:[</span><span class="mi">3</span><span class="p">,</span><span class="mf">21.3</span><span class="p">],</span>
-                  <span class="s1">&#39;Y&#39;</span><span class="p">:[</span><span class="mf">8.</span><span class="p">,</span><span class="mf">3.</span><span class="p">]</span>
-                 <span class="p">}</span>  
-                <span class="p">)</span>
-<span class="n">df</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
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-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>X</th>
-      <th>Y</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>3.0</td>
-      <td>8.0</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>21.3</td>
-      <td>3.0</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/interpolation_119_0.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Polinomio de interp. de grado <font color="blue">0</font></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">coeffs</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">polyfit</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="n">deg</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-
-<span class="n">P</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">30</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">(</span> <span class="n">x</span><span class="p">),</span><span class="s1">&#39;b-&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">12</span><span class="p">)</span>
-
-<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">coeffs</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
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-<div class="output_area">
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-<div class="output_subarea output_stream output_stdout output_text">
-<pre> 
-5.5
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/interpolation_121_1.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Polinomio de interp. de grado <font color="blue">1</font></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">coeffs</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">polyfit</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="n">deg</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-
-<span class="n">P</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">30</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">(</span> <span class="n">x</span><span class="p">),</span><span class="s1">&#39;b-&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">12</span><span class="p">)</span>
-
-<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">coeffs</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
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-<div class="jb_output_wrapper }}">
-<div class="output_area">
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-<div class="output_subarea output_stream output_stdout output_text">
-<pre> 
--0.2732 x + 8.82
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/interpolation_123_1.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Polinomio de interp. de grado <font color="blue">2</font></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">coeffs</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">polyfit</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="n">deg</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-
-<span class="n">P</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">30</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">(</span> <span class="n">x</span><span class="p">),</span><span class="s1">&#39;b-&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">12</span><span class="p">)</span>
-
-<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">coeffs</span><span class="p">))</span>
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--0.008617 x - 0.06383 x + 8.269
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-  exec(code_obj, self.user_global_ns, self.user_ns)
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-<span class="n">df</span><span class="o">.</span><span class="n">to_csv</span><span class="p">(</span><span class="s1">&#39;../data/interpolation.csv&#39;</span><span class="p">,</span><span class="n">index</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
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-      <td>10.0</td>
-      <td>6.5</td>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">coeffs</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">polyfit</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="n">deg</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-
-<span class="n">P</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">30</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">(</span> <span class="n">x</span><span class="p">),</span><span class="s1">&#39;b-&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">12</span><span class="p">)</span>
-
-<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">coeffs</span><span class="p">))</span>
-</pre></div>
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-<pre>           2
--0.005216 x - 0.1465 x + 8.486
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">coeffs</span><span class="p">))</span>
-</pre></div>
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--0.005216 x - 0.1465 x + 8.486
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-<h2 id="Derivation">Derivation<a class="anchor-link" href="#Derivation"> </a></h2>
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-<p>Let's suppose a well-behaved yet unknown function $f$ and two points $(x_0,y_0)$ and $(x_1,y_1)$ for which $f(x_0) = y_0$ and $f(x_1) = y_1$. With this information we can build a first-degree polynomial that passes through both points by using the last equation in sec. <a href="interpolation.ipynb#Linear-Interpolation">Linear Interpolation</a>, we have</p>
-$$P_1(x) = \left[ \frac{y_{1}-y_0}{x_{1}-x_0} \right]x + \left[ y_0 - \frac{y_{1}-y_0}{x_{1}-x_0}x_0 \right]$$<p>We can readily rewrite this expression like:
-\begin{align}
-P_1(x) =&amp; \frac{y_{1}}{x_{1}-x_0} x- \frac{y_0}{x_{1}-x_0} x + y_0 -\frac{y_{1}}{x_{1}-x_0}x_0 +  \frac{y_0}{x_{1}-x_0}x_0 \nonumber\\
-=&amp; \left[1 - \frac{x}{x_{1}-x_0}  +  \frac{x_0}{x_{1}-x_0}\right]y_0+
-\left[\frac{x}{x_{1}-x_0}  -\frac{x_0}{x_{1}-x_0}\right]y_1    \nonumber\\
- =&amp; \left[\frac{x_1-x_0}{x_{1}-x_0} - \frac{x}{x_{1}-x_0}  +  \frac{x_0}{x_{1}-x_0}\right]y_0+
-\left[\frac{x}{x_{1}-x_0}  -\frac{x_0}{x_{1}-x_0}\right]y_1    \nonumber\\
-=&amp; \left[\frac{x_1-x}{x_{1}-x_0}\right]y_0+
-\left[\frac{x-x_0}{x_{1}-x_0}\right]y_1    \,.
-\end{align}
-In this way
-$$P_1(x) = L_0(x)f(x_0) + L_1(x)f(x_1)$$</p>
-<p>where we define the functions $L_0(x)$ and $L_1(x)$ as:</p>
-$$L_0(x) = \frac{x-x_1}{x_0-x_1} \mbox{ and } L_1(x) = \frac{x-x_0}{x_1-x_0}$$<p>Note that</p>
-$$L_0(x_0) = 1,\ \ \ L_0(x_1) = 0,\ \ \ L_1(x_0) = 0,\ \ \ L_1(x_1) = 1$$<p>implying:</p>
-$$P_1(x_0) = f(x_0) = y_0$$$$P_1(x_1) = f(x_1) = y_1$$<p>Although all this procedure may seem unnecessary for a polynomial of degree 1, a generalization to polynomials of larger degrees is now possible.</p>
-
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-<h2 id="General-case">General case<a class="anchor-link" href="#General-case"> </a></h2><p>Let's assume again a well-behaved and unknown function $f$ sampled by using a set of $n+1$ data $(x_m,y_m)$ ($0\leq m \leq n$).
-We call the set of $[x_0,x_1,\ldots,x_n]$ as the <em>node</em> points of the <em>interpolation polynomial in the Lagrange form</em>, $P_n(x)$, where:
-$$f(x)\approx P_n(x)\,,$$</p>
-$$P_n(x) = \sum_{i=0}^n f(x_i)L_{n,i}(x) = \sum_{i=0}^n y_iL_{n,i}(x)$$<p>Such that
-$$f(x_i)= P_n(x_i)\,,$$</p>
-<p>We need to find the <em>Lagrange polynomials</em>,  $L_{n,i}(x)$, such that 
-$$L_{n,i}(x_i) = 1\,,\qquad\text{and}\,,\qquad L_{n,i}(x_j) = 0\quad\text{for $i\neq j$}$$ 
-A function that satisfies this criterion is</p>
-$$L_{n,i}(x) = \prod_{\begin{smallmatrix}m=0\\ m\neq i\end{smallmatrix}}^n \frac{x-x_m}{x_i-x_m} =\frac{(x-x_0)}{(x_i-x_0)}\frac{(x-x_1)}{(x_i-x_1)}\cdots \frac{(x-x_{i-1})}{(x_i-x_{i-1})}\underbrace{\frac{}{}}_{m\ne i}
-\frac{(x-x_{i+1})}{(x_i-x_{i+1})} \cdots \frac{(x-x_{n-1})}{(x_i-x_{n-1})}\frac{(x-x_n)}{(x_i-x_n)}  $$<p>Please note that in the expansion the term $(x-x_i)$ does not appears in both the numerator and the denominator as stablished in the productory condition $m\neq i$.</p>
-<p>Moreower
-$$L_{n,i}(x_i) = \prod_{\begin{smallmatrix}m=0\\ m\neq i\end{smallmatrix}}^n \frac{x_i-x_m}{x_i-x_m} =1$$
-and, for $j\ne i$
-$$L_{n,i}(x_j) = \prod_{\begin{smallmatrix}m=0\\ m\neq i\end{smallmatrix}}^n \frac{x_j-x_m}{x_i-x_m} =\frac{(x_j-x_0)}{(x_i-x_0)}\cdots \frac{(\boldsymbol{x_j}-\boldsymbol{x_j})}{(x_i-x_j)}\cdots\frac{(x_j-x_n)}{(x_i-x_n)}=0.$$</p>
-<p>Then, the polynomial of $n$th-degree $P_n(x)$ will satisfy the definitory property for a interpolating polynomial, i.e. $P_n(x_i) = y_i$ for any $i$ and it is called the <em>interpolation Polynomial in the Lagrange form</em>.</p>
-<p>Check <a href="./LagrangePoly.ipynb">this implementation in sympy</a> [<a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/LagrangePoly.ipynb">View in Colaboratory</a>] where both the interpolating polynomial and the Lagrange polynomials are defined.</p>
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-<p><strong>Further details at:</strong>
-<a href="https://en.wikipedia.org/wiki/Lagrange_polynomial">Wikipedia</a></p>
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-<h3 id="Example:">Example:<a class="anchor-link" href="#Example:"> </a></h3><p>Obtain the Lagrange Polynomials for a Interpolation polynomial of degree 1.</p>
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-<p>$i=0$, $n=1$
-$$ L_{1,0}=\prod_{\begin{smallmatrix}m=0\\ m\neq 0\end{smallmatrix}}^1 \frac{x-x_m}{x_i-x_m}=\prod_{\begin{smallmatrix}m=1\end{smallmatrix}}^1 \frac{x-x_m}{x_0-x_m}=\frac{x-x_1}{x_0-x_1}$$
-$i=1$, $n=1$
-$$ L_{1,1}=\prod_{\begin{smallmatrix}m=0\\ m\neq 1\end{smallmatrix}}^1 \frac{x-x_m}{x_i-x_m}=\prod_{\begin{smallmatrix}m=0\end{smallmatrix}}^0 \frac{x-x_m}{x_1-x_m}=\frac{x-x_0}{x_1-x_0}$$</p>
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-<p>Until now we can only guarantee that $P_n(x_i)=f(x_i)$. To calculate the function from any $x$ in the interpolation interval we have the following Theorem (See <a href="https://www3.nd.edu/~zxu2/acms40390F12/Lec-3.1.pdf">here</a>)</p>
-<h2 id="Theorem">Theorem<a class="anchor-link" href="#Theorem"> </a></h2><p>Suppose $X_0,\ldots,x_n$ are distinct numbers in the interval $[a,b]$ and $f\in[a,b]$.Then for each $x$ in $[a,b]$, a number $\xi(x)$ between $x_0,\ldots,x_n$, and hence in $[a,b]$, exists with
-$$
-f(x)=P_n(x)+E_n(x)\,,\qquad \text{such that } E_n(x_i)=0\,,
-$$
-where the formula for the error bound is given by
-$$
-E_n(x) = {f^{n+1}(\xi(x)) \over (n+1)!} \cdot \prod_{i=0}^{n}\left(x-x_{i}\right)\,,
-$$
-$f^{(n+1)}$ is the $n+1$ derivative of $f$</p>
-<p>For a demostration see [1d] → <a href="https://www.math.ust.hk/~mamu/courses/231/Slides/CH03_1B.pdf">https://www.math.ust.hk/~mamu/courses/231/Slides/CH03_1B.pdf</a></p>
-<p>The specific calculation of the bounded error is to find the $\xi$ and $x$ such that
-$$
-\left|f(x)-P(x)\right| \leq \max_{\xi\in[a,b]}\left|\frac{f^{(n+1)}(\xi)}{(n+1) !}\right| \cdot \max_{x\in[a,b]}\left|\prod_{i=0}^{n}\left(x-x_{i}\right)\right|
-$$</p>
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-<h3 id="Exercise-interpolation">Exercise-interpolation<a class="anchor-link" href="#Exercise-interpolation"> </a></h3><p>Obtain the Lagrange Polynomials for a Interpolation polynomial of degree 2.</p>
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-<h3 id="Implementation-in-Scipy">Implementation in Scipy<a class="anchor-link" href="#Implementation-in-Scipy"> </a></h3>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">interpolate</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#interpolate.lagrange?</span>
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-      <th>0</th>
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-      <th>1</th>
-      <td>21.3</td>
-      <td>3.0</td>
-    </tr>
-  </tbody>
-</table>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">P</span><span class="o">=</span><span class="n">interpolate</span><span class="o">.</span><span class="n">lagrange</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">P</span><span class="p">)</span>
-</pre></div>
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-<pre>           2
--0.005216 x - 0.1465 x + 8.486
-</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">30</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">(</span> <span class="n">x</span><span class="p">),</span><span class="s1">&#39;b-&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">12</span><span class="p">)</span>
-</pre></div>
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-<pre>(0.0, 12.0)</pre>
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-<h3 id="Example-of-error-calculation">Example of error calculation<a class="anchor-link" href="#Example-of-error-calculation"> </a></h3><p>See details <a href="https://math.stackexchange.com/a/1151599/790274">here</a></p>
-<p>Consider $f(x) = \operatorname{e}^{2x} - x$ interpolation in the interval $[1,1.6]$</p>
-<ol>
-<li>Construct the Lagrange Polynomial for the points $x_0=1$, $x_1=1.25$, $x_2=1.6$. </li>
-<li>Find the approximate value at $f(1.5)$ and the error bound for the approximation</li>
-</ol>
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-</div>
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-<ol>
-<li>We start by defining the function</li>
-</ol>
-<p>Be sure that the function input will be an array</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Wrong result for a list</span>
-<span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mf">1.25</span><span class="p">,</span><span class="mf">1.6</span><span class="p">]</span>
-<span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-</pre></div>
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-<pre>array([2.71828183, 3.49034296, 4.95303242, 2.71828183, 3.49034296,
-       4.95303242])</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span><span class="o">+</span><span class="n">x</span>
-</pre></div>
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-<pre>[1, 1.25, 1.6, 1, 1.25, 1.6]</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="mi">2</span><span class="o">*</span><span class="n">x</span>
-</pre></div>
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-<pre>[1, 1.25, 1.6, 1, 1.25, 1.6]</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-</pre></div>
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-<pre>array([ 7.3890561 , 12.18249396, 24.5325302 ])</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#List operations are not well defined in general</span>
-<span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">]</span><span class="o">-</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">]</span>
-</pre></div>
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-<pre>
-<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
-<span class="ansi-red-fg">TypeError</span>                                 Traceback (most recent call last)
-<span class="ansi-green-fg">&lt;ipython-input-70-641c9f212de0&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span><span class="ansi-blue-fg">()</span>
-<span class="ansi-green-intense-fg ansi-bold">      1</span> <span class="ansi-red-fg">#List operations are not well defined in general</span>
-<span class="ansi-green-fg">----&gt; 2</span><span class="ansi-red-fg"> </span><span class="ansi-blue-fg">[</span><span class="ansi-cyan-fg">1</span><span class="ansi-blue-fg">,</span><span class="ansi-cyan-fg">2</span><span class="ansi-blue-fg">,</span><span class="ansi-cyan-fg">4</span><span class="ansi-blue-fg">]</span><span class="ansi-blue-fg">-</span><span class="ansi-blue-fg">[</span><span class="ansi-cyan-fg">1</span><span class="ansi-blue-fg">,</span><span class="ansi-cyan-fg">5</span><span class="ansi-blue-fg">,</span><span class="ansi-cyan-fg">6</span><span class="ansi-blue-fg">]</span>
-
-<span class="ansi-red-fg">TypeError</span>: unsupported operand type(s) for -: &#39;list&#39; and &#39;list&#39;</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    WARNING: all the parts of the function must be </span>
-<span class="sd">    numpy arrays in order to get the element by element sum.</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="c1"># Force x → array</span>
-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">-</span><span class="n">x</span>
-</pre></div>
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-<p>The interpolation polynomial is</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mf">1.25</span><span class="p">,</span><span class="mf">1.6</span><span class="p">]</span>
-<span class="n">P_2</span><span class="o">=</span><span class="n">interpolate</span><span class="o">.</span><span class="n">lagrange</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">P_2</span><span class="p">)</span>
-</pre></div>
-
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-<pre>       2
-26.85 x - 42.25 x + 21.78
-</pre>
-</div>
-</div>
-</div>
-</div>
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-<p>Test interpolation with one of the three points</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">P_2</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-</pre></div>
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-<pre>(10.932493960703491, 10.932493960703473)</pre>
-</div>
-
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-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>For the bounded error we start with the left part
-$$
-\max_{\xi\in[1,1.6]}\left|\frac{f'''(\xi)}{(3)!}\right| 
-$$
-where
-$$f'(x)=2\operatorname{e}^{2x}-1$$
-$$f''(x)=4\operatorname{e}^{2x}$$
-$$f'''(x)=8\operatorname{e}^{2x}$$
-The maximum is obtained for the last point $x_2$</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">fppp</span><span class="o">=</span><span class="mi">8</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-<span class="n">fppp</span>
-</pre></div>
-
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-<pre>196.26024157687482</pre>
-</div>
-
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-</div>
-</div>
-</div>
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-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>For the right part 
-$$
-\max_{x\in[a,b]}\left|\prod_{i=0}^{2}\left(x-x_{i}\right)\right|=
-$$
-we need just to build the numpy polynomial with roots at $(x_0,x_1,x_2)$
-$$
-p_e(x)=(x-x_0)(x-x_1)(x-x_2)\,,
-$$</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p_e</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">poly1d</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">r</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">X</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">x</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>  <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">p_e</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">p_e</span><span class="o">.</span><span class="n">deriv</span><span class="p">()(</span><span class="n">X</span><span class="p">))</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mf">0.02</span><span class="p">,</span><span class="mf">0.01</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-</pre></div>
-
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-</div>
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-<img src="../images/features/interpolation_159_0.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>And find the maximun of the absolute value of the critical points</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p_e</span><span class="o">.</span><span class="n">deriv</span><span class="p">()</span><span class="o">.</span><span class="n">roots</span>
-</pre></div>
-
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-</div>
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-<pre>array([1.45733844, 1.10932822])</pre>
-</div>
-
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-</div>
-</div>
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-</div>
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-<p>The point in the interval $[1,1.6]$ for which $|p_e(x)|$ is maximum corresponds to the root</p>
-
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-<span class="n">xmax</span>
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-<pre>1.4573384418151778</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">p_e</span><span class="p">(</span><span class="n">xmax</span><span class="p">))</span>
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-<pre>0.013527716754363928</pre>
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-<p>Or</p>
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-<pre>0.013527716754363928</pre>
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-<p>In this way, the maximum error for $p(x)$ is expected for $x_{\text{max}}=1.45733844$</p>
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-<p>The bounded error is then
-$$
-E_2(x)=\max_{\xi\in[1,1.6]}\left|\frac{f'''(\xi)}{(3)!}\right|\cdot\max_{x\in[a,b]}\left|\prod_{i=0}^{2}\left(x-x_{i}\right)\right|
-$$</p>
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-<span class="n">E_2</span>
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-<pre>0.4424921596991669</pre>
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-<span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-</pre></div>
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-<pre>18.585536923187668</pre>
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-<pre>18.832612316478652</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">P_2</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">-</span><span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
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-<pre>0.24707539329098438</pre>
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-<p>The real bounded error is</p>
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-<span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">P_2</span><span class="p">(</span><span class="n">xlin</span><span class="p">)</span><span class="o">-</span><span class="n">f</span><span class="p">(</span><span class="n">xlin</span><span class="p">))</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
-</pre></div>
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-<pre>0.26182823963596036</pre>
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-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
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-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
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-<h3 id="Implementation-in-sympy">Implementation in <code>sympy</code><a class="anchor-link" href="#Implementation-in-sympy"> </a></h3><p>For details see <a href="./LagrangePoly.ipynb">here</a></p>
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-<pre>Populating the interactive namespace from numpy and matplotlib
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-<span class="n">LP</span><span class="o">.</span><span class="n">lagrangePolynomial</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">)</span>
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-$\displaystyle - 0.00521578136549848 x^{2} - 0.146480556534234 x + 8.48638370189219$
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-$$P_n(x) =  \sum_{i=0}^n L_{n,i}(x) \, y_i$$
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-$\displaystyle 0.0559573894841558 x^{2} - 1.44173177757974 x + 11.8215788273818$
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-<table border="1" class="dataframe">
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-      <th>0</th>
-      <td>3.0</td>
-      <td>8.0</td>
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-      <td>10.0</td>
-      <td>6.5</td>
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-      <td>21.3</td>
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-<h2 id="Steps-LP">Steps LP<a class="anchor-link" href="#Steps-LP"> </a></h2>
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-<p>Once defined the formal procedure for constructing a Lagrange Polynomial, we proceed to describe the explicit algorithm:</p>
-<ol>
-<li>Give the working dataset $(x_i, y_i)$ and stablish how many points you have.</li>
-<li>Define the functions $L_{n,i}(x)$ in a general way.</li>
-<li>Add each of those terms as shown in last expression.</li>
-<li>Evaluate your result wherever you want.</li>
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-<p><strong>Activity</strong></p>
-<p>Along with the professor, write an implementation of the previous algorithm during classtime.</p>
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-<h2 id="Activity-LP">Activity LP<a class="anchor-link" href="#Activity-LP"> </a></h2>
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-<div style="float: right;" markdown="1">
-    <img src="http://universe-review.ca/I05-28-NGC7331.jpg">
-</div><p>One of the very first evidences of the existence of dark matter was the flat rotation curves of spiral galaxies. If we assume the total budget of mass of a galaxy is entirely made of luminous matter, the orbital circular velocity of stars around the galaxy plane should decay according to a keplerian potential. However this is not the case and the circular velocity barely decreases at larger radius, thus indicating the presence of a new non-visible matter component (dark matter). When it is necessary to determine how massive is the dark matter halo embedding a galaxy, an integration of the circular velocity is required. Nevertheless, due to the finite array of a CCD camera, only a discrete set of velocities can be measured and interpolation techniques are required.</p>
-<p>In this activity we will take a discrete dataset of the circular velocity as a function of the radius for the galaxy <a href="http://es.wikipedia.org/wiki/NGC_7331">NGC 7331</a> and perform both, a linear and a Lagrange interpolation. You can download the dataset from this <a href="https://raw.githubusercontent.com/sbustamante/ComputationalMethods/master/data/NGC7331.dat">link</a>.</p>
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-<p><a href="https://upload.wikimedia.org/wikipedia/commons/transcoded/3/33/Galaxy_rotation_under_the_influence_of_dark_matter.ogv/Galaxy_rotation_under_the_influence_of_dark_matter.ogv.360p.webm">Video</a></p>
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-<p><font color='blue'>
-**TRIVIA**<br/> 
-To which of two curves the real data approach better?
-</font></p>
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-<p>import os
-os.remove('trivia_results.txt')</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f</span><span class="o">=</span><span class="nb">open</span><span class="p">(</span><span class="s1">&#39;trivia_results.txt&#39;</span><span class="p">,</span><span class="s1">&#39;a&#39;</span><span class="p">)</span>
-<span class="n">AB</span><span class="o">=</span><span class="nb">input</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;&#39;&#39;A: to the curve &quot;velocity goes to zero when distance goes to infinity&quot;</span>
-<span class="s1">B: to the curve &quot;velocity goes to high constant when distance goes to infinity&quot;</span>
-<span class="s1">&#39;&#39;&#39;</span><span class="p">)</span>
-<span class="n">f</span><span class="o">.</span><span class="n">write</span><span class="p">(</span> <span class="s1">&#39;</span><span class="si">{}</span><span class="se">\n</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">AB</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">f</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">fr</span><span class="o">=</span><span class="nb">open</span><span class="p">(</span><span class="s1">&#39;trivia_results.txt&#39;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span> <span class="n">fr</span><span class="o">.</span><span class="n">read</span><span class="p">())</span>
-<span class="n">fr</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
-</pre></div>
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-<pre>C
-A
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-<p>os.remove('trivia_results.txt')</p>
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-<h3 id="Lets-us-check!">Lets us check!<a class="anchor-link" href="#Lets-us-check!"> </a></h3>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#DATA URL: </span>
-<span class="n">url</span><span class="o">=</span><span class="s1">&#39;https://raw.githubusercontent.com/restrepo/ComputationalMethods/master/data/NGC7331.csv&#39;</span>
-<span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="n">url</span><span class="p">)</span>
-<span class="n">df</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">5</span><span class="p">]</span>
-</pre></div>
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-<style scoped>
-    .dataframe tbody tr th:only-of-type {
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-</style>
-<table border="1" class="dataframe">
-  <thead>
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-      <th></th>
-      <th>r</th>
-      <th>v</th>
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-      <th>0</th>
-      <td>0.05</td>
-      <td>33.42496</td>
-    </tr>
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-      <th>1</th>
-      <td>0.10</td>
-      <td>71.70398</td>
-    </tr>
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-      <th>2</th>
-      <td>0.15</td>
-      <td>95.14708</td>
-    </tr>
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-      <th>3</th>
-      <td>0.20</td>
-      <td>107.32276</td>
-    </tr>
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-      <th>4</th>
-      <td>0.25</td>
-      <td>117.44285</td>
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-<p>Build expected data</p>
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-<span class="c1">#dff=dff.append(pd.DataFrame( {&#39;r&#39;:[5,10],&#39;v&#39;:[60,20]} )).reset_index(drop=True) # ,15,20,25,30 ,120,105,100,98</span>
-<span class="c1">#dff=dff.append(pd.DataFrame( {&#39;r&#39;:[3.5,4,10],&#39;v&#39;:[230,200,20]} )).reset_index(drop=True) # ,15,20,25,30 ,120,105,100,98</span>
-<span class="n">dff</span><span class="o">=</span><span class="n">dff</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span> <span class="p">{</span><span class="s1">&#39;r&#39;</span><span class="p">:[</span><span class="mf">3.5</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">9</span><span class="p">],</span><span class="s1">&#39;v&#39;</span><span class="p">:[</span><span class="mi">230</span><span class="p">,</span><span class="mi">200</span><span class="p">,</span><span class="mi">22</span><span class="p">]}</span> <span class="p">))</span><span class="o">.</span><span class="n">reset_index</span><span class="p">(</span><span class="n">drop</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="c1"># ,15,20,25,30 ,120,105,100,98</span>
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-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">dff</span><span class="o">.</span><span class="n">r</span><span class="p">,</span> <span class="n">dff</span><span class="o">.</span><span class="n">v</span><span class="p">,</span><span class="s1">&#39;b.&#39;</span><span class="p">)</span>
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-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">dff</span><span class="p">[</span><span class="s1">&#39;r&#39;</span><span class="p">],</span><span class="mf">1.</span><span class="o">/</span><span class="n">dff</span><span class="p">[</span><span class="s1">&#39;v&#39;</span><span class="p">],</span><span class="s1">&#39;b.&#39;</span><span class="p">)</span>
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-<span class="n">P</span><span class="o">=</span><span class="n">poly1d</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span><span class="n">variable</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">P</span><span class="p">)</span>
-</pre></div>
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-<pre>          2
-0.001084 r - 0.00594 r + 0.01169
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">r</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="mf">0.001</span><span class="p">),</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="mi">30</span> <span class="p">),</span><span class="mi">100</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">dff</span><span class="p">[</span><span class="s1">&#39;r&#39;</span><span class="p">],</span><span class="mf">1.</span><span class="o">/</span><span class="n">dff</span><span class="p">[</span><span class="s1">&#39;v&#39;</span><span class="p">],</span><span class="s1">&#39;b.&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span><span class="n">P</span><span class="p">(</span><span class="n">r</span><span class="p">),</span><span class="s1">&#39;k-&#39;</span><span class="p">)</span>
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-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">dff</span><span class="p">[</span><span class="s1">&#39;r&#39;</span><span class="p">],</span><span class="n">dff</span><span class="p">[</span><span class="s1">&#39;v&#39;</span><span class="p">],</span><span class="s1">&#39;b.&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span><span class="mi">1</span><span class="o">/</span><span class="n">P</span><span class="p">(</span><span class="n">r</span><span class="p">),</span><span class="s1">&#39;k-&#39;</span><span class="p">)</span>
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-<h3 id="Logarithmic-interpolation">Logarithmic interpolation<a class="anchor-link" href="#Logarithmic-interpolation"> </a></h3><p>See: <a href="https://stackoverflow.com/a/29359275/2268280">https://stackoverflow.com/a/29359275/2268280</a></p>
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-<h1 id="Appendix">Appendix<a class="anchor-link" href="#Appendix"> </a></h1><p>Thecnical details of interpolation functions: <a href="./interpolation_details.ipynb">interpolation_details.ipynb</a>
-<a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/interpolation_details.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<h1 id="Divided-Differences">Divided Differences<a class="anchor-link" href="#Divided-Differences"> </a></h1>
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-<p>In spite of the good precision achieved by the Lagrange interpolating polynomials, analytical manipulation of such an expressions is rather complicated. Furthermore, when applying other polynomials-based techniques like Hermite polynomials, the algorithms present very different ways to achieve the final interpolation, making a comparison unclear.</p>
-<p>Divided differences is a way to standardize the notation for interpolating polynomials. Suppose a polynomial $P_n(x)$ and write it in the next form:</p>
-$$P_n(x) = a_0 + a_1(x-x_0)+ a_2 (x-x_0)(x-x_1)+\cdots + a_n(x-x_0)\cdots (x-x_{n-1})$$
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-<p>where $a_i$ are a set of constants to be determined from the given data $(x_i, y_i)$.</p>
-<ul>
-<li>Obtain $a_0$ <br/>Note that due to the definition of an interpolant function, previous expression should satisfy:</li>
-</ul>
-$$P_{n}\left(x_{0}\right)=a_{0}=f\left(x_{0}\right)=y_{0}$$<ul>
-<li>Obtain $a_1$
-$$\frac{P_{n}(x)-a_{0}}{x-x_{0}}=\frac{P_{n}(x)-y_{0}}{x-x_{0}}=a_{1}+a_{2}\left(x-x_{1}\right)+\cdots+a_{n}\left(x-x_{1}\right) \cdots\left(x-x_{n}\right)$$</li>
-</ul>
-$$\frac{P_{n}\left(x_{1}\right)-y_{0}}{x_1-x_{0}}=a_{1}=\frac{y_{1}-y_{0}}{x_1-x_{0}}$$<ul>
-<li>Obtain $a_2$ $$
-\frac{1}{x-x_{1}}\left(\frac{P_{n}(x)-y_{0}}{x-x_{0}}-\frac{y_{1}-y_{0}}{x_{1}-x_{0}}\right)=a_{2}+a_{3}\left(x-x_{2}\right)+\cdots+a_{n}\left(x-x_{2}\right) \cdots\left(x-x_{n}\right)$$ evaluando en $x=x_2$ $$
-a_{2}=\frac{1}{x_{2}-x_{1}}\left(\frac{y_{2}-y_{0}}{x_{2}-x_{0}}-\frac{y_{1}-y_{0}}{x_{1}-x_{0}}\right),$$ que <a href="https://bit.ly/3j018kH">podemos reescribir como</a>
-$$a_{2}=\frac{\frac{y_{2}-y_{1}}{x_{2}-x_{1}}-\frac{y_{1}-y_{0}}{x_{1}-x_{0}}}{x_{2}-x_{0}}.$$</li>
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-<p>In summary
-\begin{align}
-a_{0}=&amp; f\left[y_{0}\right] \equiv y_{0} \\
-a_{1}=&amp; f\left[y_{0}, y_{1}\right] \equiv \frac{y_{1}-y_{0}}{x_{1}-x_{0}}\\
-a_{2}=&amp;f\left[y_{0}, y_{1}, y_{2}\right] \equiv \frac{\frac{y_{2}-y_{1}}{x_{2}-x_{1}}-\frac{y_{1}-y_{0}}{x_{1}-x_{0}}}{x_{2}-x_{0}}
-\end{align}</p>
-<p>The key observation, by using the symmetry $f\left[y_{0}, y_{1}\right]=f\left[y_{1}, y_{0}\right]$, is
-$$f\left[y_{0}, y_{1}, y_{2}\right]=\frac{f\left[y_{1}, y_{2}\right]-f\left[y_{0}, y_{1}\right]}{x_{2}-x_{0}}$$</p>
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-<p>Allow us to definine the <strong>zeroth divided difference</strong>, $k=0$, of $x_i$ like</p>
-$$D_0[x_i] \equiv f[x_i] \equiv f(x_i) = y_i$$<p>the <strong>first divided difference</strong>, $k=1$ of $x_i$ like</p>
-$$D_1[x_i] \equiv f[x_i, x_{i+1}] \equiv \frac{f[x_{i+1}]-f[x_i]}{x_{i+1}-x_i}$$$$D_1[x_i] = \frac{D_{0}[x_{i+1}]-D_{0}[x_{i}]}{x_{i+1}-x_i} $$<p>the <strong>second divided difference</strong>, $k=2$ of $x_i$ like</p>
-$$D_2[x_i]\equiv f[x_i,x_{i+1},x_{i+2}] \equiv \frac{f[x_{i+1},x_{i+2}]-f[x_i,x_{i+1}]}{x_{i+2}-x_{i}}$$$$D_2[x_i]=\frac{D_1[x_{i+1}]-D_1[x_i]}{x_{i+2}-x_{i}}$$<p>successively until the <strong>kth divided difference</strong></p>
-$$D_k[x_i] \equiv f[x_i, x_{i+1},\cdots, x_{i+k-1},x_{i+k}] \equiv \frac{f[x_{i+1},x_{i+2}\cdots, x_{i+k}]-f[x_i, x_{i+1},\cdots, x_{i+k-1}]}{x_{i+k}-x_i}$$$$D_k[x_i] = \frac{D_{k-1}[x_{i+1}]-D_{k-1}[x_{i}]}{x_{i+k}-x_i}$$<p>These expressions are the fundamental bricks for any interpolating method.</p>
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-<span class="k">def</span> <span class="nf">D</span><span class="p">(</span> <span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">Xn</span><span class="p">,</span> <span class="n">Yn</span> <span class="p">):</span>
-    <span class="c1">#If k+i&gt;N</span>
-    <span class="k">if</span> <span class="n">i</span><span class="o">+</span><span class="n">k</span><span class="o">&gt;=</span><span class="nb">len</span><span class="p">(</span><span class="n">Xn</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="c1">#Zeroth divided difference</span>
-    <span class="k">elif</span> <span class="n">k</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Yn</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-    <span class="c1">#If higher divided difference</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">D</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">Xn</span><span class="p">,</span> <span class="n">Yn</span><span class="p">)</span><span class="o">-</span><span class="n">D</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">Xn</span><span class="p">,</span> <span class="n">Yn</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="n">Xn</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="n">k</span><span class="p">]</span><span class="o">-</span><span class="n">Xn</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
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-<p>As an example, Lagrange interpolation can be also derived by using divided differences, which is reached through the next equation:</p>
-$$P_n(x) = D_0[x_0] + \sum_{k=1}^n D_k[x_0] (x-x_0) \cdots (x-x_{k-1})$$<p>Note this expression is by far easier to be manipulated analytically as we can know the coefficients of each order.</p>
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-Using the previous expression and the defined function for divided differences, show both methods to calculate Lagrange interpolators are equivalents.
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-<h1 id="Hermite-Interpolation">Hermite Interpolation<a class="anchor-link" href="#Hermite-Interpolation"> </a></h1>
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-<p>From calculus we know that Taylor polynomials expand a function at a specific point $x_i$, being both functions (the original one and the Taylor function) exactly equal at any derivative-order at that point. Also, as mentioned before, a Lagrange polynomial, given a set of data points, passes through all those points at the same time. However if those points come from an unknown underlying function $f(x)$, the interpolant polynomial might (surely) differ from the real function at any superior derivative-order. So we have:</p>
-<ul>
-<li><p><strong>Taylor polynomials</strong> are exact at any order, but that only remains true at a specific point.</p>
-</li>
-<li><p><strong>Lagrange polynomials</strong> pass through all points of a give dataset, but only at zeroth-order. Derivatives are not longer equal.</p>
-</li>
-</ul>
-<p>Once established these differences, we can introduce Hermite polynomials just as a generalization of both, Taylor and Lagrange polynomials.</p>
-<p>At first, Hermite polynomials can be approximated at any desired order at all the points, as long as one has all these information. However, for the sake of simplicity and without loss of generality, we shall assume Hermite polynomials equal to the real function at zeroth and first-derivative order.</p>
-<p>Let's suppose a dataset $\{x_i\}_i$ for $i = 0,1,\cdots,n$ with the respective values $\{f(x_i)\}_i$ and $\{f'(x_i)\}_i$. If we assume two different polynomials to fit each set of data, i.e. a polynomial for $\{f(x_i)\}_i$ and another for $\{f'(x_i)\}_i$, we obtain $2n+2$ coefficients, however zeroth-order coefficients can be put together so finally there are $2n+1$ independet coefficients to be determined. In this case, we assign the respective Hermite polynimial as $H_{2n+1}(x)$.</p>
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-<h2 id="Derivation-in-terms-of-divided-differences">Derivation in terms of divided differences<a class="anchor-link" href="#Derivation-in-terms-of-divided-differences"> </a></h2>
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-<p>Remembering the divided differences expression for a Lagrange polynomial</p>
-$$P_n(x) = D_0[x_0] + \sum_{k=1}^n D_k[x_0] (x-x_0) \cdots (x-x_{k-1})$$<p>and by defining a new sequence $\{z_0, z_1, \cdots, z_{2n+1}\}$ such that</p>
-$$z_{2i} = z_{2i+1} = x_i \mbox{ for } i = 0,1,\cdots, n$$<p>However, divided differences has to be modified in order to include first-order derivatives:</p>
-<p><img src="https://raw.githubusercontent.com/sbustamante/ComputationalMethods/master/material/figures/table_coefficients.png" alt=""></p>
-<p>Note that $f[z_0,z_1]$ sould be originally</p>
-$$f[z_0,z_1] = \frac{f[z_1]-f[z_0]}{z_1-z_0}$$<p>but replacing $z_0 = z_1 = x_0$ this would lead an indetermination. In order to solve this issue, this indertemination can be readily approximated to the derivative at $z_0$, so</p>
-$$f[z_0,z_1] = f'(x_0)$$<p>or using the previously defined notation</p>
-$$D_1[z_0] = f'(x_0)$$<p>Generally, for first-order divided differences we will have</p>
-$$D_1[z_{2i}] = f'(x_i)$$$$D_1[z_{2i+1}] = D_1[x_i]$$<p>Higher order divided differences are calculated as usual.</p>
-<p>Finally, the Hermite polynomial is built using the next expression</p>
-$$H_{2n+1}(x) = D_0[z_0] + \sum_{k=1}^{2n+1} D_k[z_0] (x-z_0) \cdots (x-z_{k-1})$$
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-      <td>8.0</td>
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-    <tr>
-      <th>1</th>
-      <td>10.0</td>
-      <td>6.5</td>
-    </tr>
-    <tr>
-      <th>2</th>
-      <td>21.3</td>
-      <td>3.0</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
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-</div>
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-<h3 id="Hermite-interpolation">Hermite interpolation<a class="anchor-link" href="#Hermite-interpolation"> </a></h3><p>The recommend degree for the Hermite polynomial is $n-1$ where $n$ is the number of data points of the dataset</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">H</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">polynomial</span><span class="o">.</span><span class="n">hermite</span><span class="o">.</span><span class="n">Hermite</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="o">.</span><span class="n">max</span><span class="p">(),</span><span class="mi">100</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">H</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">X</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;$x$&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;$H(x)$&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-</pre></div>
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-<pre>Text(0,0.5,&#39;$H(x)$&#39;)</pre>
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->
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-<h2 id="Example-3">Example 3<a class="anchor-link" href="#Example-3"> </a></h2>
-</div>
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-<p>Define a routine to calculate divided differences for Hermite polynomials.</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Construction of a kth divided difference for Hermite polynomials (recursive code)</span>
-<span class="k">def</span> <span class="nf">Dh</span><span class="p">(</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">Zn</span><span class="p">,</span> <span class="n">Yn</span><span class="p">,</span> <span class="n">Ypn</span> <span class="p">):</span>
-    <span class="c1">#If k+j&gt;N</span>
-    <span class="k">if</span> <span class="n">j</span><span class="o">+</span><span class="n">k</span><span class="o">&gt;=</span><span class="nb">len</span><span class="p">(</span><span class="n">Zn</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="c1">#Zeroth divided difference</span>
-    <span class="k">elif</span> <span class="n">k</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Yn</span><span class="p">[</span><span class="n">j</span><span class="o">/</span><span class="mi">2</span><span class="p">]</span>
-    <span class="c1">#First order divided difference (even indexes)</span>
-    <span class="k">elif</span> <span class="n">k</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">j</span><span class="o">%</span><span class="k">2</span> == 0:
-        <span class="k">return</span> <span class="n">Ypn</span><span class="p">[</span><span class="n">j</span><span class="o">/</span><span class="mi">2</span><span class="p">]</span>
-    <span class="c1">#If higher divided difference</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">Dh</span><span class="p">(</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">Zn</span><span class="p">,</span> <span class="n">Yn</span><span class="p">,</span> <span class="n">Ypn</span><span class="p">)</span><span class="o">-</span><span class="n">Dh</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">Zn</span><span class="p">,</span> <span class="n">Yn</span><span class="p">,</span> <span class="n">Ypn</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="n">Zn</span><span class="p">[</span><span class="n">j</span><span class="o">+</span><span class="n">k</span><span class="p">]</span><span class="o">-</span><span class="n">Zn</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-</pre></div>
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-<h2 id="-**Activity**-"><font color="red"> **Activity** </font><a class="anchor-link" href="#-**Activity**-"> </a></h2>
-</div>
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-<p><font color='red'></p>
-<p>Calculate a routine, using the previous program for divided differences, that computes the Hermite polynomial given a dataset.</p>
-<p>Generate a set of $N$ points of the function $\sin^2(x)$ between $0$ and $2\pi$, including an array of $x$ positions, $y = f(x)$ and first derivative $y' = f'(x)$.</p>
-<p>Show which polynomial gives the best approximation to the real function, Hermite or Lagrange polynomial.</p>
-<p>&lt;/font&gt;</p>
-<p><font color='white'>
-Solution:</p>
-<p>nbviewer.ipython.org/github/sbustamante/ComputationalMethods/blob/master/activities/hermite-and-lagrange.ipynb
-&lt;/font&gt;</p>
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-  Interactive editing with Jupyter
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/ipython-notebooks.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<h1><div style="float: right;" markdown="1">     <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/3/38/Jupyter_logo.svg/245px-Jupyter_logo.svg.png" height=150px> </div>  Notebooks</h1>
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-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/ipython-notebooks.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
-<p>Jupyter notebooks are a powerful web interface designed for interactive computing in multiple programming languages. Although it was initially designed for Python, its functionalities have quickly permeated other programming languagues. It have evolved from IPython, a replacement for the official Python shell in the console</p>
-<p>It is worth mentioning the main author of IPython is Fernando Perez, a physicist graduate from the <em>Universidad de Antioquia</em> and currently Professor in University of California, Berkeley.</p>
-<p><strong><a href="http://jupyter.org/">Official page</a></strong></p>
-
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-<p><strong>Features</strong></p>
-<p>Jupyter provides a rich architecture for interactive computing with:</p>
-<ul>
-<li>Powerful interactive shells (terminal and Qt-based).</li>
-<li>A browser-based notebook with support for code, text, mathematical expressions, inline plots and other rich media.</li>
-<li>Support for interactive data visualization and use of GUI toolkits.</li>
-<li>Flexible, embeddable interpreters to load into your own projects.</li>
-<li>Easy to use, high performance tools for parallel computing.</li>
-</ul>
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-<h1 id="Jupyter-Notebooks">Jupyter Notebooks<a class="anchor-link" href="#Jupyter-Notebooks"> </a></h1>
-</div>
-</div>
-</div>
-</div>
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Since 2011, Jupyter has incorporated a new funcionality called Notebooks. For</p>
-<ol>
-<li>List item</li>
-<li>List item</li>
-</ol>
-<p>those who know notebooks of <a href="http://www.wolfram.com/mathematica/">Mathematica®</a> and <a href="http://www.sagemath.org/">SAGE</a> this feature may be familiar.</p>
-<p><strong><a href="https://jupyter-notebook.readthedocs.io/en/stable/">Official page Jupyter Notebooks</a></strong></p>
-<p>The Jupyter Notebook (or Jupyter lab) is a web-based interactive computational environment where you can combine code execution, text, mathematics, plots and rich media into a single document. These notebooks are normal files that can be shared with colleagues, converted to other formats such as HTML or PDF, etc. You can share any publicly available notebook by using the <a href="http://github.com/"><strong>GitHub viewer</strong></a> service which will render it as a static web page. This makes it easy to give your colleagues a document they can read immediately without having to install anything.</p>
-<p><a href="https://github.com/restrepo/termux-tips/blob/master/Jupyter.md">Cell phone installation</a></p>
-
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-<h1 id="How-to-(step-by-step)">How to (step by step)<a class="anchor-link" href="#How-to-(step-by-step)"> </a></h1><p>What is better than a step by step example when you want to introduce a new tool. Next it is shown a simple example of a IPython notebook for generating a family of Lissajous curves.</p>
-<hr>
-<blockquote><p>First, a suitable title for your notebook should be thought. For this, choose the style <strong>Heading 1</strong> with <code>#</code> at the beggining.</p>
-</blockquote>
-<h1 id="Lissajous-curves">Lissajous curves<a class="anchor-link" href="#Lissajous-curves"> </a></h1><blockquote><p>Then, a brief statement of the problem to solve should be done. For this, you can use the <strong>Markdown</strong> style. A useful reference for formatting markdown text can be found <a href="http://en.wikipedia.org/wiki/Markdown#Example">here</a>.</p>
-</blockquote>
-<p>Lissajous curves are a family of parametric two-dimensional curves normally obtained when solving multi-harmonic systems, like a mass-spring systems with two springs in each axis (x and y) or some circuit systems. The figures are described with the following equations:</p>
-<blockquote><p>You can use LaTex format in a Markdown cell, for this you must use double <code>$</code></p>
-</blockquote>
-<p>$x(t) = A\sin(a t +\delta)$</p>
-<p>$y(t) = B\sin(b t)$</p>
-<p>where $A$ and $B$ are the amplitudes along each axis, $a$ and $b$ the angular frequencies and $\delta$ the relative phase.</p>
-<blockquote><p>It is possible to embed figures from internet into a Markdown cell simply by writing:</p>
-</blockquote>
-
-<pre><code>![Some text](URL_to_figure)
-
-</code></pre>
-<p><img src="http://upload.wikimedia.org/wikipedia/commons/5/5d/Lissajous_animation.gif" alt="Figure"></p>
-<blockquote><p>Once established the problem, the first thing to do is to import all the libraries you should use for your calculations.</p>
-</blockquote>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#This line is always necessary when you use matplotlib in a notebook, otherwise the figures will not appear.</span>
-<span class="o">%</span><span class="k">pylab</span> inline
-<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-</pre></div>
-
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-<pre>Populating the interactive namespace from numpy and matplotlib
-</pre>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<blockquote><p>You can create subsections and subsubsections using the styles <strong>Heading 2</strong>: with <code>##</code>, and <strong>Heading 3</strong>: with <code>###</code></p>
-</blockquote>
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-<h2 id="Plotting-some-curves">Plotting some curves<a class="anchor-link" href="#Plotting-some-curves"> </a></h2>
-</div>
-</div>
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-<blockquote><p>Regarding coding, you can write Python code normally as if you were doing it in a text editor like emacs.</p>
-</blockquote>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Solutions</span>
-<span class="k">def</span> <span class="nf">Lissajous</span><span class="p">(</span> <span class="n">t</span><span class="p">,</span> <span class="n">A</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">B</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">delta</span><span class="o">=</span><span class="mf">0.0</span> <span class="p">):</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">A</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">t</span><span class="o">+</span><span class="n">delta</span><span class="p">)</span>
-    <span class="n">y</span> <span class="o">=</span> <span class="n">B</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="o">*</span><span class="n">t</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Parameters to be sampled--------------------</span>
-<span class="c1">#delta</span>
-<span class="n">Ndelta</span> <span class="o">=</span> <span class="mi">4</span>   <span class="c1">#Number of delta parameters</span>
-<span class="n">Delta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span><span class="n">Ndelta</span><span class="p">)</span>
-<span class="c1">#Ratio c=a/b</span>
-<span class="n">Nratio</span> <span class="o">=</span> <span class="mi">6</span>  <span class="c1">#Number of c parameters</span>
-<span class="n">C</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">Nratio</span><span class="p">)</span>
-<span class="c1">#Time array</span>
-<span class="n">T</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">40</span><span class="p">,</span><span class="mi">1000</span><span class="p">)</span>
-
-<span class="c1">#Initializing plotting environment</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">Ndelta</span><span class="p">,</span> <span class="mi">4</span><span class="o">*</span><span class="n">Nratio</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span> <span class="n">Nratio</span><span class="p">,</span> <span class="n">Ndelta</span><span class="p">,</span> <span class="mi">1</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">()</span>
-<span class="c1">#Sweeping all figures</span>
-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span> <span class="n">Ndelta</span> <span class="p">):</span>
-    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span> <span class="n">Nratio</span> <span class="p">):</span>
-        <span class="c1">#Creating a subfigure with the current delta and ratio c</span>
-        <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span> <span class="n">Nratio</span><span class="p">,</span> <span class="n">Ndelta</span><span class="p">,</span> <span class="n">j</span><span class="o">*</span><span class="n">Ndelta</span><span class="o">+</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span> <span class="p">)</span>
-        <span class="c1">#Plotting this Lissajous curve</span>
-        <span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">Lissajous</span><span class="p">(</span> <span class="n">T</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="n">C</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">delta</span><span class="o">=</span><span class="n">Delta</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="p">)</span>
-        <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span> <span class="s2">&quot;$\delta=$</span><span class="si">%1.2f</span><span class="se">\t</span><span class="s2">$a/b=$</span><span class="si">%1.2f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">Delta</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="n">C</span><span class="p">[</span><span class="n">j</span><span class="p">])</span> <span class="p">)</span>
-        <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span> <span class="p">)</span>
-</pre></div>
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-<pre>/usr/local/lib/python3.7/dist-packages/ipykernel_launcher.py:19: MatplotlibDeprecationWarning: Adding an axes using the same arguments as a previous axes currently reuses the earlier instance.  In a future version, a new instance will always be created and returned.  Meanwhile, this warning can be suppressed, and the future behavior ensured, by passing a unique label to each axes instance.
-</pre>
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-<h2 id="References">References<a class="anchor-link" href="#References"> </a></h2>
-</div>
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-<ul>
-<li><a href="http://en.wikipedia.org/wiki/Lissajous_curve">http://en.wikipedia.org/wiki/Lissajous_curve</a></li>
-</ul>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>For this activity, you should obtain something like <a href="http://nbviewer.ipython.org/github/sbustamante/ComputationalMethods/blob/master/activities/first_notebook.ipynb">this</a>.</p>
-
-</div>
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-<h2 id="Viewing-your-notebook-online">Viewing your notebook online<a class="anchor-link" href="#Viewing-your-notebook-online"> </a></h2>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Once you have a notebook, you can upload it in some online storage service like <a href="https://www.dropbox.com/home">Dropbox</a> or <a href="https://drive.google.com/">Google drive</a> or some repository service like <a href="https://github.com/">GitHub</a> or GitLab. In both GitHub and GitLab the notebooks are automatically displayed in their web interfaces. Also, you can put the link of your notebook directly into the <a href="http://nbviewer.ipython.org/"><strong>IPython Notebook Viewer</strong></a>.</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span><span class="p">,</span> <span class="n">Markdown</span><span class="p">,</span> <span class="n">Latex</span><span class="p">,</span> <span class="n">Image</span> 
-<span class="n">Image</span><span class="p">(</span><span class="n">filename</span><span class="o">=</span><span class="s1">&#39;./figures/nbviewer.png&#39;</span><span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">display</span><span class="p">(</span><span class="n">Markdown</span><span class="p">(</span><span class="s1">&#39;*some markdown* $\phi=</span><span class="si">{}</span><span class="s1">$&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="mi">2</span><span class="p">)))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<p><em>some markdown</em> $\phi=2$</p>
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-  - "/features/least-action-minimization"
-interact_link: content/features/least_action_minimization.ipynb
-kernel_name: python3
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-has_widgets: false
-title: |-
-  Minimization for Least Action
-pagenum: 15
-prev_page:
-  url: /features/Minimization.html
-next_page:
-  url: /features/numerical-calculus.html
-suffix: .ipynb
-search: t x action m p frac l least function s end hbox xmin plt object plot points tx right com v obtained interpolation set range ts xp height xi ti polynomial delta approx left eqnarray return lagrange np initial final define small segment free fall data mass degree energy begin g interpolate between int value such ini random txx tt minimization div raw restrepo computationalmethods master leastaction geometry following www eftaylor motion fitted figure calculate align xx problem possible correct definition lagrangian rm d code throw vertically let using calculates steps solution activity cocalc zero curve check smooth polynomials step
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-comment: "***PROGRAMMATICALLY GENERATED, DO NOT EDIT. SEE ORIGINAL FILES IN /content***"
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-    <main class="jupyter-page">
-    <div id="page-info"><div id="page-title">Minimization for Least Action</div>
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h1 id="Least-Action-with-least-squares-minimization">Least Action with least squares minimization<a class="anchor-link" href="#Least-Action-with-least-squares-minimization"> </a></h1>
-</div>
-</div>
-</div>
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-<p>Load modules</p>
-
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-<pre>Populating the interactive namespace from numpy and matplotlib
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">import</span> <span class="nn">scipy.optimize</span> 
-<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-<span class="k">global</span> <span class="n">g</span>  
-<span class="n">g</span><span class="o">=</span><span class="mf">9.8</span>
-</pre></div>
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-<div style="float: right;" markdown="1">
-    <img src="https://raw.githubusercontent.com/restrepo/ComputationalMethods/master/material/figures/leastaction3.svg?sanitize=true">
-</div><h2 id="Geometry-interpretation">Geometry interpretation<a class="anchor-link" href="#Geometry-interpretation"> </a></h2><p>Predict from initial conditions, rebuild physical path from initial and the final state.</p>
-<p>Following the geometry theory developed <a href="http://www.eftaylor.com/software/ActionApplets/LeastAction.html">here</a>, we will try to define something called the <em>Action</em> for one small segment of the free fall motion in one-dimension.</p>
-<p>For that we need the experimental data consisting on the height of an object of mass $m$ in free fall, and the height $x_i$, for each time $t_i$. This data would be fitted by a polynomial of degree two, as displayed in the figure for one of the fitted segments of the plot of $x$ as a function of $t$. We take the origin of the coordinates at ground level. For each segment we can calculate an average kinetic energy, $T$, and an averge potential energy, $V$, in the limit of $\Delta t=t_2-t_1$ small. From the figure</p>
-\begin{align}
-T_{12}=\frac12 m v^2\approx &amp;\frac12 m\left(\frac{x_2-x_1}{t_2-t_1}\right)^2\,,&amp;
-V_{12}=mgh\approx&amp; m g \frac{x_2+x_1}{2}\,.
-\end{align}<p>We can then reformulate the problem of the free fall in the following terms. From all the possible curves that can interpolate the points $(t_1,x_1)$ and $(t_2,x_2)$, which is the correct one?.</p>
-<p>The answer obtained by Euler can be obtained from the definition of the function "Lagrangian"
-$$L(t)=T(t)-V(t)$$</p>
-<p>We define the "Action" of one interpolating function between the points $(t_1,x_1)$ and $(t_2,x_2)$ as
-$$S=\int_{t_1}^{t_2} L\, {\rm d}t $$</p>
-<p>The result if that correct interpolation is the one that has a minumum value for the Action!</p>
-<p>For one segment of the action between $(t_1,x_1)$, and $(t_2,x_2)$, with $\Delta t$ sufficiently small such that $L$ can be considered constant, we have
-\begin{eqnarray}
-S_1&amp;=&amp;\int_{t_1}^{t_2} L dt \\
-&amp;\approx&amp; \left[\frac12 m v^2-m g h \right]\Delta t\\
-&amp;\approx&amp; \left[\frac12 m\left(\frac{x_2-x_1}{t_2-t_1}\right)^2-m g \frac{x_2+x_1}{2} \right](t_2-t_1)
-\end{eqnarray}
-that corresponds to Eq. (11) of Am. J. Phys, Vol. 72(2004)478: <a href="http://www.eftaylor.com/pub/Symmetries&amp;ConsLaws.pdf">http://www.eftaylor.com/pub/Symmetries&amp;ConsLaws.pdf</a></p>
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-<h2 id="Code-implementation">Code implementation<a class="anchor-link" href="#Code-implementation"> </a></h2>
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-<h3 id="The-Action">The Action<a class="anchor-link" href="#The-Action"> </a></h3><p>We define the Action $S$ such of an object of mass $m$ throw vertically upward from $x_{\hbox{ini}}$, such that $t_{\hbox{end}}$ seconds later the object return to a height $x_{\hbox{end}}$, as
-\begin{eqnarray}
-S&amp;=&amp;\int_{t_{\hbox{ini}}}^{t_{\hbox{end}}} L\, {\rm d}t \\
-&amp;=&amp;\sum_i L_i \Delta t
-\end{eqnarray}</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">S</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">tend</span><span class="o">=</span><span class="mf">3.</span><span class="p">,</span><span class="n">m</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span><span class="n">xini</span><span class="o">=</span><span class="mf">0.</span><span class="p">,</span><span class="n">xend</span><span class="o">=</span><span class="mf">0.</span><span class="p">,</span><span class="n">g</span><span class="o">=</span><span class="mf">9.8</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Calculate the Action of an object of of mass &#39;m&#39; throw vertically upward from </span>
-<span class="sd">       &#39;xini&#39;, such that &#39;tend&#39; seconds later the object return to a height &#39;xend&#39;.</span>
-<span class="sd">       Delta t must be constant.</span>
-<span class="sd">       </span>
-<span class="sd">    The defaults units for S are J.s   </span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">t</span><span class="o">=</span><span class="nb">float</span><span class="p">(</span><span class="n">tend</span><span class="p">)</span>
-    <span class="n">Dt</span><span class="o">=</span><span class="n">t</span><span class="o">/</span><span class="n">x</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">size</span>
-    <span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="c1">#Fix initial and final point</span>
-    <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">=</span><span class="n">xini</span>
-    <span class="n">x</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">=</span><span class="n">xend</span>
-    <span class="k">return</span> <span class="p">((</span><span class="mf">0.5</span><span class="o">*</span><span class="n">m</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span><span class="o">-</span><span class="n">x</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">Dt</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mf">0.5</span><span class="o">*</span><span class="n">m</span><span class="o">*</span><span class="n">g</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span><span class="o">+</span><span class="n">x</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]))</span><span class="o">*</span><span class="n">Dt</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
-</pre></div>
-
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-<h2 id="Least-Action-calculation">Least Action calculation<a class="anchor-link" href="#Least-Action-calculation"> </a></h2>
-</div>
-</div>
-</div>
-</div>
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-<p><em>Problem</em>: Let an object of mass $m=0.2$ Kg throw vertically updward and returning back to the same hand after 3 s. Find the function of distance versus time of least Action.</p>
-
-</div>
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>If we denote the height at time $t_i$ as $x_i=x(t_i)$, we can calculate the action for any set of $x_i$ points with the inititial and final points fixed at $0\ $m.</p>
-<p><strong>Example</strong>: By using the previous definition, calculates the Action for 21 steps in time from $0$ at $3\ $s for an object that does not move at all</p>
-<p><strong>Solution</strong>:</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">21</span><span class="p">)</span>
-<span class="n">S</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
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-<pre>0.0</pre>
-</div>
-
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-</div>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Activity</strong>:  <em>Brute force approach</em></p>
-<p>Open the Activity notebook in <a href="https://cocalc.com/projects/a8330cfb-9dfb-442e-9b2b-ba664e31a685/files/Activity_Least_Action.ipynb?session=default">CoCalc!</a></p>
-<p>1) calculates the Action for 21 steps in time from $0$ at $3\ $s, for an object that at a random position in each time between zero and $15\ $m, but with the initial and final positions set to zero. Make the plot for the random curve.</p>
-
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-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Let us divide the intervals in 21 parts:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Initialize with the maximum possible value</span>
-<span class="n">Smin</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">inf</span>
-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">100000</span><span class="p">):</span>
-    <span class="c1">#21 one random number between 0 and 15</span>
-    <span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">15</span><span class="p">,</span><span class="mi">21</span><span class="p">)</span>
-    <span class="c1"># Force the boundary conditions</span>
-    <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">=</span><span class="mi">0</span>
-    <span class="n">x</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">=</span><span class="mi">0</span>
-    <span class="n">Sx</span><span class="o">=</span><span class="n">S</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">Sx</span><span class="o">&lt;</span><span class="n">Smin</span><span class="p">:</span>
-        <span class="c1">#Get new minimum</span>
-        <span class="n">Smin</span><span class="o">=</span><span class="n">Sx</span>
-        <span class="n">xmin</span><span class="o">=</span><span class="n">x</span>
-        <span class="nb">print</span><span class="p">(</span><span class="n">Smin</span><span class="p">)</span>
-</pre></div>
-
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-<pre>550.3493673339286
-459.61825206303354
-342.4644127733231
-331.459689578506
-225.01316485752167
-211.21468603608093
-196.71669612765731
-186.64743929364718
-154.74301469839858
-151.59615942111134
-99.39935405252346
-72.90890773801733
-70.46213681452232
-43.20057111692605
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<!-- xmin=pd.read_csv('https://github.com/restrepo/ComputationalMethods/raw/master/data/random.csv').xrandom.values -->
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Check again the Acion, $S$, value for <code>xmin</code></p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>2) Find the set of interpolation Lagrange polinomial that a set a points as smooth as possible</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">interpolate</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
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-
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Assign a time, $t_i$ to each $x_i$ point in <code>xmin</code></p>
-
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">tx</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="nb">len</span><span class="p">(</span><span class="n">xmin</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Complete the set of interpolation Lagrange polynomials: Change <code>pe</code> to the high value that allows for on smooth curve, and uncomment the next code and repeat as many times as required</p>
-
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p</span><span class="o">=</span><span class="p">[]</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">tx</span><span class="p">,</span><span class="n">xmin</span><span class="p">,</span><span class="s1">&#39;r.&#39;</span><span class="p">)</span>
-<span class="n">pi</span><span class="o">=</span><span class="mi">0</span>
-<span class="n">pe</span><span class="o">=</span><span class="mi">3</span>
-<span class="n">L</span><span class="o">=</span><span class="n">interpolate</span><span class="o">.</span><span class="n">lagrange</span><span class="p">(</span><span class="n">tx</span><span class="p">[</span><span class="n">pi</span><span class="p">:</span><span class="n">pe</span><span class="p">],</span><span class="n">xmin</span><span class="p">[</span><span class="n">pi</span><span class="p">:</span><span class="n">pe</span><span class="p">])</span>
-<span class="n">txx</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">tx</span><span class="p">[</span><span class="n">pi</span><span class="p">],</span><span class="n">tx</span><span class="p">[</span><span class="n">pe</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">txx</span><span class="p">,</span><span class="n">L</span><span class="p">(</span><span class="n">txx</span><span class="p">))</span>
-<span class="c1">#Repeat to get a fit as smoot as possible</span>
-<span class="c1">#pi=pe-1</span>
-<span class="c1">#pe=8</span>
-<span class="c1">#L=interpolate.lagrange(tx[pi:pe],xmin[pi:pe])</span>
-<span class="c1">#txx=np.linspace(tx[pi],tx[pe-1])</span>
-<span class="c1">#plt.plot(txx,L(txx))</span>
-
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">20</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">3.3</span><span class="p">)</span>
-</pre></div>
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-<pre>(-0.1, 3.3)</pre>
-</div>
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-</div>
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-<div class="output_area">
-
-
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-<div class="output_png output_subarea ">
-<img src="../images/features/least_action_minimization_22_1.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>3) Built a step function for the full range each of the interpolation Lagrangian polynomial in each range</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<!--
-'''python
-p=[1,3,8,18,21]
-L=[]
-ts=[]
-plt.plot(tx,xmin,'r.')
-for i in range(len(p)):
-    if i<len(p)-1:
-        L.append(interpolate.lagrange(tx[p[i]-1:p[i+1]],xmin[p[i]-1:p[i+1]]))
-        txx=np.linspace(tx[p[i]-1],tx[p[i+1]-1])
-        plt.plot(txx,L[-1](txx))
-        ts.append(tx[p[i]-1])
-        plt.vlines(ts,-1,20,linestyles=':')
-
-def ff(t):
-    if t<0:
-        return 0
-    if t>=3:
-        return 0
-    for i in range(len(ts)-1):
-        if t >= ts[i] and t<ts[i+1]:
-            return L[i](t)
-f=np.vectorize(ff)
-x=f(tt)
-P=np.poly1d( np.polyfit(tx,xmin,deg=2) )
-xP=P(tt)
-xP[0]=0
-xP[-1]=0
-plt.plot(tt,xP,'--')
-plt.ylim(-1,20)
-plt.xlim(-0.1,3.3)
-plt.grid()
-'''
--->
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>4) Build a step function with <code>interpolate.interp1d</code></p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<p>5) Fit the points with a polynomial of degree 2</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span><span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
-<span class="n">p</span><span class="o">=</span><span class="p">[]</span>
-<span class="n">pi</span><span class="o">=</span><span class="mi">0</span>
-<span class="n">pe</span><span class="o">=</span><span class="mi">21</span>
-<span class="n">x1d</span><span class="o">=</span><span class="n">interpolate</span><span class="o">.</span><span class="n">interp1d</span><span class="p">(</span><span class="n">tx</span><span class="p">[</span><span class="n">pi</span><span class="p">:</span><span class="n">pe</span><span class="p">],</span><span class="n">xmin</span><span class="p">[</span><span class="n">pi</span><span class="p">:</span><span class="n">pe</span><span class="p">],</span><span class="n">kind</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-<span class="n">txx</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">tx</span><span class="p">[</span><span class="n">pi</span><span class="p">],</span><span class="n">tx</span><span class="p">[</span><span class="n">pe</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span><span class="mi">1000</span><span class="p">)</span>
-<span class="n">ax1</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">tx</span><span class="p">,</span><span class="n">xmin</span><span class="p">,</span><span class="s1">&#39;r.&#39;</span><span class="p">)</span>
-<span class="n">ax1</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">txx</span><span class="p">,</span><span class="n">x1d</span><span class="p">(</span><span class="n">txx</span><span class="p">))</span>
-<span class="n">ax1</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xlim</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">3.3</span><span class="p">),</span><span class="n">ylim</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">20</span><span class="p">),</span>
-        <span class="n">xlabel</span><span class="o">=</span><span class="s1">&#39;$t$ [s]&#39;</span><span class="p">,</span><span class="n">ylabel</span><span class="o">=</span><span class="s1">&#39;$x(t)$ (m)&#39;</span><span class="p">)</span>
-
-<span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">tx</span><span class="p">,</span><span class="n">xmin</span><span class="p">,</span><span class="s1">&#39;r.&#39;</span><span class="p">)</span>
-<span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">txx</span><span class="p">,</span><span class="n">x1d</span><span class="p">(</span><span class="n">txx</span><span class="p">))</span>
-<span class="n">ax2</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xlim</span><span class="o">=</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">1.3</span><span class="p">),</span><span class="n">ylim</span><span class="o">=</span><span class="p">(</span><span class="mf">5.7</span><span class="p">,</span><span class="mi">12</span><span class="p">),</span>
-        <span class="n">xlabel</span><span class="o">=</span><span class="s1">&#39;zoom in&#39;</span><span class="p">)</span>
-
-<span class="n">fig</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">(</span><span class="n">pad</span><span class="o">=</span><span class="mf">4.0</span><span class="p">)</span>
-</pre></div>
-
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-</div>
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-<img src="../images/features/least_action_minimization_27_0.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Plot results 3) and 4) here</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>5) Compare the Action for <code>xmin</code>, some <code>x</code> obtained from the full range function obtained from the interpolation Lagrange polynomials, and some <code>xP</code> obtained from the degree 2 polynomial</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">S</span><span class="p">(</span><span class="n">xmin</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
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-<div class="output">
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-<div class="output_area">
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-<pre>43.20057111692605</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#S for step function here</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">S</span><span class="p">(</span><span class="n">x1d</span><span class="p">(</span><span class="n">txx</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
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-<pre>56.65688265854261</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#S for degree 2 function here</span>
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-<p>Which one is the one with the least action and why?</p>
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-<h3 id="Minimization">Minimization<a class="anchor-link" href="#Minimization"> </a></h3>
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-<p>Function to find the least Action by using <code>scipy.optimize.fmin_powell</code>. It start from $\mathbf{x}=(x_{\hbox{ini}},0,0,\ldots,x_{\hbox{end}})$ and find the least action</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">scipy.optimize</span> <span class="k">as</span> <span class="nn">optimize</span>
-<span class="k">def</span> <span class="nf">xfit</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">t</span><span class="o">=</span><span class="mf">3.</span><span class="p">,</span><span class="n">m</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span><span class="n">xini</span><span class="o">=</span><span class="mf">0.</span><span class="p">,</span><span class="n">xend</span><span class="o">=</span><span class="mf">0.</span><span class="p">,</span><span class="n">ftol</span><span class="o">=</span><span class="mf">1E-8</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;Find the array of n (odd) components that minimizes the action S(x)</span>
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-<span class="sd">    :Parameters:</span>
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-<span class="sd">    n: odd integer </span>
-<span class="sd">        dimension of the ndarray x that minimizes the action  S(x,t,m)</span>
-<span class="sd">    t,m: numbers</span>
-<span class="sd">       optional parameters for the action</span>
-<span class="sd">    ftol: number</span>
-<span class="sd">        acceptable relative error in S(x) for convergence.</span>
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-<span class="sd">    :Returns: (x,xmax,Smin)</span>
-<span class="sd">    </span>
-<span class="sd">    x: ndarray</span>
-<span class="sd">        minimizer of the action S(x)</span>
-<span class="sd">        </span>
-<span class="sd">    xini:</span>
-<span class="sd">    </span>
-<span class="sd">    xend:</span>
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-<span class="sd">    xmax: number</span>
-<span class="sd">        Maximum height for the object</span>
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-<span class="sd">    Smin: number</span>
-<span class="sd">        value of function at minimum: Smin = S(x)</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="n">t</span><span class="o">=</span><span class="nb">float</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span><span class="o">%</span><span class="k">2</span>==0:
-        <span class="nb">print</span> <span class="p">(</span> <span class="s1">&#39;x array must be odd&#39;</span><span class="p">)</span>
-        <span class="n">sys</span><span class="o">.</span><span class="n">exit</span><span class="p">()</span>
-  
-    <span class="n">x0</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="n">a</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">fmin_powell</span><span class="p">(</span><span class="n">S</span><span class="p">,</span><span class="n">x0</span><span class="p">,</span><span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="n">xini</span><span class="p">,</span><span class="n">xend</span><span class="p">),</span><span class="n">ftol</span><span class="o">=</span><span class="n">ftol</span><span class="p">,</span><span class="n">full_output</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-    <span class="n">x</span><span class="o">=</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">=</span><span class="n">xini</span><span class="p">;</span><span class="n">x</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">=</span><span class="n">xend</span>
-    <span class="n">xmax</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">x</span><span class="p">)[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">Smin</span><span class="o">=</span><span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">Dt</span><span class="o">=</span><span class="n">t</span><span class="o">/</span><span class="n">x</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">size</span> <span class="c1">#  t/(n-1)</span>
-    <span class="k">return</span> <span class="n">x</span><span class="p">,</span><span class="n">xmax</span><span class="p">,</span><span class="n">Smin</span><span class="p">,</span><span class="n">Dt</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">t</span><span class="o">=</span><span class="mf">3.</span>
-<span class="n">m</span><span class="o">=</span><span class="mf">0.2</span>
-<span class="n">y</span><span class="o">=</span><span class="n">xfit</span><span class="p">(</span><span class="mi">21</span><span class="p">,</span><span class="n">t</span><span class="p">,</span><span class="n">m</span><span class="p">)</span>
-<span class="n">x</span><span class="o">=</span><span class="n">y</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="n">xmax</span><span class="o">=</span><span class="n">y</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-<span class="n">Smin</span><span class="o">=</span><span class="n">y</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-<span class="n">Dt</span><span class="o">=</span><span class="n">y</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
-<span class="n">tx</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">t</span><span class="o">+</span><span class="n">Dt</span><span class="p">,</span><span class="n">Dt</span><span class="p">)</span>
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-<pre>Optimization terminated successfully.
-         Current function value: -21.554977
-         Iterations: 28
-         Function evaluations: 5837
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-<h3 id="Plot">Plot<a class="anchor-link" href="#Plot"> </a></h3>
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-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">tx</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;HEIGHT meters&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;TIME seconds&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Worldline of least action&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">&#39;best&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
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-<p>Check the equation of motion: 
-$$x(t)=-\frac{{1}}{{2}}gt^2+v_0t$$</p>
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-<span class="nb">print</span><span class="p">(</span><span class="n">P</span><span class="p">)</span>
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-<pre>      2
--4.9 t + 14.7 t + 0.0001604
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">P</span><span class="p">(</span><span class="mf">1.5</span><span class="p">)</span>
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-<pre>11.025294114794459</pre>
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-<h3 id="Dynamics-of-the-least-action-solution">Dynamics of the least action solution<a class="anchor-link" href="#Dynamics-of-the-least-action-solution"> </a></h3>
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-<h4 id="Velocity">Velocity<a class="anchor-link" href="#Velocity"> </a></h4>
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-<span class="n">Dt</span><span class="o">=</span><span class="n">t</span><span class="o">/</span><span class="n">v</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">size</span>
-<span class="n">tx</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">t</span><span class="o">+</span><span class="n">Dt</span><span class="p">,</span><span class="n">Dt</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">tx</span><span class="p">,</span><span class="n">v</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">tx</span><span class="p">,</span><span class="n">v</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;TIME seconds&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;VELOCITY meter/seconds&#39;</span><span class="p">)</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Dt</span><span class="o">=</span><span class="n">t</span><span class="o">/</span><span class="n">x</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">size</span>
-<span class="n">a</span><span class="o">=</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span><span class="o">-</span><span class="n">v</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="o">/</span><span class="n">Dt</span>
-<span class="n">pa</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;ACELERATION meter/second$^2$&#39;</span><span class="p">)</span>
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-<pre>Text(0.5,1,&#39;ACELERATION meter/second$^2$&#39;)</pre>
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-<h4 id="Energy">Energy<a class="anchor-link" href="#Energy"> </a></h4>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">T</span><span class="o">=</span><span class="mf">0.5</span><span class="o">*</span><span class="n">m</span><span class="o">*</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span>
-<span class="n">V</span><span class="o">=</span><span class="mf">0.5</span><span class="o">*</span><span class="n">m</span><span class="o">*</span><span class="n">g</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span><span class="o">+</span><span class="n">x</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-<span class="n">E</span><span class="o">=</span><span class="n">T</span><span class="o">+</span><span class="n">V</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="n">E</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
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-<div class="output_subarea output_stream output_stdout output_text">
-<pre>[21.56 21.56 21.56 21.56 21.56 21.56 21.55 21.55 21.56 21.56 21.56 21.56
- 21.56 21.56 21.56 21.55 21.55 21.56 21.56 21.55]
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h4 id="Action">Action<a class="anchor-link" href="#Action"> </a></h4>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>The Action is minimal in each interval!</p>
-
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">SS</span><span class="o">=</span><span class="p">(</span><span class="n">T</span><span class="o">-</span><span class="n">V</span><span class="p">)</span><span class="o">*</span><span class="n">Dt</span>
-<span class="n">SS</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>array([ 2.6176807 ,  1.45086655,  0.41339493, -0.49445999, -1.27239049,
-       -1.92031545, -2.4394347 , -2.82826668, -3.08743238, -3.21713111,
-       -3.21711685, -3.08747346, -2.82816485, -2.43888968, -1.9203943 ,
-       -1.27224487, -0.49433121,  0.41338257,  1.45064061,  2.61710336])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">scipy.constants</span> <span class="k">as</span> <span class="nn">sc</span>
-</pre></div>
-
-    </div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">sc</span><span class="o">.</span><span class="n">hbar</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
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-<pre>1.0545718001391127e-34</pre>
-</div>
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-</div>
-
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;S_MINIMUM=</span><span class="si">{}</span><span class="s1">  Joules*second = </span><span class="si">{}</span><span class="s1"> ħ&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
-      <span class="n">SS</span><span class="o">.</span><span class="n">sum</span><span class="p">(),</span>
-      <span class="n">np</span><span class="o">.</span><span class="n">format_float_scientific</span><span class="p">(</span><span class="n">SS</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span><span class="o">/</span><span class="n">sc</span><span class="o">.</span><span class="n">hbar</span><span class="p">,</span><span class="n">precision</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-      <span class="p">)</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
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-<pre>S_MINIMUM=-21.55497732876534  Joules*second = -2.e+35 ħ
-</pre>
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-interact_link: content/features/linear-algebra-diagonalization.ipynb
-kernel_name: python3
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-title: |-
-  Matriz diagonalization
-pagenum: 21
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-  url: /features/linear-algebra.html
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-  url: /features/linear-algebra-extra.html
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-search: boldsymbol v u begin end matrix bmatrix m align left right operatorname c x lambda e dagger s diag t text array b alpha np vdots equation eigenvalues nu pmatrix rangle j cos sin where n mass pdf diagonalization theta delta cp w y such cdots eigenvectors example mathrm basis sqrt eigenvector order lambdai r symmetric associated therefore mu tau linalg check z d ldots l times prime rotation thetaw theorem exists lambdan arxiv org mixing terms mathcal com unitary equiv diagonal matrices python font color span h g eigenvalue not neutrino since proper zero redactivity obtain build previous angle
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-
-    <main class="jupyter-page">
-    <div id="page-info"><div id="page-title">Matriz diagonalization</div>
-</div>
-    <div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h1 id="Matrix-diagonalization">Matrix diagonalization<a class="anchor-link" href="#Matrix-diagonalization"> </a></h1><p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/linear-algebra-diagonalization.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
-
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-<h2 id="Theorem-1">Theorem 1<a class="anchor-link" href="#Theorem-1"> </a></h2><p>If $\boldsymbol{A}$ is Hermitic, e.g:
-\begin{align}
-\boldsymbol{A}^\dagger =\boldsymbol{A}\,,
-\end{align}
-Then exists an <em>unitary  matrix</em> $\boldsymbol{V}$ e.g:
-\begin{align}
-\boldsymbol{V}^{\ -1}=\boldsymbol{V}^\dagger \,,
-\end{align}
-such that
-\begin{equation}
-  \boldsymbol{V}^\dagger\boldsymbol{A}\,\boldsymbol{V}=\boldsymbol{A}_{\text{diag}}\,,
-\end{equation}
-where 
-$\boldsymbol{A}_{\text{diag}}=\operatorname{diag}(\lambda_1,\lambda_2,\ldots \lambda_n)$ is the diagonalized mass matrix.</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Corollary-1">Corollary 1<a class="anchor-link" href="#Corollary-1"> </a></h3><p>If $\boldsymbol{A}$ is symmetric, e.g:
-\begin{align}
-\boldsymbol{A}^{\operatorname{T}} =\boldsymbol{A}\,,
-\end{align}
-Then exists an <em>ortogonal matrix</em> $\boldsymbol{V}$, e.g:
-\begin{align}
-\boldsymbol{V}^{-1}=\boldsymbol{V}^{\operatorname{T}} \,,
-\end{align}
-such that
-\begin{equation}
-  \boldsymbol{V}^{\operatorname{T}}\boldsymbol{A}\,\boldsymbol{V}=\boldsymbol{A}_{\text{diag}}\,,
-\end{equation}
-where 
-$\boldsymbol{A}_{\text{diag}}=\operatorname{diag}(\lambda_1,\lambda_2,\ldots \lambda_n)$ is the diagonalized mass matrix.</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<hr>
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-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Eigenvector-problem">Eigenvector problem<a class="anchor-link" href="#Eigenvector-problem"> </a></h3><p>Note that each eigenvector, corresponding to each column of the matrix, is associated to each eigenvalue
-$$
-\boldsymbol{V}=\begin{bmatrix}
-\boldsymbol{V}_1 \vdots \boldsymbol{V}_2\cdots \vdots \boldsymbol{V}_i \vdots \boldsymbol{V}_j\vdots\cdots \vdots \boldsymbol{V}_n 
-\end{bmatrix}.
-$$
-where $\boldsymbol{V}_i$ is the $i$-th column of the matrix  $\boldsymbol{V}$.</p>
-<p>In this way, if we interchange the $i\leftrightarrow j$ columns of the diagonalization matrix $\boldsymbol{V}$, the order of the eigenvalues also change
-$$
-\begin{bmatrix}
-\boldsymbol{V}_1 \vdots \boldsymbol{V}_2\cdots \vdots \boldsymbol{V}_j \vdots \boldsymbol{V}_i\vdots\cdots \vdots \boldsymbol{V}_n 
-\end{bmatrix}^\dagger \boldsymbol{A} \begin{bmatrix}
-\boldsymbol{V}_1 \vdots \boldsymbol{V}_2\cdots \vdots \boldsymbol{V}_j \vdots \boldsymbol{V}_i\vdots\cdots \vdots \boldsymbol{V}_n 
-\end{bmatrix}=\operatorname{diag}(\lambda_1,\lambda_2,\ldots,\lambda_j,\lambda_i,\ldots \lambda_n).
-$$
-This property is very important because usually the diagonalizationn alghoritm gives not the desired ordering of the eigenvalues and eigenvectors. It is recommended to use the <code>np.c_</code> method for the eigenvector reoirdering</p>
-<p>However, the <strong>Theorem</strong> only guarantees existence. Tor really calculate the diagonalization matrix we must establish the eigenvector problem:</p>
-<p>We can use the unitary propery to write
-\begin{align}
-  \boldsymbol{V}\boldsymbol{V}^\dagger\boldsymbol{A}\,\boldsymbol{V}=&amp;\boldsymbol{V}\boldsymbol{A}_{\text{diag}}\\
-\boldsymbol{A}\,\boldsymbol{V}=&amp;\boldsymbol{V}\boldsymbol{A}_{\text{diag}}\,,
-\end{align}
-or
-$$
-\boldsymbol{A}\begin{bmatrix}
-\boldsymbol{V}_1 \vdots \boldsymbol{V}_2\cdots \vdots \boldsymbol{V}_n 
-\end{bmatrix} =\operatorname{diag}(\lambda_1,\lambda_2\ldots,\lambda_n)\begin{bmatrix}
-\boldsymbol{V}_1 \vdots \boldsymbol{V}_2\cdots  \vdots \boldsymbol{V}_n 
-\end{bmatrix}.
-$$
-Therefore, the eigenvalue equation is just:
-\begin{align}
-%$$
-  \boldsymbol{A}\,\boldsymbol{V}_i=&amp;\lambda_i\boldsymbol{V}_i \nonumber\\
-  \boldsymbol{A}\,\boldsymbol{V}_i-\lambda_i\boldsymbol{V}_i=&amp;\boldsymbol{0} \nonumber\\
-  (\boldsymbol{A}-\lambda_i \,\boldsymbol{I})\boldsymbol{V}_i=&amp;\boldsymbol{0}\,,
-%$$
-\end{align}
-where $\boldsymbol{V}_i$ is the $i$-th column of the matrix  $\boldsymbol{V}$, and $\boldsymbol{I}$ is the identity matrix.</p>
-<p>To avoid the trivial solution $\boldsymbol{V}_i=\boldsymbol{0}$, we require that $\boldsymbol{A}-\lambda_i\, \boldsymbol{I}$
-does not have an inverse, or equivalently
-$$\det( \boldsymbol{A}-\lambda_i\, \boldsymbol{I})=0\,.$$</p>
-
-</div>
-</div>
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-<h3 id="Example">Example<a class="anchor-link" href="#Example"> </a></h3><p>From <a href="https://arxiv.org/pdf/2010.06458.pdf">arXiv:2010.06458</a>:</p>
-<blockquote><p>In the three flavor neutrino oscillation, the neutrino flavor states $\left|\nu_{\alpha}\right\rangle(\alpha=e, \mu, \tau)$ are linear superposition of mass eigenstates $\left|\nu_{j}\right\rangle(j=1,2,3):\left|\nu_{\alpha}\right\rangle=\Sigma_{j} U_{\alpha j}\left|\nu_{j}\right\rangle,$ where $U_{\alpha j}$ are the elements of the lepton mixing matrix known as PMNS (Pontecorvo-Maki-Nakagawa-Sakita) matrix such that
-$$
-\left(\begin{array}{l}
-\left|\nu_{e}\right\rangle \\
-\left|\nu_{\mu}\right\rangle \\
-\left|\nu_{\tau}\right\rangle
-\end{array}\right)=\left(\begin{array}{lll}
-U_{e 1} &amp; U_{e 2} &amp; U_{e 3} \\
-U_{\mu 1} &amp; U_{\mu 2} &amp; U_{\mu 3} \\
-U_{\tau 1} &amp; U_{\tau 2} &amp; U_{\tau 3}
-\end{array}\right)\left(\begin{array}{l}
-\left|\nu_{1}\right\rangle \\
-\left|\nu_{2}\right\rangle \\
-\left|\nu_{3}\right\rangle
-\end{array}\right)
-$$
-The time evolution follows $\left|\nu_{\alpha}(t)\right\rangle=\Sigma_{j} e^{-i E_{j} t} U_{\alpha j}\left|\nu_{j}\right\rangle,$ where $E_{j}$ is the energy associated with the mass eigenstates $\left|\nu_{i}\right\rangle$ This is a superposition state.</p>
-</blockquote>
-<p>The observables associated to the three neutrinos are the entries of the $3\times 3$ unitary matrix $\boldsymbol{U}=\begin{pmatrix}\boldsymbol{U}_1\vdots\boldsymbol{U}_2\vdots\boldsymbol{U}_3\end{pmatrix}$, and the eigenvalues associated to each eigenvector $\boldsymbol{U}_1\to  m_1$, $\boldsymbol{U}_2\to m_2$, $\boldsymbol{U}_3\to m_3$. The normal ordering is $m_1&lt;m_2&lt;m_3$. The unitary matrix can be parameterized in terms of three mixing angles, $\theta_{12}$ $\theta_{13}$, $\theta_{13}$, and a complex phase, $\delta_{\text{CP}}$, such that
-$$
-\boldsymbol{U}=\left(\begin{array}{ccc}
-1 &amp; 0 &amp; 0 \\
-0 &amp; c_{23} &amp; s_{23} \\
-0 &amp; -s_{23} &amp; c_{23}
-\end{array}\right) \cdot\left(\begin{array}{ccc}
-c_{13} &amp; 0 &amp; s_{13} e^{-i \delta_{\mathrm{CP}}} \\
-0 &amp; 1 &amp; 0 \\
--s_{13} e^{i \delta_{\mathrm{CP}}} &amp; 0 &amp; c_{13}
-\end{array}\right) \cdot\left(\begin{array}{ccc}
-c_{21} &amp; s_{12} &amp; 0 \\
--s_{12} &amp; c_{12} &amp; 0 \\
-0 &amp; 0 &amp; 1
-\end{array}\right),
-$$
-where $c_{i j} \equiv \cos \theta_{i j}$ and $s_{i j} \equiv \sin \theta_{i j}$. Thus, we can write $\boldsymbol{U}$ as
-$$
-\boldsymbol{U}=\left(\begin{array}{ccc}c_{12} c_{13} &amp; s_{12} c_{13} &amp; s_{13} e^{-i \delta_{\mathrm{CP}}} \\ -s_{12} c_{23}-c_{12} s_{13} s_{23} e^{i \delta_{\mathrm{CP}}} &amp; c_{12} c_{23}-s_{12} s_{13} s_{23} e^{i \delta_{\mathrm{CP}}} &amp; c_{13} s_{23} \\ s_{12} s_{23}-c_{12} s_{13} c_{23} e^{i \delta_{\mathrm{CP}}} &amp; -c_{12} s_{23}-s_{12} s_{13} c_{23} e^{i \delta_{\mathrm{CP}}} &amp; c_{13} c_{23}\end{array}\right)
-$$
-so that
-$$
-\boldsymbol{U}_1=\begin{pmatrix}U_{e1}\\ U_{\mu 1}\\ U_{\tau 1}\end{pmatrix}=\begin{pmatrix}
-c_{12} c_{13} \\
--s_{12} c_{23}-c_{12} s_{13} s_{23} e^{i \delta_{\mathrm{CP}}} \\
-s_{12} s_{23}-c_{12} s_{13} c_{23} e^{i \delta_{\mathrm{CP}}}
-\end{pmatrix},\qquad 
-\boldsymbol{U}_2=\begin{pmatrix}U_{e2}\\ U_{\mu 2}\\ U_{\tau 2}\end{pmatrix}=\begin{pmatrix}
-s_{12} c_{13} \\
-c_{12} c_{23}-s_{12} s_{13} s_{23} e^{i \delta_{\mathrm{CP}}} \\
--c_{12} s_{23}-s_{12} s_{13} c_{23} e^{i \delta_{\mathrm{CP}}}
-\end{pmatrix},\qquad
-\boldsymbol{U}_3=\begin{pmatrix}U_{e3}\\ U_{\mu 3}\\ U_{\tau 3}\end{pmatrix}=\begin{pmatrix}
-s_{13} e^{-i \delta_{\mathrm{CP}}} \\
-c_{13} s_{23} \\
-c_{13} c_{23}
-\end{pmatrix}
-$$</p>
-<p>After decades of experimental efforts with thousands of millions of dollars in investment and two recent Nobel prizes, most of the parameters are already measured (see <a href="https://pdg.lbl.gov/2018/reviews/rpp2018-rev-neutrino-mixing.pdf">PDG22020</a> ):</p>
-<p><img src="https://github.com/restrepo/ComputationalMethods/raw/master/material/figures/nu.png" alt="IMAGE"></p>
-<p>where $\Delta m^2_{ij}=m^2_i-m^2_j$ is the squared mass difference between eigenvalues $i$ and $j$; and $\text{eV}$ is really $\text{eV}/c^2$ where $c$ is the speed of light  in vacuum in natural units with $c=1$, and $1\ \text{eV}=1.602\,176\,6208(98)\times 10^{-19}\ \text{J}$.</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>To a better measurement of $\delta_{CP}$ for example, a new large experiment called <a href="https://www.dunescience.org/">DUNE</a>, and with a cost of around <a href="https://web.fnal.gov/organization/OPSS/Projects/LBNFDUNE/LBNF%20APM%20Jul%202015/Review%20Documents/LBNF%20DUNE%20ICR_Final%20Report.pdf">$\$1\,500$ million of dollars</a>, is in construction in the United States
-<img src="https://www.dunescience.org/wp-content/uploads/2016/12/LBNE_Graphic_061615_2016.jpg" alt="IMAGE"></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Theorem-2:-Singular-value-decomposition-(SVD)">Theorem 2: Singular value decomposition (SVD)<a class="anchor-link" href="#Theorem-2:-Singular-value-decomposition-(SVD)"> </a></h2><p>See <a href="https://en.wikipedia.org/wiki/Singular_value_decomposition">SVD</a> (where the  Hermetique-conjugate is denoted with "*" instead that with "")</p>
-<p>A general complex matrix $\boldsymbol{A}$  can be diagonalized by a bi-diagonal transformation such that
-\begin{equation}
-  \boldsymbol{V}^\dagger\boldsymbol{A}\,\boldsymbol{U}=\boldsymbol{A}_{\text{diag}}\,,
-\end{equation}
-where 
-$\boldsymbol{A}_{\text{diag}}=\operatorname{diag}(\lambda_1,\lambda_2,\ldots \lambda_n)$ is the diagonalized mass matrix.</p>
-<h3 id="Demostration">Demostration<a class="anchor-link" href="#Demostration"> </a></h3>\begin{align}
-  \boldsymbol{A}_{\text{diag}}=&amp;\boldsymbol{V}^\dagger\boldsymbol{A}\,\boldsymbol{U}\\
-  \boldsymbol{A}_{\text{diag}}\boldsymbol{A}_{\text{diag}}^\dagger=&amp;\boldsymbol{V}^\dagger\boldsymbol{A}\,\boldsymbol{U}
-  \left(\boldsymbol{V}^\dagger\boldsymbol{A}\,\boldsymbol{U}\right)^\dagger\\
-  =&amp;\boldsymbol{V}^\dagger\boldsymbol{A}\,\boldsymbol{U}
-  \left(   \boldsymbol{U}^\dagger \boldsymbol{A}^\dagger\,\boldsymbol{V}\right)\\
-  =&amp;\boldsymbol{V}^\dagger
-  \left( \boldsymbol{A}  \boldsymbol{A}^\dagger\,\right) \boldsymbol{V}\,.\\
-\end{align}<p>Since $\boldsymbol{A}  \boldsymbol{A}^\dagger$ is hermitic and $\boldsymbol{A}  \boldsymbol{A}^\dagger$ is diagonal, then in fact there exists an unitary matrix $\boldsymbol{V}$.  Similarly there exists an unitary matrix $\boldsymbol{U}$ which diagonalizes  $\boldsymbol{A}^\dagger  \boldsymbol{A}$</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Scipy-implementation">Scipy implementation<a class="anchor-link" href="#Scipy-implementation"> </a></h3><p>The implementation in scipy is based in the inverted relation 
-\begin{align}
-  \boldsymbol{V}^\dagger\boldsymbol{A}\,\boldsymbol{U}=&amp;\boldsymbol{A}_{\text{diag}}\\
-  \boldsymbol{V}\boldsymbol{V}^\dagger\boldsymbol{A}\,\boldsymbol{U}\boldsymbol{U}^\dagger=&amp;\boldsymbol{V}\boldsymbol{A}_{\text{diag}}\boldsymbol{U}^\dagger\\
-  \boldsymbol{A}=&amp;\boldsymbol{V}\boldsymbol{A}_{\text{diag}}\boldsymbol{U}^\dagger\,.
-\end{align}</p>
-<p>The implementation is trough the module <code>scipy.linalg.svd</code>:</p>
-<div class="highlight"><pre><span></span>V,Adiag,Udagger<span class="o">=</span>linalg.svd<span class="o">(</span>A<span class="o">)</span>
-</pre></div>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Eigenvectors-and-eigenvalues">Eigenvectors and eigenvalues<a class="anchor-link" href="#Eigenvectors-and-eigenvalues"> </a></h3><p>If we make
-$$ \boldsymbol{U}=[\boldsymbol{U}_1\vdots\boldsymbol{U}_2\vdots\cdots\vdots\boldsymbol{U}_n], \qquad \boldsymbol{V}=[\boldsymbol{V}_1\vdots\boldsymbol{V}_2\vdots\cdots\vdots\boldsymbol{V}_n] $$</p>
-<p>We know that there exists a bi-diagonal transformación such that
-\begin{equation}
-%$$
-  \boldsymbol{A}\,\boldsymbol{U}=\boldsymbol{V}\boldsymbol{A}_{\text{diag}}
-\end{equation}
-\begin{equation}
-  \boldsymbol{A}\,\boldsymbol{U}_i=\lambda_i\boldsymbol{V}_i
-%$$
-\end{equation}
-not sum upon $i$. Here</p>
-<ul>
-<li>$\lambda_i$ are called eigenvalues</li>
-<li>$V_i$ and $U_i$ are the eigenvectors</li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>We can use this to check the proper order of the eigenvalues</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Transformation-of-a-linear-system">Transformation of a linear system<a class="anchor-link" href="#Transformation-of-a-linear-system"> </a></h3><p>We start again with the matrix equation, capitol bold letters denotes matrices
-\begin{equation}
-  \boldsymbol{A}\boldsymbol{X}=\boldsymbol{B}\,,
-\end{equation}
-where $\boldsymbol{A}$ is an $n \times n$ matrix.</p>
-<p>We know that there exists a bi-diagonal transformación such that
-\begin{equation}
-  \boldsymbol{V}^\dagger\boldsymbol{A}\,\boldsymbol{U}=\boldsymbol{A}_{\text{diag}}
-\end{equation}
-So, by doing standard operations we have
-\begin{align}
-  \boldsymbol{V}^\dagger \boldsymbol{A} \boldsymbol{U} \boldsymbol{U}^\dagger \boldsymbol{X}=&amp; \boldsymbol{V}^\dagger \boldsymbol{B}\\
-   \left( \boldsymbol{V}^\dagger \boldsymbol{A} \boldsymbol{U} \right) 
-   \left( \boldsymbol{U}^\dagger \boldsymbol{X}\right)=&amp; \boldsymbol{V}^\dagger \boldsymbol{B}\\
-  \boldsymbol{A}_{\text{diag}} \boldsymbol{X}'=&amp;\boldsymbol{B}'\,,      
-\end{align}
-where
-\begin{align}
-  \boldsymbol{X}'=&amp; \boldsymbol{U}^\dagger \boldsymbol{X}\,, &amp;    \boldsymbol{B}'=&amp; \boldsymbol{V}^\dagger \boldsymbol{B}\,,
-\end{align}
-or
-\begin{align}
-  \boldsymbol{X}=&amp; \boldsymbol{U}\boldsymbol{X}'\,, &amp;    \boldsymbol{B}=&amp; \boldsymbol{V}\boldsymbol{B}'\,.
-\end{align}</p>
-<p>If $\boldsymbol{A}_{\text{diag}}=\operatorname{diag}(\lambda_1,\lambda_2,\ldots \lambda_n)$, $\boldsymbol{X}^{\operatorname{T}}=\begin{pmatrix}x_1 &amp; x_2 &amp;\cdots &amp; x_n\end{pmatrix}^{\operatorname{T}}$ and $\boldsymbol{B}^{\operatorname{T}}=\begin{pmatrix}b_1&amp; b_2 &amp;\cdots &amp; b_n\end{pmatrix}^{\operatorname{T}}$,
-the solution of the system is given by
-\begin{equation}
-\lambda_i x'_i=b_i'\,.
-\end{equation}</p>
-<p>Note that
-\begin{align}
-   \boldsymbol{X}'=&amp;\boldsymbol{A}_{\text{diag}}^{-1}\boldsymbol{B}'\,,      
-\end{align}
-and the final solution is
-\begin{align}
-   \boldsymbol{X}=&amp;U \boldsymbol{X}'\\ 
-               =&amp;U\boldsymbol{A}_{\text{diag}}^{-1}V^\dagger \boldsymbol{B}\,,      
-\end{align}
-Therefore
-$$A^{-1}=U\boldsymbol{A}_{\text{diag}}^{-1}V^\dagger$$</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Example">Example<a class="anchor-link" href="#Example"> </a></h3>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>A suitable way to introduce this method is applying it to some basic problem. To do so, let's take the result of the <a href="#Example-1">Example 1</a>:</p>
-$$ \begin{bmatrix}
-5 &amp; -4 &amp; 0 \\
--4 &amp; 7 &amp; -3 \\ 
-0 &amp; -3 &amp; 5
-\end{bmatrix}
-\begin{bmatrix}
-x_{1} \\
-x_{2} \\
-x_{3} 
-\end{bmatrix}  =
-\begin{bmatrix}
-1 \\
-0 \\
--2
-\end{bmatrix}
-$$<p>As the matrix is symmetric $\boldsymbol{V}=\boldsymbol{U}$ and $\boldsymbol{U}^\dagger=\boldsymbol{U}^{\operatorname{T}}$</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">M1</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">5</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
-             <span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">],</span>
-             <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">]])</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">A</span><span class="o">=</span><span class="n">M1</span>
-<span class="n">A</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([[ 5, -4,  0],
-       [-4,  7, -3],
-       [ 0, -3,  5]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Check if all eigenvalues are different from zero:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>49.99999999999999</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">B</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>  <span class="p">]</span>
-<span class="n">B</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([[ 1],
-       [ 0],
-       [-2]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Also as</p>
-<div class="highlight"><pre><span></span><span class="n">B</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">],[</span><span class="mi">0</span><span class="p">],[</span><span class="o">-</span><span class="mi">2</span><span class="p">]])</span>
-<span class="c1">#or</span>
-<span class="n">B</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span>  <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">],(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span> <span class="p">)</span>
-</pre></div>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Theorem-1-in-numpy">Theorem 1 in <code>numpy</code><a class="anchor-link" href="#Theorem-1-in-numpy"> </a></h3>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">λ</span><span class="p">,</span><span class="n">V</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span> <span class="n">A</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">A_diag</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">λ</span><span class="p">)</span>
-<span class="n">A_diag</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([[11.09901951,  0.        ,  0.        ],
-       [ 0.        ,  0.90098049,  0.        ],
-       [ 0.        ,  0.        ,  5.        ]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">V</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([[-5.07191124e-01, -6.18673713e-01, -6.00000000e-01],
-       [ 7.73342141e-01, -6.33988906e-01,  1.91548674e-16],
-       [-3.80393343e-01, -4.64005285e-01,  8.00000000e-01]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>We first check the proper order of the diagonalization</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span>  <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">V</span><span class="o">.</span><span class="n">transpose</span><span class="p">(),</span><span class="n">A</span>  <span class="p">),</span> <span class="n">V</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([[11.09901951,  0.        ,  0.        ],
-       [ 0.        ,  0.90098049,  0.        ],
-       [ 0.        ,  0.        ,  5.        ]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Since</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">V</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([[-5.07191124e-01, -6.18673713e-01, -6.00000000e-01],
-       [ 7.73342141e-01, -6.33988906e-01,  1.91548674e-16],
-       [-3.80393343e-01, -4.64005285e-01,  8.00000000e-01]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>The final solution is:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">A_diag_inv</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">λ</span><span class="p">)</span>
-<span class="n">A_diag_inv</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([[0.09009805, 0.        , 0.        ],
-       [0.        , 1.10990195, 0.        ],
-       [0.        , 0.        , 0.2       ]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>check with <code>np.linalg.inv(A_diag)</code></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">X</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">V</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">λ</span><span class="p">)</span> <span class="p">),</span><span class="n">V</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span> <span class="p">),</span><span class="n">B</span><span class="p">)</span>
-<span class="n">X</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>array([[ 0.04],
-       [-0.2 ],
-       [-0.52]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong><font color="red">Activity</font></strong>: Usar np.lingalg.solve</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="input">
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">B</span><span class="o">.</span><span class="n">transpose</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>array([ 1,  0, -2])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="input">
-
-<div class="inner_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="o">.</span><span class="n">transpose</span><span class="p">()[</span><span class="mi">0</span><span class="p">])</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>array([ 0.04, -0.2 , -0.52])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>We can now check some properties.</p>
-<ul>
-<li>Obtain $\theta_{12}$ for $\lambda_1&lt;\lambda_2&lt;\lambda_3$</li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
-
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">V</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>array([[-5.07191124e-01, -6.18673713e-01, -6.00000000e-01],
-       [ 7.73342141e-01, -6.33988906e-01,  1.91548674e-16],
-       [-3.80393343e-01, -4.64005285e-01,  8.00000000e-01]])</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span> <span class="n">V</span><span class="p">[:</span> <span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="p">]</span>
-</pre></div>
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-<pre>array([[-0.50719112],
-       [ 0.77334214],
-       [-0.38039334]])</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">A_diag</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span><span class="n">λ</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
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-<pre>(11.099019513592784, 11.099019513592784)</pre>
-</div>
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-<p>We define the eigenvector $V_i$ as</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">V0</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span> <span class="n">V</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="p">]</span>
-<span class="n">V1</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span> <span class="n">V</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span> <span class="p">]</span>
-<span class="n">V2</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span> <span class="n">V</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="p">]</span>
-<span class="n">display</span><span class="p">(</span><span class="n">V0</span><span class="p">)</span>
-<span class="n">display</span><span class="p">(</span><span class="n">V1</span><span class="p">)</span>
-<span class="n">V2</span>
-</pre></div>
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-<pre>array([[-0.50719112],
-       [ 0.77334214],
-       [-0.38039334]])</pre>
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-<pre>array([[-0.61867371],
-       [-0.63398891],
-       [-0.46400528]])</pre>
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-<pre>array([[-6.00000000e-01],
-       [ 1.91548674e-16],
-       [ 8.00000000e-01]])</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;</span><span class="si">{}</span><span class="s1"> =</span><span class="se">\n</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">V0</span><span class="p">),</span><span class="n">λ</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">V0</span> <span class="p">)</span> <span class="p">)</span>
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-<pre>[[-5.62932419]
- [ 8.58333952]
- [-4.22199314]] =
-[[-5.62932419]
- [ 8.58333952]
- [-4.22199314]]
-</pre>
-</div>
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-<p>Check: $ A V_i=\lambda_i V_i$</p>
-<p>Which means the eigenvalue associated to the "operator" $A$ acting on the eigenvector $V_1$</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;</span><span class="si">{}</span><span class="s1"> =</span><span class="se">\n</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">V1</span><span class="p">),</span><span class="n">λ</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">V1</span> <span class="p">)</span> <span class="p">)</span>
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-<pre>[[-0.55741294]
- [-0.57121163]
- [-0.41805971]] =
-[[-0.55741294]
- [-0.57121163]
- [-0.41805971]]
-</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;</span><span class="si">{}</span><span class="s1"> =</span><span class="se">\n</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">V2</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">14</span><span class="p">),</span>
-                         <span class="p">(</span><span class="n">λ</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">V2</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span> <span class="p">)</span> <span class="p">)</span>
-</pre></div>
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-<pre>[[-3.]
- [ 0.]
- [ 4.]] =
-[[-3.]
- [ 0.]
- [ 4.]]
-</pre>
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-<p>The diagonalization matrix can be rebuild from the eigenvectors</p>
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-<pre>array([[-5.07191124e-01, -6.18673713e-01, -6.00000000e-01],
-       [ 7.73342141e-01, -6.33988906e-01,  1.91548674e-16],
-       [-3.80393343e-01, -4.64005285e-01,  8.00000000e-01]])</pre>
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-<p>is rebuild with</p>
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-</pre></div>
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-<p>or with: <code>np.hstack((V0,V1,V2))</code></p>
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-<p>Note that a sign of an egigenvalues can be changed:</p>
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-</pre></div>
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-</pre></div>
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-<pre>array([[11.09901951, -0.        ,  0.        ],
-       [-0.        ,  0.90098049, -0.        ],
-       [ 0.        , -0.        ,  5.        ]])</pre>
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-<h3 id="Eigenvector-reordering">Eigenvector reordering<a class="anchor-link" href="#Eigenvector-reordering"> </a></h3>
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-<p><span style="color:red">__Activity__</span>: <a href="https://beta.deepnote.com/project/17b487c8-b092-4032-94f5-438ba4eeb1e9">https://beta.deepnote.com/project/17b487c8-b092-4032-94f5-438ba4eeb1e9</a></p>
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-<p>We can use this to check the proper order of the eigenvalues</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">U</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span> <span class="n">V1</span><span class="p">,</span><span class="n">V2</span><span class="p">,</span><span class="n">V0</span><span class="p">]</span>
-<span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span>  <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">U</span><span class="o">.</span><span class="n">transpose</span><span class="p">(),</span><span class="n">A</span>  <span class="p">),</span> <span class="n">U</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
-</pre></div>
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-<pre>array([[ 0.90098049,  0.        ,  0.        ],
-       [ 0.        ,  5.        ,  0.        ],
-       [ 0.        ,  0.        , 11.09901951]])</pre>
-</div>
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-<p>The order of eigenvalues can now be changed by changing the order of the eigenvectors and redifining the diagonalization matrix. For example, from small to large. We define first the eigenvalues</p>
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-</pre></div>
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-<pre>array([[-6.18673713e-01, -6.00000000e-01, -5.07191124e-01],
-       [-6.33988906e-01,  1.91548674e-16,  7.73342141e-01],
-       [-4.64005285e-01,  8.00000000e-01, -3.80393343e-01]])</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">θ_12</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arctan</span><span class="p">(</span> <span class="n">U</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">/</span><span class="n">U</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">θ_12</span><span class="p">,</span> <span class="n">θ_12</span><span class="o">*</span><span class="mi">180</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span>
-</pre></div>
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-<pre>(0.7700763823614476, 44.122126612013574)</pre>
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-<p>Generalization of the reordering</p>
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-<pre>array([ 0.90098049,  5.        , 11.09901951])</pre>
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-<p>To reverse the order</p>
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-<pre>array([ 11.09901951,   0.90098049,   5.        ])</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">index</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">λ</span><span class="p">)</span><span class="o">.</span><span class="n">argsort</span><span class="p">()</span>
-<span class="n">index</span>
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-<pre>array([1, 2, 0])</pre>
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-<p>can be implemented in general with a <em>comprehensive</em> list</p>
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-<pre>array([[-5.07191124e-01, -6.18673713e-01, -6.00000000e-01],
-       [ 7.73342141e-01, -6.33988906e-01,  1.91548674e-16],
-       [-3.80393343e-01, -4.64005285e-01,  8.00000000e-01]])</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span> <span class="nb">tuple</span><span class="p">(</span> <span class="p">[</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">V</span><span class="p">[:,</span><span class="n">i</span><span class="p">]]</span>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="p">]</span> <span class="p">)</span> <span class="p">]</span>
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-<pre>array([[-5.07191124e-01, -6.18673713e-01, -6.00000000e-01],
-       [ 7.73342141e-01, -6.33988906e-01,  1.91548674e-16],
-       [-3.80393343e-01, -4.64005285e-01,  8.00000000e-01]])</pre>
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-<p>Changing the order to <code>index</code></p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">U</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span> <span class="nb">tuple</span><span class="p">(</span> <span class="p">[</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">V</span><span class="p">[:,</span><span class="n">i</span><span class="p">]]</span>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">λ</span><span class="p">)</span><span class="o">.</span><span class="n">argsort</span><span class="p">()</span> <span class="p">]</span> <span class="p">)</span> <span class="p">]</span>
-<span class="n">U</span>
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-<pre>array([[-6.18673713e-01, -6.00000000e-01, -5.07191124e-01],
-       [-6.33988906e-01,  1.91548674e-16,  7.73342141e-01],
-       [-4.64005285e-01,  8.00000000e-01, -3.80393343e-01]])</pre>
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-<p>or:</p>
-<div class="highlight"><pre><span></span><span class="n">n</span><span class="o">=</span><span class="mi">3</span>
-<span class="n">U</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">(</span> <span class="p">[</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span> <span class="n">V</span><span class="p">[:,</span><span class="n">i</span><span class="p">],</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span> <span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">index</span>   <span class="p">]</span> <span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span>  <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">U</span><span class="o">.</span><span class="n">transpose</span><span class="p">(),</span><span class="n">A</span>  <span class="p">),</span> <span class="n">U</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
-</pre></div>
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-<pre>array([[ 0.90098049,  0.        ,  0.        ],
-       [ 0.        ,  5.        ,  0.        ],
-       [ 0.        ,  0.        , 11.09901951]])</pre>
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-<p><font color="red">__Activity__</font>:  Build a function that diagonalize with increasing order in the eigenvalues as a replacement of <code>np.linalg.eig</code></p>
-<div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">argeig</span><span class="p">(</span><span class="n">A</span><span class="p">):</span>
-    <span class="n">l</span><span class="p">,</span><span class="n">V</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-    <span class="o">....</span>
-    <span class="k">return</span> <span class="n">argl</span><span class="p">,</span> <span class="n">argV</span>
-</pre></div>
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-<p>and check that</p>
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-<!--
-def argeig(a):
-    """
-    Diagonalize with increasing order in the eigenvalues.
-
-    See help(np.linalg.eig)
-    """
-    l,V=np.linalg.eig(a)
-    index=np.abs(l).argsort()
-    argl= np.sort( l)
-    argV=np.c_[ tuple([ np.c_[ V[:,i]] for i in index   ]) ]
-    return argl, argV
--->
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-<div class="highlight"><pre><span></span><span class="c1">#1.</span>
-<span class="n">λ</span><span class="p">,</span><span class="n">V</span><span class="o">=</span><span class="n">argeig</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="n">λ</span><span class="p">,</span><span class="n">V</span>
-<span class="n">Out</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-<span class="p">(</span><span class="n">array</span><span class="p">([</span> <span class="mf">0.90098049</span><span class="p">,</span>  <span class="mf">5.</span>        <span class="p">,</span> <span class="mf">11.09901951</span><span class="p">]),</span>
- <span class="n">array</span><span class="p">([[</span><span class="o">-</span><span class="mf">6.18673713e-01</span><span class="p">,</span> <span class="o">-</span><span class="mf">6.00000000e-01</span><span class="p">,</span> <span class="o">-</span><span class="mf">5.07191124e-01</span><span class="p">],</span>
-        <span class="p">[</span><span class="o">-</span><span class="mf">6.33988906e-01</span><span class="p">,</span>  <span class="mf">1.91548674e-16</span><span class="p">,</span>  <span class="mf">7.73342141e-01</span><span class="p">],</span>
-        <span class="p">[</span><span class="o">-</span><span class="mf">4.64005285e-01</span><span class="p">,</span>  <span class="mf">8.00000000e-01</span><span class="p">,</span> <span class="o">-</span><span class="mf">3.80393343e-01</span><span class="p">]]))</span>
-</pre></div>
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-<div class="highlight"><pre><span></span><span class="c1">#2.</span>
-<span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span>  <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">V</span><span class="o">.</span><span class="n">transpose</span><span class="p">(),</span><span class="n">A</span>  <span class="p">),</span> <span class="n">V</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
-<span class="n">Out</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-<span class="n">array</span><span class="p">([[</span> <span class="mf">0.90098049</span><span class="p">,</span>  <span class="mf">0.</span>        <span class="p">,</span>  <span class="mf">0.</span>        <span class="p">],</span>
-       <span class="p">[</span> <span class="mf">0.</span>        <span class="p">,</span>  <span class="mf">5.</span>        <span class="p">,</span>  <span class="mf">0.</span>        <span class="p">],</span>
-       <span class="p">[</span> <span class="mf">0.</span>        <span class="p">,</span>  <span class="mf">0.</span>        <span class="p">,</span> <span class="mf">11.09901951</span><span class="p">]])</span>
-</pre></div>
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-<div class="highlight"><pre><span></span><span class="c1">#3.</span>
-<span class="nb">print</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">A</span><span class="o">-</span> <span class="n">λ</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="p">)</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">A</span><span class="o">-</span> <span class="n">λ</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="p">)</span> <span class="p">)</span>
-<span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">A</span><span class="o">-</span> <span class="n">λ</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="p">)</span>
-</pre></div>
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-<h3 id="General-matrix">General matrix<a class="anchor-link" href="#General-matrix"> </a></h3>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">linalg</span>
-</pre></div>
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-<p><strong>Example</strong>:</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#A=np.array([[ 2.5       ,  6.92820323],</span>
-<span class="c1">#            [-4.33012702,  4.        ]])</span>
-</pre></div>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">A</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mf">6.66674644</span><span class="p">,</span>  <span class="mf">3.13121253</span><span class="p">],</span>
-            <span class="p">[</span><span class="o">-</span><span class="mf">0.23343505</span><span class="p">,</span>  <span class="mf">5.8902893</span> <span class="p">]])</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">linalg</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">V</span><span class="p">,</span><span class="n">diag</span><span class="p">,</span><span class="n">Udagger</span><span class="o">=</span><span class="n">linalg</span><span class="o">.</span><span class="n">svd</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
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-<pre>array([[8., 0.],
-       [0., 5.]])</pre>
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-<p>This is important to stablish that the eigenvectors are determined until ordering and permutations</p>
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-<pre>array([[-0.8660254, -0.5      ],
-       [-0.5      ,  0.8660254]])</pre>
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-<pre>array([[-0.70710678, -0.70710678],
-       [-0.70710678,  0.70710678]])</pre>
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-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[[</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">θ</span><span class="p">)</span> <span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">θ</span><span class="p">)],</span>
-                      <span class="p">[</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">θ</span><span class="p">),</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">θ</span><span class="p">)]]</span>   <span class="p">)</span>
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-<pre>array([[ 0.5      ,  0.8660254],
-       [-0.8660254,  0.5      ]])</pre>
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-<pre>array([[ 0.5      ,  0.8660254],
-       [-0.8660254,  0.5      ]])</pre>
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-<pre>array([[ 0.70710678,  0.70710678],
-       [-0.70710678,  0.70710678]])</pre>
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-<pre>array([[ 0.70710678,  0.70710678],
-       [-0.70710678,  0.70710678]])</pre>
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-<pre>array([[5., 0.],
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-<p><span style="color:red">__Activity__</span>: <a href="https://beta.deepnote.com/project/17b487c8-b092-4032-94f5-438ba4eeb1e9">https://beta.deepnote.com/project/17b487c8-b092-4032-94f5-438ba4eeb1e9</a></p>
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-<p><font color="red">__Activity__</font>: Solve the system
-$$ \boldsymbol{A} \boldsymbol{x}=\boldsymbol{B}$$
-for the previous $ \boldsymbol{A}$ matrix and
-$$\boldsymbol{B}=\begin{bmatrix}
-   1\\
-   -4\\
-   \end{bmatrix}$$</p>
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-<h3 id="Interpretations-of-Theorem-2">Interpretations of Theorem 2<a class="anchor-link" href="#Interpretations-of-Theorem-2"> </a></h3>
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-<h4 id="Single-Diagonalization-matrix">Single Diagonalization matrix<a class="anchor-link" href="#Single-Diagonalization-matrix"> </a></h4><p>In Theorem 2 establishes a unique set of a matrix $A$ with its eigenvalues and eigenvectors. However, for a fixed set of eigenvalues and eigenvectors we can have several possibilities of $A'$ matrices. For example, if we fix $U'=\boldsymbol{1}$ 
-\begin{equation}
-  {V'}^\dagger \boldsymbol{A'}=\boldsymbol{A}_{\text{diag}}\,,
-\end{equation}
-so that
-\begin{equation}
-  \boldsymbol{A}'=V'\boldsymbol{A}_{\text{diag}}
-\end{equation}</p>
-<p>Therefore: Under this conditions, for a fixed set of eigenvalues and eigenvectors (associated to the matrix $V'$) there is a unique $A'$ matrix</p>
-<p><strong>Example</strong>
-Let $V'=U^{\operatorname{T}} V$ in the previous example. Find the matrix $A'$ which gives rise to the same eigenvalues</p>
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-                 <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">8</span><span class="p">]]</span> <span class="p">)</span>
-</pre></div>
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-<span class="n">Vp</span>
-</pre></div>
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-<pre>array([[ 0.96592583,  0.25881905],
-       [-0.25881905,  0.96592583]])</pre>
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-<span class="n">Ap</span>
-</pre></div>
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-<pre>array([[ 4.82962913,  2.07055236],
-       [-1.29409523,  7.72740661]])</pre>
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-<pre>array([[5., 0.],
-       [0., 8.]])</pre>
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-<p>Finally, we can use the full procedure of hermitic matrices to obtain the bidiagonal matrices $U,V$</p>
-<p><strong>Example</strong>:
-Diagonalize the matrix $A$ by using the Theorem 1 instead</p>
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-            <span class="p">[</span><span class="o">-</span><span class="mf">0.23343505</span><span class="p">,</span>  <span class="mf">5.8902893</span> <span class="p">]])</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">A</span><span class="p">,</span><span class="n">A</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span> <span class="p">)</span>
-</pre></div>
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-<pre>array([[54.25      , 16.88749537],
-       [16.88749537, 34.74999996]])</pre>
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-<pre>array([[44.50000002, 19.50000001],
-       [19.50000001, 44.49999995]])</pre>
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-<p>→ is obtained with <code>&lt;ALT GR&gt;+i</code> or, similarly: ↓←</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">λ2</span><span class="p">,</span><span class="n">V</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">A</span><span class="p">,</span><span class="n">A</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span> <span class="p">)</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">λ2</span><span class="p">,</span><span class="s1">&#39;→&#39;</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">λ2</span><span class="p">))</span>
-<span class="c1">#V=np.c_[ -V[:,0],V[:,0]  ]</span>
-<span class="n">V</span>
-</pre></div>
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-<pre>[63.99999999 24.99999997] → [8. 5.]
-</pre>
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-<pre>array([[ 0.8660254, -0.5      ],
-       [ 0.5      ,  0.8660254]])</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">λ2p</span><span class="p">,</span><span class="n">U</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">A</span><span class="o">.</span><span class="n">transpose</span><span class="p">(),</span><span class="n">A</span> <span class="p">)</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">λ2</span><span class="p">))</span>
-<span class="n">U</span>
-</pre></div>
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-<pre>[8. 5.]
-</pre>
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-<pre>array([[ 0.70710678, -0.70710678],
-       [ 0.70710678,  0.70710678]])</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">V</span><span class="o">.</span><span class="n">transpose</span><span class="p">(),</span><span class="n">A</span> <span class="p">),</span> <span class="n">U</span> <span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
-</pre></div>
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-<pre>array([[ 8., -0.],
-       [ 0.,  5.]])</pre>
-</div>
-
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-<p><span style="color:red">__Actividad:__</span> cambiar el orden de los autovectores para obtener un orden ascendente en los autovalores</p>
-
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-<h2 id="Mixed-terms">Mixed terms<a class="anchor-link" href="#Mixed-terms"> </a></h2>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Let:
-\begin{align}
-X' = 
-\begin{bmatrix}
-B \\ 
-W \\
-\end{bmatrix}
-\end{align}</p>
-<p>Consider  the quadratic equation 
-\begin{align}
-X^{\prime\operatorname{T}} M X^\prime=&amp; \begin{bmatrix}
-B &amp; W 
-\end{bmatrix}
-\begin{bmatrix}
-M_{11} &amp; M_{12} \\
-M_{12} &amp; M_{22} \\
-\end{bmatrix}
-\begin{bmatrix}
-B \\ 
-W \\
-\end{bmatrix}\\
-=&amp; 
-\begin{bmatrix}
-B &amp; W 
-\end{bmatrix}
-\begin{bmatrix}
-M_{11}B + M_{12}W \\
-M_{12}B + M_{22}W \\
-\end{bmatrix}\\
-=&amp; 
-B( M_{11}B + M_{12}W)+  W ( M_{12}B + M_{22}W )\\
-=&amp;M_{11} B^2 + 2M_{12} BW+ M_{22} W^2\,. 
-\end{align}
-The quadratic equation is in terms of: $M_{11}$, $M_{12}$ y $M_{22}$</p>
-<p>We can simplify this expression if we change to a new basis 
-$$
-X=\begin{bmatrix}
-A\\
-Z
-\end{bmatrix}
-$$
-in which $M$ is diagonal, in such a case the crossed term would disappear. 
-The rotation from $X'\to X$ is defined by
-\begin{align}
-X'=
-\begin{bmatrix}
-B\\
-W
-\end{bmatrix}\equiv
-\begin{bmatrix}
-\cos\theta &amp; \sin\theta\\
--\sin\theta &amp; \cos\theta
-\end{bmatrix}
-\begin{bmatrix}
-A\\
-Z
-\end{bmatrix}=
-V X 
-\end{align}
-where $V$ is the rotation matrix
-$$
-V=\begin{bmatrix}
-\cos\theta &amp; \sin\theta\\
--\sin\theta &amp; \cos\theta
-\end{bmatrix}.
-$$
-Therefore
-\begin{align}
-X\equiv
-\begin{bmatrix}
-A\\
-Z
-\end{bmatrix}=V^{\operatorname{T}} X' \to X^{\operatorname{T}}= X^{\prime\operatorname{T}} V\,,
-\end{align}</p>
-<p>In the new basis
-\begin{align}
-X^{\prime\operatorname{T}} M X^\prime=&amp;X^{\prime\operatorname{T}}V V^{\operatorname{T}} M V V^{\operatorname{T}} X^\prime\\
-=&amp;(X^{\prime\operatorname{T}}V) (V^{\operatorname{T}} M V) (V^{\operatorname{T}} X^\prime)\\
-=&amp; X^{\operatorname{T}} M_{\text{diag}} X \\
-=&amp;  \lambda_1 A^2+\lambda_2 Z^2\,.
-\end{align}</p>
-
-</div>
-</div>
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-</div>
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-<p>such that $|\lambda_1|\le|\lambda_2|$, where
-\begin{align}
-M_{\text{diag}}\equiv V^{\operatorname{T}} M V=\begin{bmatrix}
-\lambda_1 &amp; 0 \\ 
-0 &amp; \lambda_2 \\
-\end{bmatrix}\,,
-\end{align}</p>
-<p>In this basis, the quadratic equation is in terms of eigenvalues and mixing angle, $\theta$.
-Therefore, there are not longer mixed terms.</p>
-<p>The diagonalization of quadratic equations can be straightforwardly  generalized to $n$-th degree equations in terms of $n\times n$ matrices</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="n">g</span> <span class="o">=</span><span class="mf">0.64996</span>
-<span class="n">gp</span><span class="o">=</span><span class="mf">0.35523</span>
-<span class="n">v</span><span class="o">=</span><span class="mf">246.22046</span>
-</pre></div>
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-<h3 id="Example:-Electroweak-interactions">Example: Electroweak interactions<a class="anchor-link" href="#Example:-Electroweak-interactions"> </a></h3><p>To understand the electromagnetic and weak fundamental interactions, the mathematical formulation need to be done in one basis where the photon field, denoted with a symbol $A$, is still not well defined. Instead, the field $B$, the precursor of $A$, appears along with the weak field $W$, the precursor of the electroweak field $Z$. In the <em>mathematical</em> basis we have then
-\begin{align}
-X' = 
-\begin{bmatrix}
-B \\ 
-W \\
-\end{bmatrix},
-\end{align}
-and there, the symmetric mass matrix is calculated as (Details here: <a href="https://github.com/restrepo/TCC/releases/latest">PDF</a>)</p>
-
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-<!-- 
-sin2θ=0.23
-θ=np.arcsin( np.sqrt(sin2θ)  )
-print(θ)
-MZ=91.1876
-GF=1.166371E-5
-v=1/np.sqrt(np.sqrt(2)*GF)
-g=2*MZ*np.cos(θ)/v
-gp=g*np.tan(θ)
-gp
--->
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">M</span><span class="o">=</span><span class="p">(</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="n">gp</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="n">g</span><span class="o">*</span><span class="n">gp</span><span class="p">],</span>
-                     <span class="p">[</span><span class="o">-</span><span class="n">g</span><span class="o">*</span><span class="n">gp</span><span class="p">,</span> <span class="n">g</span><span class="o">**</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">M</span>
-</pre></div>
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-<pre>array([[ 1912.52692086, -3499.32718938],
-       [-3499.32718938,  6402.67629426]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
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-<p>Checking that the determinant is zero</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">M</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">8</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
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-<pre>0.0</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<p>This imply that one egivanlue is zero</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigvals</span><span class="p">(</span><span class="n">M</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">12</span><span class="p">)</span>
-</pre></div>
-
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-<pre>array([   0.        , 8315.20321512])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<p>which means that the matrix rank, the number of non-zero eigenvalues, is 1</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_rank</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-</pre></div>
-
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-</div>
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-<pre>1</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>To make the change to the <em>phyisical</em> basis,
-$$
-X=\begin{bmatrix}
-A\\
-Z
-\end{bmatrix}
-$$
-through the rotation matrix
-$$
-V=\begin{bmatrix}
-\cos\theta_W &amp; \sin\theta_W\\
--\sin\theta_W &amp; \cos\theta_W
-\end{bmatrix},
-$$
-the following transformation need to be established
-\begin{align}
-X'=
-\begin{bmatrix}
-B\\
-W
-\end{bmatrix}\equiv
-\begin{bmatrix}
-\cos\theta_W &amp; \sin\theta_W\\
--\sin\theta_W &amp; \cos\theta_W
-\end{bmatrix}
-\begin{bmatrix}
-A\\
-Z
-\end{bmatrix}=
-V X\,,
-\end{align}
-such that $V$ is the diagonalization matrix of $M$</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
-
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-<div class="text_cell_render border-box-sizing rendered_html">
-$$M_{\text{diag}}=V^{\operatorname{T}} M V$$<p>with the normal ordering: $|\lambda_1|\le|\lambda_2|$, and</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="text_cell_render border-box-sizing rendered_html">
-<ul>
-<li>$V$ → Diagonalization matrix</li>
-<li>$V$ → Ortogonal matrix</li>
-<li>$V$ → Rotation matrix</li>
-</ul>
-<p>To obtain the eigensystem, we use</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">λ</span><span class="p">,</span><span class="n">V</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
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-</div>
-</div>
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
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-<p>which in fact satisfy</p>
-
-</div>
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-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">V</span><span class="o">.</span><span class="n">transpose</span><span class="p">(),</span><span class="n">M</span><span class="p">)</span> <span class="p">,</span> <span class="n">V</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">12</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
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-<pre>array([[  -0.        ,    0.        ],
-       [  -0.        , 8315.20321512]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Since the first (zero) eigenvalue is the one associated to $A$, we can interpret directly the rotation matrix without changing the order of the eigenvectors</p>
-
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">V</span>
-</pre></div>
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-<pre>array([[-0.87749437,  0.47958694],
-       [-0.47958694, -0.87749437]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Note that in fact, the absolute value of the first element of the first eigenvector is the larger one and corresponds to the component along the $B$-axis.</p>
-
-</div>
-</div>
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-</div>
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
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-<p>Therefore</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">θ_W</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arcsin</span><span class="p">(</span> <span class="n">V</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span> <span class="p">)</span>
-<span class="n">θ_W</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
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-<pre>0.5001839211647364</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>The eigenvalue associated to $Z$ is</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">mZ</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">λ</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-<span class="n">mZ</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<pre>91.18773610041056</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<p>corresponding to the  $Z$ mass in units of $\text{GeV}/c^2$. As a reference, the proton mass is approximately $1\ \text{GeV}/c^2$</p>
-<p>The physical observable associated to the <em>weak mixing angle</em>, $\theta_W$, is (see <a href="https://pdg.lbl.gov/2019/reviews/rpp2018-rev-phys-constants.pdf">PDF</a>, along with the $Z^0$ <em>boson mass</em>, $m_Z$)</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">θ_W</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<pre>0.23000362966286086</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Neutrino-mass-matrix">Neutrino mass matrix<a class="anchor-link" href="#Neutrino-mass-matrix"> </a></h2><p>Returning back to the neutrino mixing discussion, it is worth noticing that in the mathematical basis
-$$
-N=\left(\begin{array}{l}
-\left|\nu_{e}\right\rangle \\
-\left|\nu_{\mu}\right\rangle \\
-\left|\nu_{\tau}\right\rangle
-\end{array}\right)
-$$
-the mass matrix, $\mathcal{M}_\nu$ is non-diagonal. However, if we assume that it is symmetric, then the previous rotation matrix there
-$$
-\left(\begin{array}{l}
-\left|\nu_{e}\right\rangle \\
-\left|\nu_{\mu}\right\rangle \\
-\left|\nu_{\tau}\right\rangle
-\end{array}\right)= \boldsymbol{U}\left(\begin{array}{l}
-\left|\nu_{1}\right\rangle \\
-\left|\nu_{2}\right\rangle \\
-\left|\nu_{3}\right\rangle
-\end{array}\right),
-$$
-corresponds to the diagonalization matrix
-$$
-\boldsymbol{U}^\dagger \mathcal{M}_\nu \boldsymbol{U}=\operatorname{diag}(m_1,m_2,m_3).
-$$</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Casas-Ibarra-parameterization">Casas-Ibarra parameterization<a class="anchor-link" href="#Casas-Ibarra-parameterization"> </a></h3><p>Consider a $n\times n$ symmetric matrix $A$. We can assumme without lost of generality that this can be generated from a matrix $Y$ such that
-$$
-A=Y^{\operatorname{T}}Y
-$$
-Theorem 1 gurantees that exists an ortogonal matrix $U$ such that
-$$
-U^{\operatorname{T}} A U=U^{\operatorname{T}} Y^{\operatorname{T}}Y U=D_\lambda
-$$
-where
-$$
-D_{\lambda}=A_{\text{diag}}=\operatorname{diag}\left(\lambda_1,\lambda_2,\ldots,\lambda_n\right)
-$$
-where $\lambda_i$ are the eigenvalues of $A$. Therefore
-\begin{align}
- Y^{\operatorname{T}}Y =&amp;U D_\lambda U^{\operatorname{T}}\\
- =&amp;U D_{\sqrt{\lambda}} D_{\sqrt{\lambda}} U^{\operatorname{T}}\\
-\end{align}
-where
-$$
-D_{\sqrt{\lambda}}=\operatorname{diag}\left(\sqrt{\lambda_1},\sqrt{\lambda_2},\ldots \sqrt{\lambda_n}\right)
-$$
-Therefore, exists an ortogonal arbitrary matrix $R$, such that
-$$
- Y^{\operatorname{T}}Y =U D_{\sqrt{\lambda}}R^{\operatorname{T}}R D_{\sqrt{\lambda}} U^{\operatorname{T}}\\
-$$</p>
-<p>In this way, the matrix $Y$ can be parameterized in terms of $R$ as
-$$
-Y=R D_{\sqrt{\lambda}} U^{\operatorname{T}}
-$$</p>
-<p>1) By using the previous equations, build a $2\times 2$  $Y$ matrix  with the following conditions</p>
-<ul>
-<li>$R$ is an orthogonal matrix with a rotation angle as a random number between $(0,2\pi)$. Use your identification number as the seed of the random number generator.</li>
-<li>The eigenvalues are $\lambda_1=2$ and $\lambda_2=4$. </li>
-<li>$U$ is a diagonalization matrix with mixing angle $\pi/4$</li>
-</ul>
-<p>2) Build the matrix $A$ and check that has the proper eigenvalues and eigenvectors</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>1)</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-
-<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">98554576</span><span class="p">)</span>
-
-<span class="n">θ</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span>
-
-<span class="k">def</span> <span class="nf">orthogonal</span><span class="p">(</span><span class="n">θ</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[[</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">θ</span><span class="p">),</span>  <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">θ</span><span class="p">)],</span>
-                      <span class="p">[</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">θ</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">θ</span><span class="p">)</span> <span class="p">]]</span>    <span class="p">)</span>
-</pre></div>
-
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-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">λ1</span><span class="o">=</span><span class="mi">2</span><span class="p">;</span> <span class="n">λ2</span><span class="o">=</span><span class="mi">4</span>
-<span class="n">R</span><span class="o">=</span><span class="n">orthogonal</span><span class="p">(</span><span class="n">θ</span><span class="p">)</span>
-<span class="n">U</span><span class="o">=</span><span class="n">orthogonal</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span>
-<span class="n">D_sqrtλ</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">([</span><span class="n">λ1</span><span class="p">,</span><span class="n">λ2</span><span class="p">])</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;R=&#39;</span><span class="p">)</span>
-<span class="n">R</span>
-</pre></div>
-
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-</div>
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-<pre>R=
-</pre>
-</div>
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-<pre>array([[-0.81433017,  0.58040191],
-       [-0.58040191, -0.81433017]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>From the equation for $Y$</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Y</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">R</span><span class="p">,</span><span class="n">D_sqrtλ</span><span class="p">),</span> <span class="n">U</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>The symmetric matrix is</p>
-
-</div>
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-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">A</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">Y</span><span class="o">.</span><span class="n">transpose</span><span class="p">(),</span> <span class="n">Y</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>2)</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">λ</span><span class="p">,</span><span class="n">Unew</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Eigenvalues=</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">λ</span><span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;U=&#39;</span><span class="p">)</span>
-<span class="n">Unew</span>
-</pre></div>
-
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-</div>
-</div>
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-<pre>Eigenvalues=[4. 2.]
-U=
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([[ 0.70710678, -0.70710678],
-       [ 0.70710678,  0.70710678]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>After the reordering of the eigenvalues, we would obtain the original $U$ and therefore</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">U</span><span class="o">.</span><span class="n">transpose</span><span class="p">(),</span><span class="n">A</span> <span class="p">),</span> <span class="n">U</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">15</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
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-<div class="output">
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-<div class="output_area">
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-<pre>array([[2., 0.],
-       [0., 4.]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>In this way, there is an infinity set of symmetric matrices $A$ which have the same eigenvalues and eigenvectors</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="THDM-CP-even-masses-and-mixing">THDM CP-even masses and mixing<a class="anchor-link" href="#THDM-CP-even-masses-and-mixing"> </a></h3><p>In this case we have the rotation between an interaction basis and a physical basis defined as
-$$
-\begin{pmatrix}
-R_1\\
-R_2\\
-\end{pmatrix}=
-\begin{pmatrix}
-\cos\alpha &amp; -\sin\alpha \\
-\sin\alpha &amp; \cos\alpha \\
-\end{pmatrix}
-\begin{pmatrix}
-H\\
-h\\
-\end{pmatrix}.
-$$
-Since the diagonalization of a symmetric $2\times 2$ matrix can be obtained analitically,
-here we build a $2\times 2$ symmetric matrix from the eigenvalues, $m_H$ and $m_h$, and the rotation angle, $\alpha$, and check with the numerical results by using a benchmark point:</p>
-<p>→ <a href="https://twiki.cern.ch/twiki/pub/LHCPhysics/LHCHXSWG3Benchmarks2HDM/Exotic_Benchmarks.pdf">Benchmark point BP9</a></p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">optimize</span>
-<span class="n">G_F</span><span class="o">=</span><span class="mf">1.1663787E-5</span> <span class="c1">#GeV^-2</span>
-<span class="n">v</span><span class="o">=</span><span class="mi">1</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mf">2.</span><span class="p">)</span><span class="o">*</span><span class="n">G_F</span><span class="p">)</span> <span class="c1"># GeV</span>
-
-<span class="c1">#*********  BP9 ************</span>
-<span class="n">tanβ</span><span class="o">=</span><span class="mi">2</span>
-<span class="n">sinβ_α</span><span class="o">=</span><span class="mf">0.6</span>   <span class="c1"># sin(β-α)</span>
-<span class="n">m_h</span> <span class="o">=</span> <span class="mi">125</span>    <span class="c1"># GeV/c^2 (c→1)</span>
-<span class="n">m_H</span> <span class="o">=</span> <span class="mi">300</span>    <span class="c1"># GeV/c^2 </span>
-<span class="n">m_A</span> <span class="o">=</span> <span class="mi">300</span>    <span class="c1"># GeV/c^2</span>
-<span class="n">m_Hp</span><span class="o">=</span> <span class="mi">400</span>    <span class="c1"># GeV/c^2</span>
-<span class="n">m2_12</span><span class="o">=</span><span class="mi">100</span><span class="o">**</span><span class="mi">2</span> <span class="c1"># (GeV/c^2)^2</span>
-<span class="c1">#***********************</span>
-</pre></div>
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-<p>The formula for the mass matrix in terms of the previous parameters can be found in [<a href="https://arxiv.org/pdf/hep-ph/0207010.pdf">hep-ph/0207010</a>,<a href="https://arxiv.org/pdf/1507.00933.pdf">arXiv:1507.00933</a>]. We assumme here that $\lambda_6=\lambda_7=0$. See Appendix D of  [<a href="https://arxiv.org/pdf/hep-ph/0207010.pdf">hep-ph/0207010</a>] for full formulas</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">β</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arctan</span><span class="p">(</span><span class="n">tanβ</span><span class="p">)</span>
-<span class="n">α</span><span class="o">=</span><span class="n">β</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">arcsin</span><span class="p">(</span><span class="n">sinβ_α</span><span class="p">)</span> <span class="c1"># &lt;= np.pi/2~1.57</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;α=&#39;</span><span class="p">,</span><span class="n">α</span><span class="p">)</span>
-<span class="n">λ1</span>  <span class="o">=</span><span class="p">(</span><span class="n">m_H</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">α</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">m_h</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">α</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">m2_12</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">tan</span><span class="p">(</span><span class="n">β</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="n">λ_2</span> <span class="o">=</span><span class="p">(</span><span class="n">m_H</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">α</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">m_h</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">α</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">m2_12</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">tan</span><span class="p">(</span><span class="n">β</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="n">λ3</span>  <span class="o">=</span><span class="p">((</span><span class="n">m_H</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">m_h</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">α</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">α</span><span class="p">)</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">m_Hp</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">-</span><span class="n">m2_12</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">β</span><span class="p">))</span>
-<span class="n">λ4</span>  <span class="o">=</span><span class="p">((</span><span class="n">m_A</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">m_Hp</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">+</span><span class="n">m2_12</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">β</span><span class="p">))</span>
-<span class="n">λ5</span>  <span class="o">=</span><span class="p">(</span><span class="n">m2_12</span><span class="o">-</span><span class="n">m_A</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">β</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">β</span><span class="p">))</span>
-<span class="n">λ345</span><span class="o">=</span><span class="n">λ3</span><span class="o">+</span><span class="n">λ4</span><span class="o">+</span><span class="n">λ5</span>
-<span class="n">λ345</span>
-</pre></div>
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-<pre>α= 0.46364760900080604
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-<pre>1.63919914780058</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">M2</span><span class="p">(</span><span class="n">β</span><span class="p">,</span><span class="n">λ1</span><span class="p">,</span><span class="n">λ_2</span><span class="p">,</span><span class="n">λ5</span><span class="p">,</span><span class="n">λ345</span><span class="p">,</span><span class="n">m2_12</span><span class="p">,</span><span class="n">v</span><span class="o">=</span><span class="mf">246.2</span><span class="p">):</span>
-    <span class="n">mA2</span><span class="o">=</span><span class="n">m2_12</span><span class="o">/</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">β</span><span class="p">))</span><span class="o">-</span><span class="n">λ5</span><span class="o">*</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span>
-    <span class="n">M11</span><span class="o">=</span><span class="n">λ1</span><span class="o">*</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="p">(</span><span class="n">mA2</span><span class="o">+</span><span class="n">λ5</span><span class="o">*</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-    <span class="n">M12</span><span class="o">=</span><span class="p">(</span><span class="n">λ345</span><span class="o">*</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="p">(</span><span class="n">mA2</span><span class="o">+</span><span class="n">λ5</span><span class="o">*</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">β</span><span class="p">)</span>
-    <span class="n">M22</span><span class="o">=</span><span class="n">λ_2</span><span class="o">*</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="p">(</span><span class="n">mA2</span><span class="o">+</span><span class="n">λ5</span><span class="o">*</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">β</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[[</span><span class="n">M11</span><span class="p">,</span> <span class="n">M12</span><span class="p">],</span>
-                      <span class="p">[</span><span class="n">M12</span><span class="p">,</span> <span class="n">M22</span><span class="p">]])</span>
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-<p>In this way, from the BP point, we can build the $2\times2$ symmetric mass $\mathcal{M}$:</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ℳ2</span><span class="o">=</span><span class="n">M2</span><span class="p">(</span><span class="n">β</span><span class="p">,</span><span class="n">λ1</span><span class="p">,</span><span class="n">λ_2</span><span class="p">,</span><span class="n">λ5</span><span class="p">,</span><span class="n">λ345</span><span class="p">,</span><span class="n">m2_12</span><span class="p">,</span><span class="n">v</span><span class="o">=</span><span class="n">v</span><span class="p">)</span>
-<span class="n">ℳ2</span>
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-<pre>array([[75125., 29750.],
-       [29750., 30500.]])</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">m2</span><span class="p">,</span><span class="n">V</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">ℳ2</span><span class="p">)</span>
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-<p>By using the proper basis
-$$
-\left(\begin{array}{cc}
-m_{H}^{2} &amp; 0 \\
-0 &amp; m_{h}^{2}
-\end{array}\right)=\left(\begin{array}{rr}
-\cos{\alpha} &amp; \sin{\alpha} \\
--\sin{\alpha} &amp; \cos{\alpha}
-\end{array}\right)\left(\begin{array}{ll}
-\mathcal{M}_{11}^{2} &amp; \mathcal{M}_{12}^{2} \\
-\mathcal{M}_{12}^{2} &amp; \mathcal{M}_{22}^{2}
-\end{array}\right)\left(\begin{array}{lr}
-\cos{\alpha} &amp; -\sin{\alpha} \\
-\sin{\alpha} &amp; \cos{\alpha}
-\end{array}\right).
-$$
-Therefore
-$$
-U=\left(\begin{array}{lr}
-\cos{\alpha} &amp; -\sin{\alpha} \\
-\sin{\alpha} &amp; \cos{\alpha}
-\end{array}\right).
-$$
-Since $-\pi/2\le\alpha\le\pi/2$ then $\cos\alpha\ge 0$. In this way, we must obtain $\alpha$ from $\tan\alpha$  to avoid ambiguities with a global sign in the second eigenvector 
-$$
-\tan\alpha=\frac{\sin\alpha}{\cos\alpha}=-\frac{U_{01}}{U_{11}}\,.
-$$</p>
-<p>The eigenvector with large projection in the second component along $R_2$ ( Interaction basis $(R_1,R_2)$  → Mass basis $(H^0,h^0)$) is the associated with the standard model Higgs mass, $m_h$</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">if</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">V</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span><span class="o">&lt;=</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">V</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]):</span>
-    <span class="n">H</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span> <span class="n">h</span><span class="o">=</span><span class="mi">1</span>
-<span class="k">else</span><span class="p">:</span>
-    <span class="n">H</span><span class="o">=</span><span class="mi">1</span><span class="p">;</span> <span class="n">h</span><span class="o">=</span><span class="mi">0</span>
-<span class="p">(</span><span class="n">H</span><span class="p">,</span><span class="n">h</span><span class="p">)</span>
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-<pre>(0, 1)</pre>
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-<span class="n">U</span>
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-<pre>array([[ 0.89442719, -0.4472136 ],
-       [ 0.4472136 ,  0.89442719]])</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">M2diag</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">U</span><span class="o">.</span><span class="n">transpose</span><span class="p">(),</span><span class="n">ℳ2</span><span class="p">),</span> <span class="n">U</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">9</span><span class="p">)</span>
-<span class="n">mH</span><span class="p">,</span><span class="n">mh</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">M2diag</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]),</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">M2diag</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="n">mH</span><span class="p">,</span><span class="n">mh</span>
-</pre></div>
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-<pre>(300.0, 125.0)</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">tanα</span><span class="o">=-</span><span class="n">U</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">/</span><span class="n">U</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>
-
-<span class="n">α</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arctan</span><span class="p">(</span><span class="n">tanα</span><span class="p">)</span>
-<span class="k">if</span> <span class="n">α</span><span class="o">&gt;=-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span> <span class="ow">and</span> <span class="n">α</span><span class="o">&lt;=</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span>
-    <span class="nb">print</span><span class="p">(</span><span class="n">α</span><span class="p">)</span>
-<span class="k">else</span><span class="p">:</span>
-    <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Bad α range&#39;</span><span class="p">)</span>
-</pre></div>
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-<pre>0.4636476090008061
-</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">β</span><span class="o">-</span><span class="n">α</span><span class="p">)</span>
-</pre></div>
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-<pre>0.8</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">β</span><span class="o">-</span><span class="n">α</span><span class="p">)</span>
-</pre></div>
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-<pre>0.5999999999999999</pre>
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-<p>Analytical diagonalization from [<a href="https://arxiv.org/pdf/1507.00933.pdf">arXiv:1507.00933</a>]</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Δ</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span>  <span class="p">(</span><span class="n">ℳ2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="n">ℳ2</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="mi">4</span><span class="o">*</span><span class="n">ℳ2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span> <span class="p">[</span><span class="mf">0.5</span><span class="o">*</span><span class="p">(</span> <span class="n">ℳ2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">ℳ2</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">Δ</span> <span class="p">),</span>
-          <span class="mf">0.5</span><span class="o">*</span><span class="p">(</span> <span class="n">ℳ2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">ℳ2</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">Δ</span> <span class="p">)</span> <span class="p">])</span>
-</pre></div>
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-<pre>array([300., 125.])</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Cosα</span><span class="p">,</span><span class="n">Sinα</span><span class="o">=</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">((</span><span class="n">Δ</span><span class="o">+</span><span class="n">ℳ2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="n">ℳ2</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">Δ</span><span class="p">)),</span>
-            <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">ℳ2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">Δ</span><span class="o">*</span><span class="p">(</span><span class="n">Δ</span><span class="o">+</span><span class="n">ℳ2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="n">ℳ2</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]))</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;U=&#39;</span><span class="p">)</span>
-<span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="n">Cosα</span><span class="p">,</span><span class="o">-</span><span class="n">Sinα</span><span class="p">],[</span><span class="n">Sinα</span><span class="p">,</span><span class="n">Cosα</span><span class="p">]])</span>
-</pre></div>
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-<pre>U=
-</pre>
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-<pre>array([[ 0.89442719, -0.4472136 ],
-       [ 0.4472136 ,  0.89442719]])</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;cos(β-α)=</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span> <span class="n">β</span><span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">arctan</span><span class="p">(</span> <span class="n">Sinα</span><span class="o">/</span><span class="n">Cosα</span><span class="p">)</span> <span class="p">)))</span>
-</pre></div>
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-<pre>cos(β-α)=0.8
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-  Custum Gaussian elimination
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-  url: /features/linear-algebra-diagonalization.html
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-    <div id="page-info"><div id="page-title">Custum Gaussian elimination</div>
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-<h1 id="Extra-material-about-Linear-Algebra">Extra material about Linear Algebra<a class="anchor-link" href="#Extra-material-about-Linear-Algebra"> </a></h1>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="n">xrange</span><span class="o">=</span><span class="nb">range</span>
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-<h2 id="Custom-implementation-of-general-gaussian-method">Custom implementation of general gaussian method<a class="anchor-link" href="#Custom-implementation-of-general-gaussian-method"> </a></h2>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Gaussian Elimination</span>
-<span class="k">def</span> <span class="nf">row_lamb</span><span class="p">(</span> <span class="n">i</span><span class="p">,</span> <span class="n">lamb</span><span class="p">,</span> <span class="n">A</span> <span class="p">):</span>
-    <span class="n">B</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">copy</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-    <span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">lamb</span><span class="o">*</span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">B</span>
-
-<span class="c1">#Combination function</span>
-<span class="k">def</span> <span class="nf">row_comb</span><span class="p">(</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">lamb</span><span class="p">,</span> <span class="n">A</span> <span class="p">):</span>
-    <span class="n">B</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">copy</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-    <span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">lamb</span><span class="o">*</span><span class="n">B</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">B</span>
-
-<span class="c1">#Swap function</span>
-<span class="k">def</span> <span class="nf">row_swap</span><span class="p">(</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">A</span> <span class="p">):</span>
-    <span class="n">B</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">copy</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-    <span class="n">B</span><span class="p">[[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]]</span> <span class="o">=</span> <span class="n">B</span><span class="p">[[</span><span class="n">j</span><span class="p">,</span><span class="n">i</span><span class="p">]]</span>
-    <span class="k">return</span> <span class="n">B</span>
-
-<span class="k">def</span> <span class="nf">Gaussian_Elimination</span><span class="p">(</span> <span class="n">A0</span> <span class="p">):</span>
-    <span class="c1">#Making local copy of matrix</span>
-    <span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">copy</span><span class="p">(</span><span class="n">A0</span><span class="p">)</span>
-    <span class="c1">#Detecting size of matrix</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-    
-    <span class="c1">#Sweeping all the columns in order to eliminate coefficients of the i-th variable</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">xrange</span><span class="p">(</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="p">):</span>
-        
-        <span class="c1">#Sweeping all the rows for the i-th column in order to find the first non-zero coefficient</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">xrange</span><span class="p">(</span> <span class="n">i</span><span class="p">,</span> <span class="n">n</span> <span class="p">):</span>
-            <span class="k">if</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="c1">#Normalization coefficient</span>
-                <span class="n">Norm</span> <span class="o">=</span> <span class="mf">1.0</span><span class="o">*</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span>
-                <span class="k">break</span>
-                
-        <span class="c1">#Applying swap operation to put the non-zero coefficient in the i-th row</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="n">row_swap</span><span class="p">(</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">A</span> <span class="p">)</span>
-        
-        <span class="c1">#Eliminating the coefficient associated to the i-th variable</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">xrange</span><span class="p">(</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="p">):</span>
-            <span class="n">A</span> <span class="o">=</span> <span class="n">row_comb</span><span class="p">(</span> <span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="o">-</span><span class="n">A</span><span class="p">[</span><span class="n">j</span><span class="p">,</span><span class="n">i</span><span class="p">]</span><span class="o">/</span><span class="n">Norm</span><span class="p">,</span> <span class="n">A</span> <span class="p">)</span>
-            
-    <span class="c1">#Normalizing n-th variable</span>
-    <span class="n">A</span> <span class="o">=</span> <span class="n">row_lamb</span><span class="p">(</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mf">1.0</span><span class="o">/</span><span class="n">A</span><span class="p">[</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">A</span> <span class="p">)</span>
-    
-    <span class="c1">#Finding solution</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span> <span class="n">n</span> <span class="p">)</span>
-    <span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">A</span><span class="p">[</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">n</span><span class="p">]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">xrange</span><span class="p">(</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span> <span class="p">):</span>
-        <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">n</span><span class="p">]</span> <span class="o">-</span> <span class="nb">sum</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">:</span><span class="n">n</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">:</span><span class="n">n</span><span class="p">])</span> <span class="p">)</span><span class="o">/</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">i</span><span class="p">]</span>
-    
-    <span class="c1">#Upper diagonal matrix and solutions x</span>
-    <span class="k">return</span> <span class="n">A</span><span class="p">,</span> <span class="n">x</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="n">M</span> <span class="o">=</span>  <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span> <span class="p">)</span>
-<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Augmented Matrix M:</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="s2">&quot;</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
-<span class="c1">#Solving using Gaussian Elimination</span>
-<span class="n">D</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="n">Gaussian_Elimination</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Augmented Diagonal Matrix D:</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="s2">&quot;</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
-<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Obtained solution:</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="s2">&quot;</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
-<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Exact solution:</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">,</span> <span class="p">(</span><span class="n">M</span><span class="p">[:,:</span><span class="mi">4</span><span class="p">]</span><span class="o">**-</span><span class="mi">1</span><span class="o">*</span><span class="n">M</span><span class="p">[:,</span><span class="mi">4</span><span class="p">])</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="s2">&quot;</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
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-</div>
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-<pre>Augmented Matrix M:
- [[0.5507979  0.70814782 0.29090474 0.51082761 0.89294695]
- [0.89629309 0.12558531 0.20724288 0.0514672  0.44080984]
- [0.02987621 0.45683322 0.64914405 0.27848728 0.6762549 ]
- [0.59086282 0.02398188 0.55885409 0.25925245 0.4151012 ]] 
-
-Augmented Diagonal Matrix D:
- [[ 5.50797903e-01  7.08147823e-01  2.90904739e-01  5.10827605e-01
-   8.92946954e-01]
- [ 0.00000000e+00 -1.02675749e+00 -2.66135662e-01 -7.79783697e-01
-  -1.01224976e+00]
- [ 0.00000000e+00  0.00000000e+00  5.24909828e-01 -6.69967089e-02
-   2.15310020e-01]
- [ 0.00000000e+00  0.00000000e+00 -1.70372042e-16  1.00000000e+00
-   9.32073626e-03]] 
-
-Obtained solution:
- [0.27395594 0.8721633  0.41137442 0.00932074] 
-
-Exact solution:
- [[ 1.62118801  0.9962667  29.88822639  1.51125934]
- [ 0.62248281  3.51004303  0.9649251  18.38095262]
- [ 2.324661    3.26310322  1.041764    1.21007418]
- [ 0.81260526  8.06535367  1.4905571   1.60114669]] 
-
-</pre>
-</div>
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-</div>
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-  Linear algebra
-pagenum: 20
-prev_page:
-  url: /features/numerical-calculus-integration-methods.html
-next_page:
-  url: /features/linear-algebra-diagonalization.html
-suffix: .ipynb
-search: n matrix vdots mathbf b cdots x m left right e computing j ij frac l times linear ei matrices det hat gaussian span equations elimination system operations determinant using r k operation determinants factorization example previous inverse mbox systems row order ej boldsymbol where lu rows lambda given equation nn diagonal algebra general array arrays next activity u numpy both style color red solve align coefficients upper necessary algorithm method operatorname associated obtain xn y colab not large errors pivoting also column begin end leq routine psi com vector python properties integer possible set new function return ib problem
-
-comment: "***PROGRAMMATICALLY GENERATED, DO NOT EDIT. SEE ORIGINAL FILES IN /content***"
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-
-    <main class="jupyter-page">
-    <div id="page-info"><div id="page-title">Linear algebra</div>
-</div>
-    <div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/linear-algebra.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
-
-</div>
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
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-<h1 id="Linear-Algebra">Linear Algebra<a class="anchor-link" href="#Linear-Algebra"> </a></h1><p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/linear-algebra.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
-
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Linear Algebra is a discipline where vector spaces and linear mapping between them are studied. In physics and astronomy, several phenomena can be readily written in terms of linear variables, what makes Computational Linear Algebra a very important topic to be covered throughout this course. We shall cover linear systems of equations, techniques for calculating inverses and determinants and factorization methods.</p>
-<p>An interesting fact of Computational Linear Algebra is that it does not comprises numerical approaches as most of the methods are exact. The usage of a computer is then necessary because of the large number of calculations rather than the non-soluble nature of the problems. Numerical errors come then from round-off approximations.</p>
-<p>See: <a href="http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture14.pdf">http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture14.pdf</a></p>
-
-</div>
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-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<hr>
-<ul>
-<li><a href="#Linear-Systems-of-Equations">Linear Systems of Equations</a> <ul>
-<li><a href="#Matrices-and-vectors">Matrices and vectors</a></li>
-<li><a href="#Example-1">Example 1</a></li>
-<li><a href="#Matrices-in-Python">Matrices in Python</a></li>
-<li><a href="#Basic-operations-with-matrices">Basic operations with matrices</a></li>
-</ul>
-</li>
-<li><a href="#Gaussian-Elimination">Gaussian Elimination</a><ul>
-<li><a href="#General-Gaussian-elimination">General Gaussian elimination</a></li>
-<li><a href="#Computing-time">Computing time</a></li>
-<li><a href="#Example-2">Example 2</a></li>
-</ul>
-</li>
-<li><a href="./linear-algebra-diagonalization.ipynb">Matrix diagonalization</a>    </li>
-<li><a href="#Pivoting-Strategies">Pivoting Strategies</a><ul>
-<li><a href="#Partial-pivoting">Partial pivoting</a></li>
-</ul>
-</li>
-<li><a href="#Matrix-Inversion">Matrix Inversion</a></li>
-<li><a href="#Determinant-of-a-Matrix">Determinant of a Matrix</a><ul>
-<li><a href="#Calculating-determinants">Calculating determinants</a></li>
-<li><a href="#Computing-time-of-determinants">Computing time of determinants</a></li>
-<li><a href="#Properties-of-determinants">Properties of determinants</a></li>
-</ul>
-</li>
-<li><a href="#LU-Factorization">LU Factorization</a><ul>
-<li><a href="#Derivation-of-LU-factorization">Derivation of LU factorization</a></li>
-<li><a href="#Algorithm-for-LU-factorization">Algorithm for LU factorization</a></li>
-</ul>
-</li>
-</ul>
-<hr>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">HTML</span><span class="p">,</span> <span class="n">Markdown</span><span class="p">,</span> <span class="n">YouTubeVideo</span><span class="p">,</span><span class="n">Latex</span>
-<span class="o">%</span><span class="k">pylab</span> inline
-
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
-<span class="c1"># JSAnimation import available at https://github.com/jakevdp/JSAnimation</span>
-<span class="c1">#from JSAnimation import IPython_display</span>
-<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">animation</span>
-<span class="c1">#Interpolation add-on</span>
-<span class="kn">import</span> <span class="nn">scipy.interpolate</span> <span class="k">as</span> <span class="nn">interp</span>
-<span class="n">xrange</span><span class="o">=</span><span class="nb">range</span>
-</pre></div>
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-<pre>Populating the interactive namespace from numpy and matplotlib
-</pre>
-</div>
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-</div>
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-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p><a href="https://stackoverflow.com/a/50748163">https://stackoverflow.com/a/50748163</a></p>
-
-</div>
-</div>
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">try</span><span class="p">:</span>
-    <span class="kn">from</span> <span class="nn">google.colab.output._publish</span> <span class="kn">import</span> <span class="n">javascript</span>
-    <span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Math</span> <span class="k">as</span> <span class="n">math</span>
-    <span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Latex</span> <span class="k">as</span> <span class="n">latex</span>
-    <span class="n">url</span> <span class="o">=</span> <span class="s2">&quot;https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.3/latest.js?config=default&quot;</span>
-    <span class="k">def</span> <span class="nf">Math</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span><span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">javascript</span><span class="p">(</span><span class="n">url</span><span class="o">=</span><span class="n">url</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">math</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">Latex</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span><span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="c1">#print(args[0].replace(&#39;$&#39;,&#39;&#39;))</span>
-        <span class="n">javascript</span><span class="p">(</span><span class="n">url</span><span class="o">=</span><span class="n">url</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">math</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-<span class="k">except</span><span class="p">:</span>
-    <span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Math</span><span class="p">,</span> <span class="n">Latex</span>
-</pre></div>
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-<pre>/usr/local/lib/python3.7/dist-packages/IPython/utils/traitlets.py:5: UserWarning: IPython.utils.traitlets has moved to a top-level traitlets package.
-  warn(&#34;IPython.utils.traitlets has moved to a top-level traitlets package.&#34;)
-</pre>
-</div>
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-<h1 id="Matrices-in-Python">Matrices in Python<a class="anchor-link" href="#Matrices-in-Python"> </a></h1>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>One of the most useful advantages of high level languages like Python, is the manipulation of complex objects like matrices and vectors. For this part we are going to use advanced capabilities for handling matrices, provided by the library NumPy.</p>
-<p>NumPy, besides the extreme useful NumPy array objects, also provides the Matrix objects that are overloaded with proper matrix operations.</p>
-<h3 id="Numpy-arrays">Numpy arrays<a class="anchor-link" href="#Numpy-arrays"> </a></h3><ul>
-<li>A matrix can be represented by an array of two indices</li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#NumPy Arrays</span>
-<span class="n">M1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[[</span> <span class="mi">5</span> <span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-                <span class="p">[</span><span class="o">-</span><span class="mi">4</span> <span class="p">,</span> <span class="mi">7</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">],</span>
-                <span class="p">[</span> <span class="mi">0</span> <span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">]]</span> <span class="p">)</span>
-<span class="n">M2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[[</span><span class="mi">3</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">],[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">]]</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;M1=</span><span class="se">\n</span><span class="si">{}</span><span class="s1">, with shape=</span><span class="si">{}</span><span class="s1">,</span><span class="se">\n\n</span><span class="s1">M2=</span><span class="se">\n</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">M1</span><span class="p">,</span><span class="n">M1</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span><span class="n">M2</span><span class="p">)</span> <span class="p">)</span>
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-<pre>M1=
-[[ 5 -4  0]
- [-4  7 -3]
- [ 0 -3  5]], with shape=(3, 3),
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-M2=
-[[ 3 -2  1]
- [-1  5  4]
- [ 1 -2  3]]
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-<p>Note tha the previous definitions correspond to integer arrays.</p>
-<p><strong>WARNING</strong>: Be careful with integer arrays. In the next example, a single float entry convert the full matrix from an integer array to a float array:</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">M1</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
-</pre></div>
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-<pre>5</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mi">5</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">,</span>  <span class="mi">0</span><span class="p">],</span>
-          <span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span>  <span class="mi">7</span><span class="p">,</span> <span class="o">-</span><span class="mf">3.</span><span class="p">],</span>
-          <span class="p">[</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span>  <span class="mi">5</span><span class="p">]])[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
-</pre></div>
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-<pre>5.0</pre>
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-<h4 id="Special-arrays:">Special arrays:<a class="anchor-link" href="#Special-arrays:"> </a></h4><p>Let $n$ the range of the matrix</p>
-<ul>
-<li>zero matrix</li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">n</span><span class="o">=</span><span class="mi">3</span>
-<span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">n</span><span class="p">)</span> <span class="p">)</span>
-</pre></div>
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-<pre>array([[0., 0., 0.],
-       [0., 0., 0.],
-       [0., 0., 0.]])</pre>
-</div>
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-<ul>
-<li>Ones matrix</li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">n</span><span class="p">)</span> <span class="p">)</span>
-</pre></div>
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-<pre>array([[1., 1., 1.],
-       [1., 1., 1.],
-       [1., 1., 1.]])</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">n</span><span class="p">)</span> <span class="p">)</span><span class="o">*</span><span class="mi">1</span><span class="n">j</span>
-</pre></div>
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-<pre>array([[0.+1.j, 0.+1.j, 0.+1.j],
-       [0.+1.j, 0.+1.j, 0.+1.j],
-       [0.+1.j, 0.+1.j, 0.+1.j]])</pre>
-</div>
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-</div>
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-</div>
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-<ul>
-<li>Identity matrix</li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-</pre></div>
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-<pre>array([[1., 0., 0.],
-       [0., 1., 0.],
-       [0., 0., 1.]])</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
-</pre></div>
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-<pre>array([[1, 0, 0],
-       [0, 2, 0],
-       [0, 0, 4]])</pre>
-</div>
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-<ul>
-<li>Random matrix with entries between 0 and 1</li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="n">n</span><span class="p">))</span>
-</pre></div>
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-<pre>array([[0.03759466, 0.03529442, 0.77509671],
-       [0.65252241, 0.71648007, 0.09995053],
-       [0.11837179, 0.08144293, 0.62915985]])</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">,(</span><span class="n">n</span><span class="p">,</span><span class="n">n</span><span class="p">))</span>
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-<pre>array([[0.69568594, 5.55397727, 2.52884654],
-       [0.92956248, 7.03556212, 7.69677769],
-       [3.17346739, 1.12515801, 2.84483672]])</pre>
-</div>
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-<ul>
-<li>Integer Random matrix</li>
-</ul>
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-<pre>array([[9, 0, 5],
-       [4, 5, 7],
-       [0, 0, 6]])</pre>
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-<h3 id="Analytical-matrices">Analytical matrices<a class="anchor-link" href="#Analytical-matrices"> </a></h3><p>Some times it is necessary to work with analytical Matrices. In such a case we can use the symbolic module <code>Sympy</code></p>
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-<span class="n">x</span> <span class="o">=</span><span class="n">sympy</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span> <span class="c1"># declare analytical varibles</span>
-<span class="n">sympy</span><span class="o">.</span><span class="n">init_printing</span><span class="p">()</span> <span class="c1"># Use LaTeX to print sympy obejects</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">(</span><span class="n">M2</span><span class="p">)</span>
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-$\displaystyle \left[\begin{matrix}3 & -2 & 1\\-1 & 5 & 4\\1 & -2 & 3\end{matrix}\right]$
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-<p>In the following we will focus only in numerical matrices as <code>numpy</code> arrays, but also use <code>sympy</code> to print the matrices in a more readable way</p>
-<h3 id="Matrix-operations-(numpy)">Matrix operations (numpy)<a class="anchor-link" href="#Matrix-operations-(numpy)"> </a></h3><p>Both as functions or array's methods</p>
-<ul>
-<li>Transpose</li>
-</ul>
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-<span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">(</span> <span class="n">M2</span><span class="o">.</span><span class="n">transpose</span><span class="p">())</span>
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-$\displaystyle \left[\begin{matrix}3 & -1 & 1\\-2 & 5 & -2\\1 & 4 & 3\end{matrix}\right]$
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-$\displaystyle \left[\begin{matrix}8 & -6 & 1\\-5 & 12 & 1\\1 & -5 & 8\end{matrix}\right]$
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Mc</span><span class="o">=</span><span class="n">M1</span><span class="o">+</span><span class="n">M2</span><span class="o">*</span><span class="mi">1</span><span class="n">j</span>
-<span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">(</span> <span class="n">M1</span><span class="o">+</span><span class="n">M2</span><span class="o">*</span><span class="mi">1</span><span class="n">j</span> <span class="p">)</span>
-</pre></div>
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-$\displaystyle \left[\begin{matrix}5.0 + 3.0 i & -4.0 - 2.0 i & 1.0 i\\-4.0 - 1.0 i & 7.0 + 5.0 i & -3.0 + 4.0 i\\1.0 i & -3.0 - 2.0 i & 5.0 + 3.0 i\end{matrix}\right]$
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-<p>with the corresponding complex matrix operations</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">display</span> <span class="p">(</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">(</span>  <span class="n">Mc</span><span class="o">.</span><span class="n">conjugate</span><span class="p">()</span> <span class="p">)</span> <span class="p">)</span>
-<span class="n">display</span> <span class="p">(</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">(</span>  <span class="n">Mc</span><span class="o">.</span><span class="n">conjugate</span><span class="p">()</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span> <span class="p">)</span> <span class="p">)</span> <span class="c1"># hermítico-conjugado</span>
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-$\displaystyle \left[\begin{matrix}5.0 - 3.0 i & -4.0 + 2.0 i & - 1.0 i\\-4.0 + 1.0 i & 7.0 - 5.0 i & -3.0 - 4.0 i\\- 1.0 i & -3.0 + 2.0 i & 5.0 - 3.0 i\end{matrix}\right]$
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-$\displaystyle \left[\begin{matrix}5.0 - 3.0 i & -4.0 + 1.0 i & - 1.0 i\\-4.0 + 2.0 i & 7.0 - 5.0 i & -3.0 + 2.0 i\\- 1.0 i & -3.0 - 4.0 i & 5.0 - 3.0 i\end{matrix}\right]$
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-<ul>
-<li>Matrix multiplication
-<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Matrix_multiplication_diagram_2.svg/200px-Matrix_multiplication_diagram_2.svg.png" alt="MP"></li>
-</ul>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<div class="highlight"><pre><span></span><span class="n">dot</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">out</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
-</pre></div>
-<p>Dot product of two arrays. Specifically,</p>
-<ul>
-<li><p>If both <code>a</code> and <code>b</code> are 1-D arrays, it is inner product of vectors
-(without complex conjugation).</p>
-</li>
-<li><p>If both <code>a</code> and <code>b</code> are 2-D arrays, it is matrix multiplication,
-but using :func:<code>matmul</code> or <code>a @ b</code> is preferred.</p>
-</li>
-<li><p>If either <code>a</code> or <code>b</code> is 0-D (scalar), it is equivalent to :func:<code>multiply</code>
-and using <code>numpy.multiply(a, b)</code> or <code>a * b</code> is preferred.</p>
-</li>
-<li><p>...</p>
-</li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Multiplication</span>
-<span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span> <span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">M1</span><span class="p">,</span> <span class="n">M2</span> <span class="p">)</span> <span class="p">)</span>
-</pre></div>
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-$\displaystyle \left[\begin{matrix}19 & -30 & -11\\-22 & 49 & 15\\8 & -25 & 3\end{matrix}\right]$
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Multiplication is not commutative</span>
-<span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span> <span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">M2</span><span class="p">,</span> <span class="n">M1</span> <span class="p">)</span> <span class="p">)</span>
-</pre></div>
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-$\displaystyle \left[\begin{matrix}23 & -29 & 11\\-25 & 27 & 5\\13 & -27 & 21\end{matrix}\right]$
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-<ul>
-<li>Complex matrices</li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">σ_2</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="n">j</span><span class="p">],[</span><span class="mi">1</span><span class="n">j</span><span class="p">,</span><span class="mi">0</span><span class="p">]])</span>
-<span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">(</span><span class="n">σ_2</span><span class="p">)</span>
-</pre></div>
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-$\displaystyle \left[\begin{matrix}0 & - 1.0 i\\1.0 i & 0\end{matrix}\right]$
-</div>
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-<ul>
-<li>Trace</li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">trace</span><span class="p">(</span><span class="n">σ_2</span><span class="p">)</span>
-</pre></div>
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-<pre>0j</pre>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<ul>
-<li>Determinant</li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">σ_2</span><span class="p">)</span>
-</pre></div>
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-<pre>(-1+0j)</pre>
-</div>
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-<p>Raise a square matrix to the (integer) power <code>n</code>.</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span> <span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_power</span><span class="p">(</span><span class="n">σ_2</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="p">)</span>
-</pre></div>
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-$\displaystyle \left[\begin{matrix}1.0 & 0\\0 & 1.0\end{matrix}\right]$
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<p>odd power</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span> <span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_power</span><span class="p">(</span><span class="n">σ_2</span><span class="p">,</span><span class="mi">7</span><span class="p">)</span> <span class="p">)</span>
-</pre></div>
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-$\displaystyle \left[\begin{matrix}0 & - 1.0 i\\1.0 i & 0\end{matrix}\right]$
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<p>even power</p>
-
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span> <span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_power</span><span class="p">(</span><span class="n">σ_2</span><span class="p">,</span><span class="mi">14</span><span class="p">)</span> <span class="p">)</span>
-</pre></div>
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-$\displaystyle \left[\begin{matrix}1.0 & 0\\0 & 1.0\end{matrix}\right]$
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Inverse of a matrix</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">M2</span><span class="p">)</span> <span class="p">)</span>
-</pre></div>
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-</div>
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-$\displaystyle \left[\begin{matrix}0.442307692307692 & 0.0769230769230769 & -0.25\\0.134615384615385 & 0.153846153846154 & -0.25\\-0.0576923076923077 & 0.0769230769230769 & 0.25\end{matrix}\right]$
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span> <span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">M2</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">M2</span><span class="p">)</span> <span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">15</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span> <span class="p">)</span>
-</pre></div>
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-$\displaystyle \left[\begin{matrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{matrix}\right]$
-</div>
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-</div>
-</div>
-</div>
-</div>
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-</div>
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-<h3 id="Element-by-element-operations">Element by element operations<a class="anchor-link" href="#Element-by-element-operations"> </a></h3>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">M1</span><span class="p">)</span>
-<span class="n">M2</span>
-</pre></div>
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-<pre>[[ 5 -4  0]
- [-4  7 -3]
- [ 0 -3  5]]
-</pre>
-</div>
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-<pre>array([[ 3, -2,  1],
-       [-1,  5,  4],
-       [ 1, -2,  3]])</pre>
-</div>
-
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-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">M1</span><span class="o">*</span><span class="n">M2</span>
-</pre></div>
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-<pre>array([[ 15,   8,   0],
-       [  4,  35, -12],
-       [  0,   6,  15]])</pre>
-</div>
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-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Division</span>
-<span class="nb">print</span><span class="p">(</span> <span class="n">M1</span><span class="o">/</span><span class="n">M2</span> <span class="p">)</span>
-</pre></div>
-
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-<pre>[[ 1.66666667  2.          0.        ]
- [ 4.          1.4        -0.75      ]
- [ 0.          1.5         1.66666667]]
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-$\displaystyle \left[\begin{matrix}1.673 & -0.231 & -0.25\\-0.654 & 0.538 & -1.5\\-0.692 & -0.077 & 2.0\end{matrix}\right]$
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-                <span class="p">[</span><span class="o">-</span><span class="mi">4</span> <span class="p">,</span> <span class="mi">7</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">],</span>
-                <span class="p">[</span> <span class="mi">0</span> <span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">]]</span> <span class="p">)</span>
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-<h3 id="Extract-rows">Extract rows<a class="anchor-link" href="#Extract-rows"> </a></h3>
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-       [ 7],
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-<pre>array([[ 0, -3,  5],
-       [-4,  7, -3],
-       [ 5, -4,  0]])</pre>
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-<pre>array([[ 0, -3,  5],
-       [-4,  7, -3],
-       [ 5, -4,  0]])</pre>
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-<pre>array([[ 0, -4,  5],
-       [-3,  7, -4],
-       [ 5, -3,  0]])</pre>
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-<h3 id="It-is-possible-to-reasign-a-full-set-of-rows-to-a-matrix">It is possible to reasign a full set of rows to a matrix<a class="anchor-link" href="#It-is-possible-to-reasign-a-full-set-of-rows-to-a-matrix"> </a></h3>
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-                <span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">],</span>
-                <span class="p">[</span> <span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">]]</span> <span class="p">)</span>
-</pre></div>
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-           <span class="p">[</span><span class="mf">0.3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span>  <span class="p">]]</span>
-<span class="n">M1</span>
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-<pre>array([[ 0,  0,  0],
-       [-4,  7, -3],
-       [ 0,  0,  0]])</pre>
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-<p>Note that because <code>M1</code> has only integer numbers, it is assumed to be an <em>integer matrix</em>. In order to allow float assignments, it is first necessary to convert to a <em>float matrix</em>:</p>
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-<span class="n">M1</span><span class="p">[[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">]]</span><span class="o">=</span><span class="p">[[</span><span class="mi">0</span>  <span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.3</span><span class="p">],</span>
-           <span class="p">[</span><span class="mf">0.3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span>  <span class="p">]]</span>
-<span class="n">M1</span>
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-       [-4. ,  7. , -3. ],
-       [ 0.3,  0. ,  0. ]])</pre>
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-<p><strong>WARNING</strong><br/>
-In some cases a  reasignment of a list or array points to the same space in memory. To really copy an array to other variable it is always recommended to use the <code>.copy()</code> method.</p>
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-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;A=</span><span class="si">{}</span><span class="s1"> → C=</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">C</span><span class="p">))</span>
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-<pre>A=[5 2] → C=[5 2]
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-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;A=</span><span class="si">{}</span><span class="s1"> → C=</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">C</span><span class="p">))</span>
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-<pre>A=[5 8] → C=[5 8]
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-<p>To keep the first memory space</p>
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-<span class="n">B</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">=</span><span class="mi">4</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;A =&#39;</span><span class="p">,</span><span class="n">A</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;B =&#39;</span><span class="p">,</span><span class="n">B</span><span class="p">)</span>
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-<pre>A = [5 8]
-B = [4 8]
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-<h3 id="Matrix-object">Matrix object<a class="anchor-link" href="#Matrix-object"> </a></h3>
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-<p>numpy also has a Matrix object with simplified operations. However it is recommended to use the general array object even for specific matrix operations. This helps to avoid incompatibilities with usual array operations. For example, as shown below the multiplication symbol, <code>*</code>, behaves different for arrays that for matrices, and could induce to errors.</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#NumPy Matrix</span>
-<span class="n">M1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span> <span class="p">[[</span><span class="mi">5</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">]]</span> <span class="p">)</span>
-<span class="n">M2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span> <span class="p">[[</span><span class="mi">3</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">],[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">]]</span> <span class="p">)</span>
-
-<span class="nb">print</span> <span class="p">(</span><span class="n">M1</span><span class="p">,</span> <span class="s2">&quot;</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
-<span class="nb">print</span> <span class="p">(</span><span class="n">M2</span><span class="p">)</span>
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-<pre>[[ 5 -4  0]
- [-4  7 -3]
- [ 0 -3  5]] 
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-[[ 3 -2  1]
- [-1  5  4]
- [ 1 -2  3]]
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-<span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">(</span> <span class="n">M1</span><span class="o">+</span><span class="n">M2</span> <span class="p">)</span>
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-$\displaystyle \left[\begin{matrix}8 & -6 & 1\\-5 & 12 & 1\\1 & -5 & 8\end{matrix}\right]$
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-<p><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Matrix_multiplication_diagram_2.svg/200px-Matrix_multiplication_diagram_2.svg.png" alt="MP"></p>
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-<span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">(</span> <span class="n">M1</span><span class="o">*</span><span class="n">M2</span> <span class="p">)</span>
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-$\displaystyle \left[\begin{matrix}19 & -30 & -11\\-22 & 49 & 15\\8 & -25 & 3\end{matrix}\right]$
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-<h2 id="Basic-operations-with-matrices">Basic operations with matrices<a class="anchor-link" href="#Basic-operations-with-matrices"> </a></h2>
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-<p>In order to simplify the following methods, we introduce here 3 basic operations over linear systems:</p>
-<ol>
-<li>The $i$-th row $E_i$ can be multiplied by a non-zero constant $\lambda$, and then a new row used in place of $E_i$, i.e. $(E_i)\rightarrow (\lambda E_i)$. We denote this operation as $\operatorname{Lamb}(E_i,\lambda)$.</li>
-<li>The $j$-th row $E_j$ can be multiplied by a non-zero constant $\lambda$ and added to some row $E_i$. The resulting value take the place of $E_i$, i.e. $(E_i) \rightarrow (E_i + \lambda E_j)$. We denote this operation as $\operatorname{Comb}(E_i,E_j,\lambda)$.</li>
-<li>Finally, we define a swapping of two rows as $(E_i)\leftrightarrow (E_j)$, denoted as $\operatorname{Swap}(E_i,E_j)$</li>
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-<h3 id="Extract-rows-from-an-array">Extract rows from an array<a class="anchor-link" href="#Extract-rows-from-an-array"> </a></h3>
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-<h2 id="-Activity-"><span style="color:red"> Activity </span><a class="anchor-link" href="#-Activity-"> </a></h2>
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-<p>Write three routines that perform, given a matrix $A$, the previous operations over matrices:</p>
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-<li><code>row_lamb(i, λ, A)</code>: <code>i</code> is the row to be changed, <code>λ</code> the multiplicative factor and <code>A</code> the matrix. This function should return the new matrix with the performed operation $(\lambda E_i)\rightarrow (E_i)$.</li>
-<li><code>row_comb(i, j, λ, A)</code>: <code>i</code> is the row to be changed, <code>j</code> the row to be added, <code>λ</code> the multiplicative factor and <code>A</code> the matrix. This function should return the new matrix with the performed operation $(E_i + \lambda E_j)\rightarrow (E_i)$.</li>
-<li><code>row_swap(i, j, A)</code>: <code>i</code> and <code>j</code> are the rows to be swapped. This function should return the new matrix with the performed operation $(E_i)\leftrightarrow (E_j)$.</li>
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-<!--
-#Lambda function
-def row_lamb( i, lamb, A ):
-    B = np.copy(A)
-    B[i] = lamb*B[i]
-    return B
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-#Combination function
-def row_comb( i, j, lamb, A ):
-    B = np.copy(A)
-    B[i] = lamb*B[j] + B[i]
-    return B
-
-#Swap function
-def row_swap( i, j, A ):
-    B = np.copy(A)
-    B[[i,j]] = B[[j,i]]
-    return B
--->
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-<span class="k">def</span> <span class="nf">row_lamb</span><span class="p">(</span> <span class="n">i</span><span class="p">,</span> <span class="n">λ</span><span class="p">,</span> <span class="n">A</span> <span class="p">):</span>
-    <span class="n">B</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">B</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span><span class="n">int64</span><span class="p">):</span>
-        <span class="n">B</span><span class="o">=</span><span class="n">B</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span>
-    <span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">λ</span><span class="o">*</span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">B</span>
-
-<span class="k">def</span> <span class="nf">row_comb</span><span class="p">(</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">λ</span><span class="p">,</span> <span class="n">A</span> <span class="p">):</span>
-    <span class="n">B</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">B</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span><span class="n">int64</span><span class="p">):</span>
-        <span class="n">B</span><span class="o">=</span><span class="n">B</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span>
-    <span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">λ</span><span class="o">*</span><span class="n">B</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">B</span>
-
-<span class="k">def</span> <span class="nf">row_swap</span><span class="p">(</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">A</span> <span class="p">):</span>
-    <span class="n">B</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-    <span class="n">B</span><span class="p">[[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]]</span> <span class="o">=</span> <span class="n">B</span><span class="p">[[</span><span class="n">j</span><span class="p">,</span><span class="n">i</span><span class="p">]]</span>
-    <span class="k">return</span> <span class="n">B</span>
-</pre></div>
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-<p><strong>Example</strong><br/></p>
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-             <span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span>  <span class="mi">7</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">],</span>
-             <span class="p">[</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span>  <span class="mi">5</span><span class="p">]])</span>
-</pre></div>
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-<h1 id="Gaussian-Elimination">Gaussian Elimination<a class="anchor-link" href="#Gaussian-Elimination"> </a></h1>
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-<p>Using Kirchhoff's circuit laws, it is possible to solve the next system:</p>
-<p><img src="https://raw.githubusercontent.com/sbustamante/ComputationalMethods/master/material/figures/circuit.png" alt=""></p>
-<p>obtaining the next equations:</p>
-\begin{align}
-A:&amp;&amp;I_A( R+4R ) - I_B( 4R ) =&amp; E \\
-C:&amp;&amp;-I_A( 4R ) + I_B( 4R + 3R ) - I_C(3R) =&amp; 0 \\
-B:&amp;&amp;-I_B( 3R ) + I_C(3R+2R) =&amp; -2E\,
-\end{align}<p>or
-\begin{align}
-5\frac{I_A R }{E} - 4\frac{I_B R }{E} =&amp; 1 \\
--4 \frac{I_A R }{E} + 7\frac{I_B R }{E} - 3 \frac{I_C R }{E} =&amp; 0 \\
--3 \frac{I_B R }{E} + 5 \frac{I_C R }{E} =&amp; -2\,.
-\end{align}</p>
-<p>Defining the variables $x_1 = R I_A/E$, $x_2 = R I_B/E$ and $x_3 = R I_C/E$ we have
-\begin{align}
-5x_1 - 4x_2 =&amp; 1 \\
--4 x_1 + 7x_2 - 3 x_3 =&amp; 0 \\
--3x_2 + 5 x_3 =&amp; -2\,.
-\end{align}</p>
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-<p>A first method to solve linear systems of equations is the Gaussian elimination. This procedure consists of a set of recursive steps performed in order to diagonalise the matrix of the problem. A suitable way to introduce this method is applying it to some basic problem. To do so, let's take the result of the <a href="#Example-1">Example 1</a>:</p>
-$$ \left[ \matrix{
-5 &amp; -4 &amp; 0 \\
--4 &amp; 7 &amp; -3 \\ 
-0 &amp; -3 &amp; 5
-}\right] 
-\left[ \matrix{
-x_{1} \\
-x_{2} \\
-x_{3} 
-}\right]  =
-\left[ \matrix{
-1 \\
-0 \\
--2
-}\right]$$<p>Constructing the associated augmented matrix, we obtain</p>
-$$ \left[ \matrix{
-5 &amp; -4 &amp; 0 &amp; \vdots &amp; 1 \\
--4 &amp; 7 &amp; -3 &amp; \vdots &amp; 0 \\
-0 &amp; -3 &amp; 5 &amp; \vdots &amp; -2
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-<h3 id="-Activity-"><span style="color:red"> Activity </span><a class="anchor-link" href="#-Activity-"> </a></h3><p>Use numpy functions to build the augmented matrix M1</p>
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-<span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">(</span><span class="n">M1</span><span class="p">)</span>
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-$\displaystyle \left[\begin{matrix}5 & -4 & 0\\-4 & 7 & -3\\0 & -3 & 5\end{matrix}\right]$
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-<span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">(</span><span class="n">M1_aug</span><span class="p">)</span>
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-$\displaystyle \left[\begin{matrix}5 & -4 & 0 & 1\\-4 & 7 & -3 & 0\\0 & -3 & 5 & -2\end{matrix}\right]$
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-<span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">(</span><span class="n">M1_aug</span><span class="p">)</span>
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-$\displaystyle \left[\begin{matrix}5 & -4 & 0 & 1\\-4 & 7 & -3 & 0\\0 & -3 & 5 & -2\end{matrix}\right]$
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-<p>At this point, we can eliminate the coefficients of the variable $x_1$ in the second and third equations. For this, we apply the operations $\operatorname{Comb}(E_2,E_1,4/5)$.</p>
-<p>$E_2$ → <code>[-4,7,-3⋮0]+[4,-16/5,0⋮4/5]=[0,19/5,-3⋮4/5]</code></p>
-<p>The coefficient in the third equation is already null. We then obtain:</p>
-$$ \left[ \matrix{
-5 &amp; -4 &amp; 0 &amp; \vdots &amp; 1 \\
-0 &amp; 19/5 &amp; -3 &amp; \vdots &amp; 4/5 \\
-0 &amp; -3 &amp; 5 &amp; \vdots &amp; -2
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-<pre>array([[ 5. , -4. ,  0. ,  1. ],
-       [ 0. ,  3.8, -3. ,  0.8],
-       [ 0. , -3. ,  5. , -2. ]])</pre>
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-<p>Now, we proceed to eliminate the coefficient of $x_2$ in the third equation. For this, we apply again $\operatorname{Comb}(E_3,E_2,3\cdot 5/19)$</p>
-<p>$E_3$ → <code>[0,-3,5⋮-2]+[0,(3*5/19)*(19/5),-(3*5/19)*3⋮(3*5/19)*4/5]→[0,0,50/19⋮-26/19]</code></p>
-$$ \left[ \matrix{
-5 &amp; -4 &amp; 0 &amp; \vdots &amp; 1 \\
-0 &amp; 16/5 &amp; -3 &amp; \vdots &amp; 4/5 \\
-0 &amp; 0 &amp; 50/19 &amp; \vdots &amp; -26/19
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-<pre>array([[ 5.        , -4.        ,  0.        ,  1.        ],
-       [ 0.        ,  3.8       , -3.        ,  0.8       ],
-       [ 0.        ,  0.        ,  2.63157895, -1.36842105]])</pre>
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-<p>The final step is to solve for $x_3$ in the last equation, doing the operation $\operatorname{Lamb}(E_3,19/50)$, yielding:</p>
-$$ \left[ \matrix{
-5 &amp; -4 &amp; 0 &amp; \vdots &amp; 1 \\
-0 &amp; 3.8 &amp; -3 &amp; \vdots &amp; 0.8 \\
-0 &amp; 0 &amp; 1 &amp; \vdots &amp; -26/50
-}\right] $$<p>From this, it is direct to conclude that $x_3 = -26/50=-0.52$, for $x_2$ and $x_1$ it is only necessary to replace the found value of $x_3$.</p>
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-       [ 0.  ,  0.  ,  1.  , -0.52]])</pre>
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-<h1 id="Linear-Systems-of-Equations">Linear Systems of Equations<a class="anchor-link" href="#Linear-Systems-of-Equations"> </a></h1>
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-<p>A linear system is a set of equations of $n$ variables that can be written in a general form:</p>
-$$ a_{11}x_1 + a_{12}x_2 + \cdots +a_{1n}x_n = b_1 $$$$\vdots$$$$ a_{m1}x_1 + a_{m2}x_2 + \cdots +a_{mn}x_n = b_m $$<p>where $n$ is, again, the number of variables, and $m$ the number of equations.</p>
-<p>A linear system has solution if and only if $m\geq n$. This leads us to the main objetive of a linear system, find the set $\{x_i\}_{i=0}^n$ that fulfills all the equations.</p>
-<p>Although there is an intuitive way to solve this type of systems, by just adding and subtracting equations until reaching the desire result, the large number of variables of some systems found in physics and astronomy makes necessary to develop iterative and general approaches. Next, we shall introduce matrix and vector notation and some basic operations that will be the basis of the methods to be developed in this section.</p>
-<p>Quadratic eqution...</p>
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-<p>A $m\times n$ matrix can be defined as a set of numbers arranged in columns and rows such as:</p>
-$$ A = [a_{ij}] = \left[ \matrix{
-a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} \\
-a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} \\
-\vdots &amp; \vdots &amp; &amp; \vdots\\
-a_{m1} &amp; a_{m2} &amp; \cdots &amp; a_{mn} 
-}\right] $$<p>In the same way, it is possible to define a $n$-dimensional column vector as</p>
-$$ \boldsymbol{x} = \left[ \matrix{
-x_{1} \\
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-\vdots\\
-x_{n} 
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-$$ \boldsymbol{y} = \left[ \matrix{
-y_{1} &amp;
-y_{2} &amp;
-\cdots &amp;
-y_{n} 
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-<p>The system of equations</p>
-$$ a_{11}x_1 + a_{12}x_2 + \cdots a_{1n}x_n = b_1 $$$$\vdots$$$$ a_{m1}x_1 + a_{m2}x_2 + \cdots a_{mn}x_n = b_m $$<p>can be then written in a more convenient way as</p>
-$$ A \textbf{x} =  \left[ \matrix{
-a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} \\
-a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} \\
-\vdots &amp; \vdots &amp; &amp; \vdots\\
-a_{m1} &amp; a_{m2} &amp; \cdots &amp; a_{mn} 
-}\right] 
-\left[ \matrix{
-x_{1} \\
-x_{2} \\
-\vdots\\
-x_{n} 
-}\right]  = \boldsymbol{b} = 
-\left[ \matrix{
-b_{1} \\
-b_{2} \\
-\vdots\\
-b_{n} 
-}\right]$$<p>We can also introducing the $n\times (n+1)$ <strong>augmented matrix</strong> as</p>
-$$[A:\textbf{b}] =  \left[ \matrix{
-a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} &amp; \vdots &amp; b_1 \\
-a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} &amp; \vdots &amp; b_2 \\
-\vdots &amp; \vdots &amp; &amp; \vdots &amp; \vdots &amp; \vdots \\
-a_{m1} &amp; a_{n2} &amp; \cdots &amp; a_{nn} &amp; \vdots &amp; b_n
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-<h3 id="Implementation-in-Scipy">Implementation in Scipy<a class="anchor-link" href="#Implementation-in-Scipy"> </a></h3>
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-<span class="s1">&#39;x3=-26/50=</span><span class="si">{:.2f}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">x3</span><span class="p">)</span>
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-<span class="nb">round</span><span class="p">(</span><span class="n">x2</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
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-$\displaystyle -0.2$
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-<span class="nb">round</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
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-$\displaystyle 0.04$
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-<h2 id="General-Gaussian-elimination">General Gaussian elimination<a class="anchor-link" href="#General-Gaussian-elimination"> </a></h2>
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-<p>Now, we shall describe the general procedure for Gaussian elimination:</p>
-<ol>
-<li>Give an augmented matrix $\hat{A}=[A:\textbf{b}]$.</li>
-<li>Find the first non-zero coefficient $a_{i1}$ associated to $x_1$. This element is called <em>pivot</em>.</li>
-<li>Apply the operation $\operatorname{Swap}(E_1,E_i)$. This guarantee the first row has a non-zero coefficient $a_{11}$.</li>
-<li>Apply the operation $\operatorname{Comb}(E_j,E_1,-a_{j1}/a_{11})$. This eliminates the coefficients associated to $x_1$ in all the rows but in the first one.</li>
-<li>Repeat steps 2. to 4. but for the coefficients of $x_2$ and then $x_3$ and so up to $x_n$. When iterating the coefficients of $x_{k}$, do not take into account the first $k$-th rows as they are already sorted.</li>
-<li>Once you obtain a diagonal form of the matrix, apply the operation $\operatorname{Lamb}(E_n,1/a_{nn})$. This will make the coefficient of $x_n$ in the last equation equal to 1.</li>
-<li>The final result should be an augmented matrix of the form:
-$$\left[ \matrix{
-a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1(n-1)} &amp; a_{1n} &amp; \vdots &amp; \hat b_1 \\
-0 &amp; a_{22} &amp; \cdots &amp; a_{2(n-1)} &amp; a_{2n} &amp; \vdots &amp; \hat b_2 \\
-\vdots &amp; \vdots &amp;  &amp; \vdots &amp; \vdots &amp; \vdots &amp; \vdots \\
-0 &amp; 0 &amp; \cdots &amp; a_{(n-1)(n-1)} &amp; a_{(n-1)n} &amp; \vdots &amp; \hat b_{n-1} \\
-0 &amp; 0 &amp; \cdots &amp; 0 &amp; 1 &amp; \vdots &amp; \hat b_n
-}\right]$$</li>
-<li>The solution to the problem is then obtained through backward substitutions, i.e.
-$$x_n = \hat b_n$$
-$$ x_{n-1} =  \frac{\hat b_{n-1} - a_{(n-1)n}x_n}{a_{(n-1)(n-1)}}$$
-$$\vdots$$
-$$x_i = \frac{\hat b_i - \sum_{j=i+1}^n a_{ij}x_j}{a_{ii}}\ \ \ \ \mbox{for}\ \ \ i=n-1, n-2, \cdots, 1$$</li>
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-<p>See custom implementation of <a href="./linear-algebra-extra.ipynb">general Gaussian elimination</a></p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Gaussian Elimination</span>
-<span class="kn">import</span> <span class="nn">scipy</span>
-<span class="k">def</span> <span class="nf">Gaussian_Elimination</span><span class="p">(</span> <span class="n">A0</span> <span class="p">):</span>
-    <span class="c1">#Making local copy of matrix</span>
-    <span class="n">P</span><span class="p">,</span><span class="n">L</span><span class="p">,</span><span class="n">A</span><span class="o">=</span><span class="n">scipy</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">lu</span><span class="p">(</span><span class="n">A0</span><span class="p">)</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>    
-    <span class="c1">#Finding solution</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span> <span class="n">n</span> <span class="p">)</span>
-    <span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">A</span><span class="p">[</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">n</span><span class="p">]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">xrange</span><span class="p">(</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span> <span class="p">):</span>
-        <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">n</span><span class="p">]</span> <span class="o">-</span> <span class="nb">sum</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">:</span><span class="n">n</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">:</span><span class="n">n</span><span class="p">])</span> <span class="p">)</span><span class="o">/</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">i</span><span class="p">]</span>
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-    <span class="c1">#Upper diagonal matrix and solutions x</span>
-    <span class="k">return</span> <span class="n">A</span><span class="p">,</span> <span class="n">x</span>
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-<span class="n">M</span> <span class="o">=</span>  <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span> <span class="p">)</span>
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-<pre>array([[0.5507979 , 0.70814782, 0.29090474, 0.51082761, 0.89294695],
-       [0.89629309, 0.12558531, 0.20724288, 0.0514672 , 0.44080984],
-       [0.02987621, 0.45683322, 0.64914405, 0.27848728, 0.6762549 ],
-       [0.59086282, 0.02398188, 0.55885409, 0.25925245, 0.4151012 ]])</pre>
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-       [ 0.        ,  0.63097203,  0.16354802,  0.47919953,  0.62205662],
-       [ 0.        ,  0.        ,  0.52490983, -0.06699671,  0.21531002],
-       [ 0.        ,  0.        ,  0.        ,  0.32582312,  0.00303691]])</pre>
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-<pre>(array([[ 5.        , -4.        ,  0.        ,  1.        ],
-        [ 0.        ,  3.8       , -3.        ,  0.8       ],
-        [ 0.        ,  0.        ,  2.63157895, -1.36842105]]),
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-<p>The previous method of Gaussian Elimination for finding solutions of linear systems is mathematically exact, however round-off errors that appear in computational arithmetics can represent a real problem when high accuracy is required.</p>
-<p>In order to illustrate this, consider the next situation:</p>
-$$ E_1: 0.00300x_1 + 59.14x_2 = 59.17 $$$$ E_2: 5.291x_1 - 6.130x_2 = 46.78 $$<p>Using four-digit arithmetic we obtain:</p>
-<p><strong>1.</strong> Constructing the augmented matrix:</p>
-$$ \left[ \matrix{
-0.003 &amp; 59.14 &amp; \vdots &amp; 59.17 \\
-5.291 &amp; -6.130 &amp; \vdots &amp; 46.78
-}\right] $$<p><strong>2.</strong> Applying the reduction with the first pivot, we obtain:</p>
-$$(E_1 + m E_2)\rightarrow (E_1)$$<p>where:</p>
-$$m = -\frac{a_{21}}{a_{11}} = -\frac{5.291}{0.003} = 1763.666\cdots \approx 1764$$<p>In this step, we have taken the first four digits. This leads us to:</p>
-$$ \left[ \matrix{
-0.003 &amp; 59.14 &amp; \vdots &amp; 59.17 \\
-0 &amp; -104300 &amp; \vdots &amp; -104400
-}\right] $$<p>The exact system is instead</p>
-$$ \left[ \matrix{
-0.003 &amp; 59.14 &amp; \vdots &amp; 59.17 \\
-0 &amp; -104309.37\bar{6} &amp; \vdots &amp; -104309.37\bar{6}
-}\right] $$<p>Using the solution $x_2 \approx 1.001$, we obtain</p>
-$$ x_1 \approx \frac{59.17 - (59.14)(1.001)}{0.00300} = -10 $$<p>The exact solution is however:</p>
-$$ x_1 = 10.00 $$<p>The source of such a large error is that the factor $59.14/0.00300 \approx 20000$. This quantity is propagated through the combination steps of the Gaussian Elimination, yielding a complete wrong result.</p>
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-<p>As mentioned before, Algebra Linear methods do not invole numerical approximations excepting round-off errors. This implies the computing time depends on the number of involved operations (multiplications, divisions, additions and subtractions). It is possible to demonstrate that the numbers of required divisions/multiplications is given by:</p>
-$$\frac{n^3}{3}+n^2-\frac{n}{3}$$<p>and the required additions/subtractions:</p>
-$$\frac{n^3}{3}+\frac{n^2}{2}-\frac{5n}{6}$$<p>These numbers are calculated separately as the computing time per operation for divisions and multiplications are similar and much larger than computing times for additions and subtractions. According to this, the overall computing time, for large $n$, scales as $n^3$/3.</p>
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-<p>Using the library <code>datetime</code> of python, evaluate the computing time required for <code>Gaussian_Elimination</code> to find the solution of a system of $n=20$ equations and $20$ unknowns. For this purpose you can generate random systems. For a more robust result, repeat $500$ times, give a mean value and plot an histrogram of the computing times.</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Importing datatime</span>
-<span class="kn">from</span> <span class="nn">datetime</span> <span class="kn">import</span> <span class="n">datetime</span>
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-<span class="c1">#Number of repeats</span>
-<span class="n">Nrep</span> <span class="o">=</span> <span class="mi">500</span>
-<span class="c1">#Size of matrix</span>
-<span class="n">n</span> <span class="o">=</span> <span class="mi">20</span>
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-<span class="k">def</span> <span class="nf">Gaussian_Time</span><span class="p">(</span> <span class="n">n</span><span class="p">,</span> <span class="n">Nrep</span> <span class="p">):</span>
-    <span class="c1">#Arrays of times</span>
-    <span class="n">Times</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="c1">#Cicle for number of repeats</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">xrange</span><span class="p">(</span><span class="n">Nrep</span><span class="p">):</span>
-        <span class="c1">#Generating random matrix</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="p">)</span> <span class="p">)</span>
-        
-        <span class="c1">#Starting time counter</span>
-        <span class="n">tstart</span> <span class="o">=</span> <span class="n">datetime</span><span class="o">.</span><span class="n">now</span><span class="p">()</span>
-        <span class="c1">#Invoking Gaussian Elimination routine</span>
-        <span class="n">Gaussian_Elimination</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-        <span class="c1">#Killing time counter</span>
-        <span class="n">tend</span> <span class="o">=</span> <span class="n">datetime</span><span class="o">.</span><span class="n">now</span><span class="p">()</span>
-        
-        <span class="c1">#Saving computing time</span>
-        <span class="n">Times</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="p">(</span><span class="n">tend</span><span class="o">-</span><span class="n">tstart</span><span class="p">)</span><span class="o">.</span><span class="n">microseconds</span> <span class="p">)</span>
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-    <span class="c1">#Numpy Array</span>
-    <span class="n">Times</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">Times</span><span class="p">)</span>
-    
-    <span class="nb">print</span> <span class="s2">&quot;The mean computing time for a </span><span class="si">%d</span><span class="s2">x</span><span class="si">%d</span><span class="s2"> matrix is: </span><span class="si">%lf</span><span class="s2"> microseconds&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">n</span><span class="p">,</span><span class="n">Times</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span>
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-    <span class="c1">#Histrogram</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span> <span class="p">)</span>
-    <span class="n">histo</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span> <span class="n">Times</span><span class="p">,</span> <span class="n">bins</span> <span class="o">=</span> <span class="mi">30</span> <span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span> <span class="s2">&quot;Computing Time [microseconds]&quot;</span> <span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span> <span class="s2">&quot;Ocurrences&quot;</span> <span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
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-    <span class="k">return</span> <span class="n">Times</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
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-<span class="n">Gaussian_Time</span><span class="p">(</span> <span class="n">n</span><span class="p">,</span> <span class="n">Nrep</span> <span class="p">)</span>
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-<pre>The mean computing time for a 20x20 matrix is: 2762.628000 microseconds
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-<pre>2762.6280000000002</pre>
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-<p>Using the previous code, estimate the computing time for random matrices of size $n=5,10,50,100,500,1000$. For each size, compute $500$ times in order to reduce spurious errors. Plot the results in a figure of $n$ vs computing time. Is it verified the some of scaling laws (Multiplication/Division or Addition/Sustraction). Note that for large values of $n$, both scaling laws converge to the same.</p>
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-<h2 id="Partial-pivoting">Partial pivoting<a class="anchor-link" href="#Partial-pivoting"> </a></h2>
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-<p>A suitable method to reduce round-off errors is to choose a pivot more conveniently. As we saw before, a small pivot generally implies larger propagated errors as they appear usually as dividends. The partial pivoting method consists then of a choosing of the largest absolute coefficient associated to $x_i$ instead of the first non-null one, i.e.</p>
-$$ |a_{ii}| = \max_{i\leq j\leq n}|a_{ji}| $$<p>This way, propagated multiplication errors would be minimized.</p>
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-<p>Create a new routine <code>Gaussian_Elimination_Pivoting</code> from <code>Gaussian_Elimination</code> in order to include the partial pivoting method. Compare both routines with some random matrix and with the exact solution.</p>
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-<h1 id="Matrix-Inversion">Matrix Inversion<a class="anchor-link" href="#Matrix-Inversion"> </a></h1>
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-<p>Asumming a nonsingular matrix $A$, if a matrix $A^{-1}$ exists, with $AA^{-1} = I$ and $A^{-1}A = I$, where $I$ is the identity matrix, then $A^{-1}$ is called the inverse matrix of $A$. If such a matrix does not exist, $A$ is said to be a singular matrix.</p>
-<p>A corollary of this definition is that $A$ is also the inverse matrix of $A^{-1}$.</p>
-<p>Once defined the Gaussian Elimination method, it is possible to extend it in order to find the inverse of any nonsingular matrix.
-Let's consider the next equation:</p>
-$$ AA^{-1} = AB= \left[ \matrix{
-a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} \\
-a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} \\
-\vdots &amp; \vdots &amp; &amp; \vdots\\
-a_{m1} &amp; a_{n2} &amp; \cdots &amp; a_{nn} 
-}\right]\left[ \matrix{
-b_{11} &amp; b_{12} &amp; \cdots &amp; b_{1n} \\
-b_{21} &amp; b_{22} &amp; \cdots &amp; b_{2n} \\
-\vdots &amp; \vdots &amp; &amp; \vdots\\
-b_{n1} &amp; b_{n2} &amp; \cdots &amp; b_{nn} 
-}\right] = 
-\left[ \matrix{
-1 &amp; 0 &amp; \cdots &amp; 0 \\
-0 &amp; 1 &amp; \cdots &amp; 0 \\
-\vdots &amp; \vdots &amp; \ddots &amp; \vdots\\
-0 &amp; 0 &amp; \cdots &amp; 1
-}\right]$$<p>This can be rewritten as a set of $n$ systems of equations, i.e.</p>
-$$ \left[ \matrix{
-a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} \\
-a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} \\
-\vdots &amp; \vdots &amp; &amp; \vdots\\
-a_{n1} &amp; a_{n2} &amp; \cdots &amp; a_{nn} 
-}\right]\left[ \matrix{
-b_{11} \\
-b_{21} \\
-\vdots \\
-b_{n1}
-}\right] = 
-\left[ \matrix{
-1 \\
-0 \\
-\vdots \\
-0
-}\right],$$$$\left[ \matrix{
-a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} \\
-a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} \\
-\vdots &amp; \vdots &amp; &amp; \vdots\\
-a_{n1} &amp; a_{n2} &amp; \cdots &amp; a_{nn} 
-}\right]\left[ \matrix{
-b_{12} \\
-b_{22} \\
-\vdots \\
-b_{n2}
-}\right] = 
-\left[ \matrix{
-0 \\
-1 \\
-\vdots \\
-0
-}\right]$$$$\vdots$$<p>
-$$\left[ \matrix{
-a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} \\
-a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} \\
-\vdots &amp; \vdots &amp; &amp; \vdots\\
-a_{n1} &amp; a_{n2} &amp; \cdots &amp; a_{nn} 
-}\right]\left[ \matrix{
-b_{1n} \\
-b_{2n} \\
-\vdots \\
-b_{nn}
-}\right] = 
-\left[ \matrix{
-0 \\
-0 \\
-\vdots \\
-1
-}\right]$$</p>
-<p>These systems can be solved individually by using Gaussian Elimination, however we can mix all the problems, obtaining the augmented matrix:</p>
-$$\left[ \matrix{
-a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} &amp; \vdots &amp; 1 &amp; 0 &amp; \cdots &amp; 0 \\
-a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} &amp; \vdots &amp; 0 &amp; 1 &amp; \cdots &amp; 0 \\
-\vdots &amp; \vdots &amp; &amp; \vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\
-a_{n1} &amp; a_{n2} &amp; \cdots &amp; a_{nn} &amp; \vdots &amp; 0 &amp; 0 &amp; \cdots &amp; 1
-}\right]$$<p>Now, applying Gaussian Elimination we can obtain a upper diagonal form for the first matrix. Completing the steps using forwards elimination we can convert the first matrix into the identity matrix, obtaining</p>
-$$\left[ \matrix{
-1 &amp; 0 &amp; \cdots &amp; 0 &amp; \vdots &amp; b_{11} &amp; b_{12} &amp; \cdots &amp; b_{1n} \\
-0 &amp; 1 &amp; \cdots &amp; 0 &amp; \vdots &amp; b_{21} &amp; b_{22} &amp; \cdots &amp; b_{2n} \\
-\vdots &amp; \vdots &amp; \ddots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; &amp; \vdots\\
-0 &amp; 0 &amp; \cdots &amp; 1 &amp; \vdots &amp; b_{n1} &amp; b_{n2} &amp; \cdots &amp; b_{nn}
-}\right]$$<p>Where the second matrix is then the inverse $B=A^{-1}$.</p>
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-<p>Using the previous routine <code>Gaussian_Elimination_Pivoting</code>, create a new routine <code>Inverse</code> that calculates the inverse of any given squared matrix.</p>
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-<h1 id="Determinant-of-a-Matrix">Determinant of a Matrix<a class="anchor-link" href="#Determinant-of-a-Matrix"> </a></h1>
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-<p>The determinant of a matrix is a scalar quantity calculated for square matrix. This provides important information about the matrix of coefficients of a system of linear of equations. For example, any system of $n$ equations and $n$ unknowns has an unique solution if the associated determinant is nonzero. This also implies the determinant allows to evaluate whether a matrix is singular or nonsingular.</p>
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-<p>Next, we shall define some properties of determinants that will allow us to calculate determinants by using a recursive code:</p>
-<p><strong>1.</strong> If $A = [a]$ is a $1\times 1$ matrix, its determinant is then $\det A = a$.</p>
-<p><strong>2.</strong> If $A$ is a $n\times n$ matrix, the minor matrix $M_{ij}$ is the determinant of the $(n-1)\times(n-1)$ matrix obtained by deleting the $i$th row and the $j$th column.</p>
-<p><strong>3.</strong> The cofactor $A_{ij}$ associated with $M_{ij}$ is defined by $A_{ij} = (-1)^{i+j}M_{ij}$.</p>
-<p><strong>4.</strong> The determinant of a $n\times n$ matrix $A$ is given by:</p>
-$$ \det A = \sum_{j=1}^n a_{ij}A_{ij} $$<p>or</p>
-$$ \det A = \sum_{i=1}^n a_{ij}A_{ij} $$<p>This is, it is possible to use both, a row or a column for calculating the determinant.</p>
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-<h2 id="Computing-time-of-determinants">Computing time of determinants<a class="anchor-link" href="#Computing-time-of-determinants"> </a></h2>
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-<p>Using the previous recurrence, we can calculate the computing time of the previous algorithm. First, let's consider the number of required operations for a $2\times 2$ matrix: let $A$ be a $2\times 2$ matrix given by:</p>
-$$ A = \left[ \matrix{
-a_{11} &amp; a_{12} \\
-a_{21} &amp; a_{22}}\right]$$<p>The determinant is then given by:</p>
-$$\det(A) = a_{11}a_{22} - a_{12}a_{21}$$<p>the number of required multiplications was $2$ and subtractions is $1$.</p>
-<p>Now, using the previous formula for the determinant</p>
-$$ \det A = \sum_{j=1}^n a_{ij}A_{ij} $$<p>For a $3\times 3$ matrix, it is necessary to calculate $3$ times $2\times 2$ determinants. Besides, it is necessary to multiply the cofactor $A_{ij}$ with the coefficient $a_{ij}$, that leads us with $t_{n=3}=3\times 2 + 3$ multiplications. Additions are not important as their computing time is far less than multiplications.</p>
-<p>For a $4\times 4$ matrix, we need four deteminants of $3\times 3$ submatrices, leading $t_{n=4} = 4\times( 3\times 2 + 3 ) + 4 = 4! + \frac{4!}{2!} + \frac{4!}{3!}$. In general, for a $n\times n$ matrix, we have then:</p>
-$$ t_{n} = \frac{n!}{(n-1)!} + \frac{n!}{(n-2)!} + \cdots + \frac{n!}{1!} = n!\left( \sum_{i=1}^{n-1}\frac{1}{i!} \right)$$<p>If $n$ is large enough, we can approximate $t_{n}\approx n!$.</p>
-<p>In computers, this is a prohibitive computing time so other schemes have to be proposed.</p>
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-<p>Evaluate the computing time of the <code>Determinant</code> routine for matrix sizes of $n=1,2,3,\cdots,10$ and doing several repeats. Plot your result ($n$ vs $t_n$). What can you conclude about the behaviour of the computing time?</p>
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-<h2 id="Properties-of-determinants">Properties of determinants<a class="anchor-link" href="#Properties-of-determinants"> </a></h2>
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-<p>Determinants have a set of properties that can reduce considerably computing times. Suppose $A$ is a $n\times n$ matrix:</p>
-<p><strong>1.</strong> If any row or column of $A$ has only zero entries, then $\det A = 0$.</p>
-<p><strong>2.</strong> If two rows or columns of $A$ are the same, then $\det A = 0$.</p>
-<p><strong>3.</strong> If $\hat A$ is obtained from $A$ by using a swap operation $(E_i)\leftrightarrow (E_j)$, then $\det \hat A=-\det A$.</p>
-<p><strong>4.</strong> If $\hat A$ is obtained from $A$ by using a escalation operation $(\lambda E_i)\leftrightarrow (E_i)$, then $\det \hat A=\lambda \det A$.</p>
-<p><strong>5.</strong> If $\hat A$ is obtained from $A$ by using a combination operation $(E_i+\lambda E_j) \leftrightarrow (E_i)$, then $\det \hat A=\det A$.</p>
-<p><strong>6.</strong> If $B$ is also a $n\times n$ matrix, then $\det(AB)=(\det A)(\det B).$</p>
-<p><strong>7.</strong> $\det A^t=\det A.$</p>
-<p><strong>8.</strong> $\det A^{-1}=(\det A)^{-1}$</p>
-<p><strong>9.</strong> Finally and most importantly: if $A$ is an upper, lower or diagonal matrix, then:</p>
-$$ \det A = \prod_{i=1}^n a_{ii} $$
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-<p>As we analysed before, Gaussian Elimination takes a computing time scaling like $n^3$ for large matrix sizes. According to the previous properties, the determinant of a upper diagonal matrix just takes $n-1$ multiplications, far less than a nondiagonal matrix. Combining these properties, we can track back and relate the determinant of the resulting upper diagonal matrix and the original one. Leading us to a computing time scaling like $n^3$, much better than the original $n!$.</p>
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-<p>Using the <code>Gaussian_Elimination</code> routine and tracking back the performed operations, construct a new routine called <code>Gaussian_Determinant</code>. Make the same analysis of the computing time as the previous activity. Compare both results.</p>
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-<p>If the matrix $A$ has an inverse, then
-\begin{align}
-A \boldsymbol{x}=&amp;\boldsymbol{b}\\
-A^{-1}A\boldsymbol{x}=&amp;A^{-1}\boldsymbol{b}\\
-\boldsymbol{x}=&amp;A^{-1}\boldsymbol{b}\,,
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-From the previous example</p>
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-       [0.89629309, 0.12558531, 0.20724288, 0.0514672 , 0.44080984],
-       [0.02987621, 0.45683322, 0.64914405, 0.27848728, 0.6762549 ],
-       [0.59086282, 0.02398188, 0.55885409, 0.25925245, 0.4151012 ]])</pre>
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-       [0.89629309, 0.12558531, 0.20724288, 0.0514672 ],
-       [0.02987621, 0.45683322, 0.64914405, 0.27848728],
-       [0.59086282, 0.02398188, 0.55885409, 0.25925245]])</pre>
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-<p>such that
-\begin{align}
-A \boldsymbol{x}=&amp;\boldsymbol{b}\,,
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-<p>Check that $A$ has an inverse and calculate $\boldsymbol{x}$</p>
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-$$\det(A)=-0.097\ne 0$$
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-$\displaystyle \left[\begin{matrix}0.274\\0.8722\\0.4114\\0.0093\end{matrix}\right]$
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-<h3 id="Numpy-implementation">Numpy implementation<a class="anchor-link" href="#Numpy-implementation"> </a></h3>
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-<pre>array([[0.27395594],
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-<h1 id="Matrix-diagonalization">Matrix diagonalization<a class="anchor-link" href="#Matrix-diagonalization"> </a></h1><p>See <a href="./linear-algebra-diagonalization.ipynb">linear-algebra-diagonalization</a></p>
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-<h1 id="LU-Factorization">LU Factorization<a class="anchor-link" href="#LU-Factorization"> </a></h1>
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-<p>As we saw before, the Gaussian Elimination algorithm takes a computing time scaling as $\mathcal{O}(n^3/3)$ in order to solve a system of $n$ equations and $n$ unknowns. Let's assume a system of equations $\mathbf{A}\mathbf{x} = \mathbf{b}$ where $\mathbf{b}$ is already in upper diagonal form.</p>
-$$\left[ \matrix{
-a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1(n-1)} &amp; a_{1n} &amp; \vdots &amp; \hat b_1 \\
-0 &amp; a_{22} &amp; \cdots &amp; a_{2(n-1)} &amp; a_{2n} &amp; \vdots &amp; \hat b_2 \\
-\vdots &amp; \vdots &amp;  &amp; \vdots &amp; \vdots &amp; \vdots &amp; \vdots \\
-0 &amp; 0 &amp; \cdots &amp; a_{(n-1)(n-1)} &amp; a_{(n-1)n} &amp; \vdots &amp; \hat b_{n-1} \\
-0 &amp; 0 &amp; \cdots &amp; 0 &amp; a_{nn} &amp; \vdots &amp; \hat b_n
-}\right]$$<p>The Gauss-Jordan algorithm can reduce even more this problem in order to solve it directly, yielding:</p>
-$$\left[ \matrix{
-1 &amp; 0 &amp; \cdots &amp; 0 &amp; 0 &amp; \vdots &amp; x_1 \\
-0 &amp; 1 &amp; \cdots &amp; 0 &amp; 0 &amp; \vdots &amp; x_2 \\
-\vdots &amp; \vdots &amp;  &amp; \vdots &amp; \vdots &amp; \vdots &amp; \vdots \\
-0 &amp; 0 &amp; \cdots &amp; 1 &amp; 0 &amp; \vdots &amp; x_{n-1} \\
-0 &amp; 0 &amp; \cdots &amp; 0 &amp; 1 &amp; \vdots &amp; x_n
-}\right]$$<p>From the upper diagonal form to the completely reduced one, it is necessary to perform $n+(n-1)+(n-2)+\cdots\propto n(n-1)$ backwards substitutions. The computing time for solving a upper diagonal system is then $\mathcal{O}(n^2)$.</p>
-<p>Now, let $\mathbf{A}\mathbf{x} = \mathbf{b}$ be a general system of equations of $n$ dimensions. Let's assume $\mathbf{A}$ can be written as a multiplication of two matrices, one lower diagonal $\mathbf{L}$ and other upper diagonal $\mathbf{U}$, such that $\mathbf{A}=\mathbf{L}\mathbf{U}$. Defining a vector $\mathbf{y} = \mathbf{U}\mathbf{x}$, it is obtained for the original system</p>
-$$ \mathbf{A} \mathbf{x}=\mathbf{L}(\mathbf{U}\mathbf{x}) = \mathbf{L}\mathbf{y} = \mathbf{b}$$<p>For solving this system we can then:</p>
-<p><strong>1.</strong> Solve the equivalent system $\mathbf{L}\mathbf{y} = \mathbf{b}$, what takes a computing time of $\mathcal{O}(n^2)$.</p>
-<p><strong>2.</strong> Once we know $\mathbf{y}$, we can solve the system $\mathbf{U}\mathbf{x} = \mathbf{y}$, with a computing time of $\mathcal{O}(n^2)$.</p>
-<p>The global computing time is then $\mathcal{O}(2n^2)$</p>
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-<p>In order to compare the computing time that Gaussian Elimination takes and the previous time for the LU factorization, make a plot of both computing times. What can you conclude when $n$ becomes large enough?</p>
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-<h2 id="Derivation-of-LU-factorization">Derivation of LU factorization<a class="anchor-link" href="#Derivation-of-LU-factorization"> </a></h2>
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-<p>Although the LU factorization seems to be a far better method for solving linear systems as compared with say Gaussian Elimination, we was assuming we already knew the matrices $\mathbf{L}$ and $\mathbf{U}$. Now we are going to see the algorithm for perfoming this reduction takes a computing time of $\mathcal{O}(n^3/3)$.</p>
-<p>You may wonder then, what advantage implies the use of this factorization? Well, matrices $\mathbf{L}$ and $\mathbf{U}$ do not depend on the specific system to be solved, i.e. there is not dependence on the $\mathbf{b}$ vector, so once we have both matrices, we can use them to solve any system we want, just taking a $\mathcal{O}(2n^2)$ computing time.</p>
-<p>First, let's assume a matrix $\mathbf{A}$ with all its pivots are nonzero, so there is not need to swap rows. Now, when we want to eliminate all the coefficients associated to $x_1$, we perform the next operations:</p>
-$$ (E_j-m_{j1}E_1)\rightarrow (E_j), \ \ \ \ \mbox{where}\ \ \ \ m_{j1} = \frac{a^{(1)}_{j1}}{a^{(1)}_{11}} $$<p>henceforth, $a^{(1)}_{ij}$ denotes the components of the original matrix $\mathbf{A}=\mathbf{A}^{(1)}$, $a^{(2)}_{ij}$ the components of the matrix after eliminating the coefficients of $x_1$, and generally, $a^{(k)}_{ij}$ the components of the matrix after eliminating the coefficients of $x_{k-1}$.</p>
-<p>The previous operation over the matrix $\mathbf{A}$ can be also reproduced defining the matrix $\mathbf{M}^{(1)}$</p>
-$$\mathbf{M}^{(1)} = \left[ \matrix{
-1 &amp; 0 &amp; \cdots &amp; 0 &amp; 0 \\
--m_{21} &amp; 1 &amp; \cdots &amp; 0 &amp; 0 \\
-\vdots &amp; \vdots &amp; \ddots &amp; \vdots &amp; \vdots \\
--m_{(n-1)1} &amp; 0 &amp; \cdots &amp; 1 &amp; 0 \\
--m_{n1} &amp; 0 &amp; \cdots &amp; 0 &amp; 1
-}\right]$$<p>This is called the <strong>first Gaussian transformation matrix</strong>. From this, we have</p>
-$$ \mathbf{A}^{(2)}\mathbf{x} = \mathbf{M}^{(1)}\mathbf{A}^{(1)}\mathbf{x} = \mathbf{M}^{(1)}\mathbf{b}^{(1)} = \mathbf{b}^{(2)} $$<p>where $\mathbf{A}^{(2)}$ is matrix with null coefficients associated to $x_1$ but the first one.</p>
-<p>Repeating the same procedure for the next pivots, we obtain then</p>
-$$ \mathbf{A}^{(n)} = \mathbf{M}^{(n-1)}\mathbf{M}^{(n-2)}\cdots \mathbf{M}^{(1)}\mathbf{A}^{(1)} $$<p>where the <strong>$k$th Gaussian transformation matrix</strong> is defined as</p>
-$$\mathbf{M}^{(k)}_{ij} = \left\{ \matrix{
-1 &amp; \mbox{if}\ \ i=j \\
--m_{ij} &amp; \mbox{if}\ \ j=k\ \ \mbox{and}\ \ k+1\leq i \leq n \\
-0 &amp; \mbox{otherwise}
-}  \right.$$<p>and</p>
-$$m_{ij} = \frac{a^{(j)}_{ij}}{a^{(j)}_{jj}} $$<p>Note $\mathbf{A}^{(n)}$ is a upper diagonal matrix given by</p>
-$$\mathbf{A}^{(n)} = \left[ \matrix{
-a_{11}^{(n)} &amp; a_{12}^{(n)} &amp; \cdots &amp; a_{1(n-1)}^{(n)} &amp; a_{1n}^{(n)}\\
-0 &amp; a_{22}^{(n)} &amp; \cdots &amp; a_{2(n-1)}^{(n)} &amp; a_{2n}^{(n)} \\
-\vdots &amp; \vdots &amp;  &amp; \vdots &amp; \vdots &amp;\\
-0 &amp; 0 &amp; \cdots &amp; a_{(n-1)(n-1)}^{(n)} &amp; a_{(n-1)n}^{(n)} \\
-0 &amp; 0 &amp; \cdots &amp; 0 &amp; a_{nn}^{(n)}
-}\right]$$<p>so we can define $\mathbf{U}\equiv \mathbf{A}^{(n)}$.</p>
-<p>Now, taking the equation</p>
-$$ \mathbf{A}^{(n)} = \mathbf{M}^{(n-1)}\mathbf{M}^{(n-2)}\cdots \mathbf{M}^{(1)}\mathbf{A}^{(1)} $$<p>and defining the inverse of $\mathbf{M}^{(k)}$ as</p>
-$$ \mathbf{L}^{(k)}_{ij} = \left(\mathbf{M}^{(k)}\right)^{-1}_{ij} =  \left\{ \matrix{
-1 &amp; \mbox{if}\ \ i=j \\
-m_{ij} &amp; \mbox{if}\ \ j=k\ \ \mbox{and}\ \ k+1\leq i \leq n \\
-0 &amp; \mbox{otherwise}
-}  \right.$$<p>we obtain</p>
-$$ \mathbf{L}^{(1)} \cdots \mathbf{L}^{(n-2)}\mathbf{L}^{(n-1)}\mathbf{A}^{(n)} = \mathbf{L}^{(1)} \cdots \mathbf{L}^{(n-2)}\mathbf{L}^{(n-1)}\mathbf{M}^{(n-1)}\mathbf{M}^{(n-2)}\cdots \mathbf{M}^{(1)}\mathbf{A}^{(1)} = \mathbf{L}\mathbf{U} $$<p>where the lower diagonal matrix $\mathbf{L}$ is given by:</p>
-$$ \mathbf{L} = \mathbf{L}^{(1)} \cdots \mathbf{L}^{(n-2)}\mathbf{L}^{(n-1)} $$<p>.</p>
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-<p>The algorithm is then given by:</p>
-<p><strong>1.</strong> Give a square matrix $\mathbf{A}$ where the pivots are nonzero.</p>
-<p><strong>2.</strong> Apply the operation $Comb(E_j,E_1,-a^{(1)}_{j1}/aa^{(1)}_{11})$. This eliminates the coefficients associated to $x_1$ in all the rows but in the first one.</p>
-<p><strong>3.</strong> Construct the matrix $\mathbf{L}^{(1)}$ given by</p>
-$$ \mathbf{L}^{(k)}_{ij} =  \left\{ \matrix{
-1 &amp; \mbox{if}\ \ i=j \\
-m_{ij} = \frac{a^{(j)}_{ij}}{a^{(j)}_{jj}} &amp; \mbox{if}\ \ j=k\ \ \mbox{and}\ \ k+1\leq i \leq n \\
-0 &amp; \mbox{otherwise}
-}  \right.$$<p>with $k=1$.</p>
-<p><strong>4.</strong> Repeat the steps <strong>2</strong> and <strong>3</strong> for the next column until reaching the last one.</p>
-<p><strong>5.</strong> Return the matrices $\mathbf{U} = \mathbf{A}^{(n)}$ and $ \mathbf{L} = \mathbf{L}^{(1)} \cdots \mathbf{L}^{(n-2)}\mathbf{L}^{(n-1)} $.</p>
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-<p>Create a routine called <code>LU_Factorization</code> that, given a matrix $\mathbf{A}$ and the previous algorithm, calculate the LU factorization of the matrix. Test your routine with a random square matrix, verify that $\mathbf{A} = \mathbf{L}\mathbf{U}$.</p>
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-       [ 0.        ,  0.        ,  2.63157895]])</pre>
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-<h2 id="-Eigenvalues-and-Eigenvectors-activity--"><span style="color:red"> Eigenvalues and Eigenvectors activity  </span><a class="anchor-link" href="#-Eigenvalues-and-Eigenvectors-activity--"> </a></h2>
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-<h3 id="Electron-interacting-with-a-magnetic-field">Electron interacting with a magnetic field<a class="anchor-link" href="#Electron-interacting-with-a-magnetic-field"> </a></h3><p>An electron is placed to interact with an uniform magnetic field. To give account of the possible allowed levels of the electron in the presence of the magnetic field it is necessary to solve the next equation</p>
-\begin{equation}
-\hat{H}|\Psi\rangle = E|\Psi\rangle
-\end{equation}<p>where the hamiltonian is equal to $H = -\mu \cdot B = -\gamma B \cdot S$, with the gyromagnetic ratio $\gamma$, $\textbf{B}$ the magnetic field and $\textbf{S}$ the spin. It can be shown that the hamiltonian expression is transformed in</p>
-\begin{equation}
-\hat{\textbf{H}} = - \frac{\gamma \hbar}{2} \left( \begin{array}{cc}
-B_z  &amp; B_x -i B_y \\
-B_x + i B_y  &amp; -B_z  \end{array} \right) 
-\end{equation}<p>Then, by solving the problem $|H - EI|=0$ is found the allowed energy levels, i.e., finding the determinant of the 
-matrix $H - EI$ allows to get the values $E_1$ and $E_2$.</p>
-<p>And solving the problem $\hat{H}\Psi$ - E$\Psi = 0$ gives the autofunctions $\Psi$, i.e., the column vector $\Psi= \{\Psi_1, \Psi_2\}$.</p>
-<p>The function scipy.optimize.root can be used to get roots of a given equation. The experimental value of $\gamma$ for the electron is 2. The order of magnitude of the magnetic field is $1g$.</p>
-<p>1)  Find the allowed energy levels.</p>
-<p>2)  Find the autofunctions and normalize them.</p>
-<p><strong>Hint:</strong> An imaginary number in python can be written as 1j.</p>
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-<h1><div style="float: right;" markdown="1">     <img src="http://matplotlib.org/_static/logo2.png"> </div>  Matplotlib</h1>
-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/matplotlib.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<p>Matplotlib is a python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms. matplotlib can be used in python scripts, the python and ipython shell, web application servers, and six graphical user interface toolkits.</p>
-<p>Matplotlib tries to make easy things easy and hard things possible. You can generate plots, histograms, power spectra, bar charts, errorcharts, scatterplots, etc, with just a few lines of code. For a sampling, see the screenshots, thumbnail gallery, and examples directory</p>
-<p>For simple plotting the pyplot interface provides a MATLAB-like interface, particularly when combined with IPython. For the power user, you have full control of line styles, font properties, axes properties, etc, via an object oriented interface or via a set of functions familiar to MATLAB users.</p>
-<p><a href="http://matplotlib.org/"><strong>Official page</strong></a></p>
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-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="c1">#Ignore me!! (for scripts)</span>
-<span class="o">%</span><span class="k">pylab</span> inline
-<span class="c1">#Importing matplotlib</span>
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-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="p">)</span>
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-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
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-<span class="c1">#Ploting a function</span>
-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="mi">100</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span> <span class="n">x</span> <span class="p">)</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span> <span class="n">x</span> <span class="p">)</span> <span class="p">)</span>
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-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">10</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">semilogy</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span> <span class="n">x</span> <span class="p">)</span> <span class="p">)</span>
-<span class="n">p</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">semilogy</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span> <span class="n">x</span> <span class="p">),</span><span class="s1">&#39;.&#39;</span> <span class="p">)</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
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-<pre>array([2.71828183e+00, 7.38905610e+00, 2.00855369e+01, 5.45981500e+01,
-       1.48413159e+02, 4.03428793e+02, 1.09663316e+03, 2.98095799e+03,
-       8.10308393e+03, 2.20264658e+04])</pre>
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-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">semilogx</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span> <span class="n">x</span> <span class="p">)</span> <span class="p">)</span>
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-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">16.5</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="mi">200</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">loglog</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span> <span class="n">x</span> <span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">loglog</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span> <span class="n">x</span> <span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span><span class="s1">&#39;.&#39;</span> <span class="p">)</span>
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-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="mf">0.1</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="mf">16.5</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">),</span> <span class="mi">200</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">loglog</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span> <span class="n">x</span> <span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">loglog</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span> <span class="n">x</span> <span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span><span class="s1">&#39;.&#39;</span> <span class="p">)</span>
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-<span class="n">r</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">,</span> <span class="mf">0.01</span> <span class="p">)</span>
-<span class="n">theta</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">r</span>
-
-<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">,</span> <span class="n">polar</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">theta</span><span class="p">,</span> <span class="n">r</span> <span class="p">)</span>
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-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">0.1</span> <span class="p">)</span>
-<span class="n">y1</span> <span class="o">=</span> <span class="n">x</span><span class="o">/</span><span class="mf">2.</span>
-<span class="n">y2</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="n">y1</span><span class="p">,</span> <span class="n">y2</span><span class="p">,</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.6</span> <span class="p">)</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Multiple figures</span>
-<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mf">0.1</span> <span class="p">)</span>
-<span class="c1">#First plot</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">X</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="mi">1</span> <span class="p">)</span>
-<span class="c1">#Second plot</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">semilogy</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">X</span><span class="o">-</span><span class="mi">10</span><span class="p">)</span> <span class="p">)</span>
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-<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span> <span class="mi">500</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span> <span class="n">X</span> <span class="p">)</span>
-</pre></div>
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-<pre>(array([54., 54., 48., 53., 40., 53., 47., 46., 51., 54.]),
- array([0.0026303 , 0.10221054, 0.20179078, 0.30137101, 0.40095125,
-        0.50053148, 0.60011172, 0.69969195, 0.79927219, 0.89885243,
-        0.99843266]),
- &lt;BarContainer object of 10 artists&gt;)</pre>
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Animmations">Animmations<a class="anchor-link" href="#Animmations"> </a></h2><p>See: <a href="http://louistiao.me/posts/notebooks/embedding-matplotlib-animations-in-jupyter-as-interactive-javascript-widgets/">http://louistiao.me/posts/notebooks/embedding-matplotlib-animations-in-jupyter-as-interactive-javascript-widgets/</a></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
-<span class="kn">import</span> <span class="nn">matplotlib.animation</span>
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-
-<span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span>
-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-
-<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
-<span class="n">ax</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="n">l</span><span class="p">,</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">([],[])</span>
-
-<span class="k">def</span> <span class="nf">animate</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-    <span class="n">l</span><span class="o">.</span><span class="n">set_data</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">i</span><span class="p">],</span> <span class="n">x</span><span class="p">[:</span><span class="n">i</span><span class="p">])</span>
-
-<span class="c1">#Avoid intial plot</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
-<span class="n">ani</span> <span class="o">=</span> <span class="n">matplotlib</span><span class="o">.</span><span class="n">animation</span><span class="o">.</span><span class="n">FuncAnimation</span><span class="p">(</span><span class="n">fig</span><span class="p">,</span> <span class="n">animate</span><span class="p">,</span> <span class="n">frames</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">))</span>
-
-<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">HTML</span>
-<span class="n">HTML</span><span class="p">(</span><span class="n">ani</span><span class="o">.</span><span class="n">to_jshtml</span><span class="p">())</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-
-<link rel="stylesheet"
-href="https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/css/font-awesome.min.css">
-<script language="javascript">
-  function isInternetExplorer() {
-    ua = navigator.userAgent;
-    /* MSIE used to detect old browsers and Trident used to newer ones*/
-    return ua.indexOf("MSIE ") > -1 || ua.indexOf("Trident/") > -1;
-  }
-
-  /* Define the Animation class */
-  function Animation(frames, img_id, slider_id, interval, loop_select_id){
-    this.img_id = img_id;
-    this.slider_id = slider_id;
-    this.loop_select_id = loop_select_id;
-    this.interval = interval;
-    this.current_frame = 0;
-    this.direction = 0;
-    this.timer = null;
-    this.frames = new Array(frames.length);
-
-    for (var i=0; i<frames.length; i++)
-    {
-     this.frames[i] = new Image();
-     this.frames[i].src = frames[i];
-    }
-    var slider = document.getElementById(this.slider_id);
-    slider.max = this.frames.length - 1;
-    if (isInternetExplorer()) {
-        // switch from oninput to onchange because IE <= 11 does not conform
-        // with W3C specification. It ignores oninput and onchange behaves
-        // like oninput. In contrast, Mircosoft Edge behaves correctly.
-        slider.setAttribute('onchange', slider.getAttribute('oninput'));
-        slider.setAttribute('oninput', null);
-    }
-    this.set_frame(this.current_frame);
-  }
-
-  Animation.prototype.get_loop_state = function(){
-    var button_group = document[this.loop_select_id].state;
-    for (var i = 0; i < button_group.length; i++) {
-        var button = button_group[i];
-        if (button.checked) {
-            return button.value;
-        }
-    }
-    return undefined;
-  }
-
-  Animation.prototype.set_frame = function(frame){
-    this.current_frame = frame;
-    document.getElementById(this.img_id).src =
-            this.frames[this.current_frame].src;
-    document.getElementById(this.slider_id).value = this.current_frame;
-  }
-
-  Animation.prototype.next_frame = function()
-  {
-    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));
-  }
-
-  Animation.prototype.previous_frame = function()
-  {
-    this.set_frame(Math.max(0, this.current_frame - 1));
-  }
-
-  Animation.prototype.first_frame = function()
-  {
-    this.set_frame(0);
-  }
-
-  Animation.prototype.last_frame = function()
-  {
-    this.set_frame(this.frames.length - 1);
-  }
-
-  Animation.prototype.slower = function()
-  {
-    this.interval /= 0.7;
-    if(this.direction > 0){this.play_animation();}
-    else if(this.direction < 0){this.reverse_animation();}
-  }
-
-  Animation.prototype.faster = function()
-  {
-    this.interval *= 0.7;
-    if(this.direction > 0){this.play_animation();}
-    else if(this.direction < 0){this.reverse_animation();}
-  }
-
-  Animation.prototype.anim_step_forward = function()
-  {
-    this.current_frame += 1;
-    if(this.current_frame < this.frames.length){
-      this.set_frame(this.current_frame);
-    }else{
-      var loop_state = this.get_loop_state();
-      if(loop_state == "loop"){
-        this.first_frame();
-      }else if(loop_state == "reflect"){
-        this.last_frame();
-        this.reverse_animation();
-      }else{
-        this.pause_animation();
-        this.last_frame();
-      }
-    }
-  }
-
-  Animation.prototype.anim_step_reverse = function()
-  {
-    this.current_frame -= 1;
-    if(this.current_frame >= 0){
-      this.set_frame(this.current_frame);
-    }else{
-      var loop_state = this.get_loop_state();
-      if(loop_state == "loop"){
-        this.last_frame();
-      }else if(loop_state == "reflect"){
-        this.first_frame();
-        this.play_animation();
-      }else{
-        this.pause_animation();
-        this.first_frame();
-      }
-    }
-  }
-
-  Animation.prototype.pause_animation = function()
-  {
-    this.direction = 0;
-    if (this.timer){
-      clearInterval(this.timer);
-      this.timer = null;
-    }
-  }
-
-  Animation.prototype.play_animation = function()
-  {
-    this.pause_animation();
-    this.direction = 1;
-    var t = this;
-    if (!this.timer) this.timer = setInterval(function() {
-        t.anim_step_forward();
-    }, this.interval);
-  }
-
-  Animation.prototype.reverse_animation = function()
-  {
-    this.pause_animation();
-    this.direction = -1;
-    var t = this;
-    if (!this.timer) this.timer = setInterval(function() {
-        t.anim_step_reverse();
-    }, this.interval);
-  }
-</script>
-
-<style>
-.animation {
-    display: inline-block;
-    text-align: center;
-}
-input[type=range].anim-slider {
-    width: 374px;
-    margin-left: auto;
-    margin-right: auto;
-}
-.anim-buttons {
-    margin: 8px 0px;
-}
-.anim-buttons button {
-    padding: 0;
-    width: 36px;
-}
-.anim-state label {
-    margin-right: 8px;
-}
-.anim-state input {
-    margin: 0;
-    vertical-align: middle;
-}
-</style>
-
-<div class="animation">
-  <img id="_anim_img6cf4ad09491040a289750ac0522fb4d3">
-  <div class="anim-controls">
-    <input id="_anim_slider6cf4ad09491040a289750ac0522fb4d3" type="range" class="anim-slider"
-           name="points" min="0" max="1" step="1" value="0"
-           oninput="anim6cf4ad09491040a289750ac0522fb4d3.set_frame(parseInt(this.value));"></input>
-    <div class="anim-buttons">
-      <button title="Decrease speed" onclick="anim6cf4ad09491040a289750ac0522fb4d3.slower()">
-          <i class="fa fa-minus"></i></button>
-      <button title="First frame" onclick="anim6cf4ad09491040a289750ac0522fb4d3.first_frame()">
-        <i class="fa fa-fast-backward"></i></button>
-      <button title="Previous frame" onclick="anim6cf4ad09491040a289750ac0522fb4d3.previous_frame()">
-          <i class="fa fa-step-backward"></i></button>
-      <button title="Play backwards" onclick="anim6cf4ad09491040a289750ac0522fb4d3.reverse_animation()">
-          <i class="fa fa-play fa-flip-horizontal"></i></button>
-      <button title="Pause" onclick="anim6cf4ad09491040a289750ac0522fb4d3.pause_animation()">
-          <i class="fa fa-pause"></i></button>
-      <button title="Play" onclick="anim6cf4ad09491040a289750ac0522fb4d3.play_animation()">
-          <i class="fa fa-play"></i></button>
-      <button title="Next frame" onclick="anim6cf4ad09491040a289750ac0522fb4d3.next_frame()">
-          <i class="fa fa-step-forward"></i></button>
-      <button title="Last frame" onclick="anim6cf4ad09491040a289750ac0522fb4d3.last_frame()">
-          <i class="fa fa-fast-forward"></i></button>
-      <button title="Increase speed" onclick="anim6cf4ad09491040a289750ac0522fb4d3.faster()">
-          <i class="fa fa-plus"></i></button>
-    </div>
-    <form title="Repetition mode" action="#n" name="_anim_loop_select6cf4ad09491040a289750ac0522fb4d3"
-          class="anim-state">
-      <input type="radio" name="state" value="once" id="_anim_radio1_6cf4ad09491040a289750ac0522fb4d3"
-             >
-      <label for="_anim_radio1_6cf4ad09491040a289750ac0522fb4d3">Once</label>
-      <input type="radio" name="state" value="loop" id="_anim_radio2_6cf4ad09491040a289750ac0522fb4d3"
-             checked>
-      <label for="_anim_radio2_6cf4ad09491040a289750ac0522fb4d3">Loop</label>
-      <input type="radio" name="state" value="reflect" id="_anim_radio3_6cf4ad09491040a289750ac0522fb4d3"
-             >
-      <label for="_anim_radio3_6cf4ad09491040a289750ac0522fb4d3">Reflect</label>
-    </form>
-  </div>
-</div>
-
-
-<script language="javascript">
-  /* Instantiate the Animation class. */
-  /* The IDs given should match those used in the template above. */
-  (function() {
-    var img_id = "_anim_img6cf4ad09491040a289750ac0522fb4d3";
-    var slider_id = "_anim_slider6cf4ad09491040a289750ac0522fb4d3";
-    var loop_select_id = "_anim_loop_select6cf4ad09491040a289750ac0522fb4d3";
-    var frames = new Array(50);
-
-  frames[0] = "data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbAAAAEgCAYAAADVKCZpAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90\
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-<h2 id="Easy-animmations-with-video-output">Easy animmations with video output<a class="anchor-link" href="#Easy-animmations-with-video-output"> </a></h2>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#See </span>
-<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">HTML</span>
-<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
-<span class="kn">from</span> <span class="nn">celluloid</span> <span class="kn">import</span> <span class="n">Camera</span>
-
-<span class="n">plt</span><span class="o">.</span><span class="n">ioff</span><span class="p">()</span>
-<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
-<span class="n">camera</span> <span class="o">=</span> <span class="n">Camera</span><span class="p">(</span><span class="n">fig</span><span class="p">)</span>
-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="mi">10</span><span class="p">)</span>
-    <span class="n">camera</span><span class="o">.</span><span class="n">snap</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
-<span class="n">animation</span> <span class="o">=</span> <span class="n">camera</span><span class="o">.</span><span class="n">animate</span><span class="p">()</span>
-<span class="n">animation</span><span class="o">.</span><span class="n">save</span><span class="p">(</span><span class="s1">&#39;animation.mp4&#39;</span><span class="p">)</span>
-<span class="n">HTML</span><span class="p">(</span><span class="n">animation</span><span class="o">.</span><span class="n">to_html5_video</span><span class="p">())</span>
-</pre></div>
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-
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-
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-<p>Matplotlib supports complex formatting, even allowing LaTeX expressions. These features makes Matplotlib a very appealing package for making professional figures, for example for a scientific paper. Next it is shown an example (<strong>script</strong>) where you can see some of the capabilities of Matplotlib</p>
-
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-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="ch">#! /usr/bin/python</span>
-<span class="c1">#====================================================================</span>
-<span class="c1">#Importing libraries</span>
-<span class="c1">#====================================================================</span>
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
-
-<span class="c1">#Arrays</span>
-<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</span> <span class="p">)</span>
-<span class="n">Y</span> <span class="o">=</span> <span class="n">X</span><span class="o">**</span><span class="mf">0.5</span>
-<span class="n">Y1</span> <span class="o">=</span> <span class="n">Y</span><span class="o">+</span><span class="mf">0.5</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">100</span><span class="p">))</span>
-<span class="n">Y2</span> <span class="o">=</span> <span class="n">Y</span><span class="o">-</span><span class="mf">0.5</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">100</span><span class="p">))</span>
-
-<span class="c1">#Setting the figure environment</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">16</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span> <span class="p">)</span>
-
-<span class="c1">#====================================================================</span>
-<span class="c1">#First plot</span>
-<span class="c1">#====================================================================</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span> <span class="s2">&quot;This is an unformatted plot&quot;</span> <span class="p">)</span>
-
-<span class="c1">#====================================================================</span>
-<span class="c1">#Second plot</span>
-<span class="c1">#====================================================================</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span> <span class="p">)</span>
-<span class="c1">#2D Plots</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">4</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;green&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Function&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y1</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;green&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y2</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;green&quot;</span> <span class="p">)</span>
-<span class="c1">#Error region</span>
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-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span> <span class="s2">&quot;This is a formatted plot&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span> <span class="p">)</span>
-
-<span class="c1">#====================================================================</span>
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-<span class="c1">#====================================================================</span>
-<span class="c1">#You can store this figure in any format, even in vectorial formats </span>
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-title: |-
-  SIR Model
-pagenum: 24
-prev_page:
-  url: /features/differential-equations.html
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-  url: /features/statistics.html
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-search: d infected com t colab r google cdot model github blob master beta frac research ipynb s restrepo sir n person gamma href computationalmethods material mediummodellingpartone target parentimg src assets badge svg alt open hf infectiousdiseasemodelling license scholar compartmental population susceptible recovered total infects per disease used covid understanding align partone bibliography introduction mathematical epidemiology m martcheva springer co scholarlookup title anintroductiontomathematicalepidemiologyauthor mmartcheva pdf drive file ggeqrkxada eqozemflbcsqmkrk view usp sharing separates into several compartments example still healthy already cannot get again require following parameters expected amount days spread proportion recovering basic reproduction just intuitive formula check colombia models coronavirus
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-    <div id="page-info"><div id="page-title">SIR Model</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/medium_modelling_part_one.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<h1 id="SIR-Model">SIR Model<a class="anchor-link" href="#SIR-Model"> </a></h1><p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/medium_modelling_part_one.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
-<p>From: <a href="https://github.com/hf2000510/infectious_disease_modelling/blob/master/part_one.ipynb">https://github.com/hf2000510/infectious_disease_modelling/blob/master/part_one.ipynb</a></p>
-<p>LICENSE: <a href="https://github.com/hf2000510/infectious_disease_modelling/blob/master/LICENSE">https://github.com/hf2000510/infectious_disease_modelling/blob/master/LICENSE</a></p>
-<hr>
-<h2 id="Bibliography">Bibliography<a class="anchor-link" href="#Bibliography"> </a></h2><p>An introduction to mathematical epidemiology, M Martcheva - 2015 - Springer <a href="https://scholar.google.co.in/scholar_lookup?title=An+Introduction+to+Mathematical+Epidemiology&amp;author=M+Martcheva">[Google Scholar]</a> <a href="https://drive.google.com/file/d/1gGEqrkxadA1_5-eqOZEMFl6bCS6qMKrk/view?usp=sharing">[PDF]</a></p>
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-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
-<span class="o">%</span><span class="k">matplotlib</span> inline 
-<span class="c1">#!pip install mpld3</span>
-<span class="kn">import</span> <span class="nn">mpld3</span>
-<span class="n">mpld3</span><span class="o">.</span><span class="n">enable_notebook</span><span class="p">()</span>
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-<h1 id="compartmental-model.">compartmental model.<a class="anchor-link" href="#compartmental-model."> </a></h1><p>A compartmental model separates the population into several compartments, for example:</p>
-<ul>
-<li><strong>S</strong>usceptible (can still be infected, “healthy”)</li>
-<li><strong>I</strong>nfected</li>
-<li><strong>R</strong>ecovered (were already infected, cannot get infected again)</li>
-</ul>
-<p>We require the following parameters</p>
-<ul>
-<li>$N$: total population</li>
-<li>$S(t)$: number of people susceptible on day t</li>
-<li>$I(t)$: number of people infected on day t</li>
-<li>$R(t)$: number of people recovered on day t</li>
-<li>$\beta$: expected amount of people an infected person infects per day</li>
-<li>$D$: number of days an infected person has and can spread the disease</li>
-<li>$\gamma=1/D$: <!--γ --> the proportion of infected recovering per day ( = 1/D)</li>
-</ul>
-<p>the basic reproduction number $R_0$, is the total number of people an infected person infects. We just used an intuitive formula: 
-$$R_0 = \beta \cdot D\,.$$</p>
-<p>Check $R_0$ for Colombia: <a href="https://github.com/restrepo/COVID-19/blob/master/covid.ipynb">https://github.com/restrepo/COVID-19/blob/master/covid.ipynb</a></p>
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-<p>See <a href="https://towardsdatascience.com/infectious-disease-modelling-part-i-understanding-sir-28d60e29fdfc">Understanding the models that are used to model Coronavirus</a>: Explaining the background and deriving the formulas of the SIR model from scratch. Coding and visualizing the model in Python.</p>
-\begin{align}
-\frac{d S}{d t}=&amp;-\beta \cdot I \cdot \frac{S}{N} \\
-\frac{d I}{d t}=&amp;\beta \cdot I \cdot \frac{S}{N}-\gamma \cdot I \\
-\frac{d R}{d t}=&amp;\gamma \cdot I
-\end{align}
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">deriv</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">N</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">β</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">γ</span><span class="o">=</span><span class="mi">1</span><span class="o">/</span><span class="mi">4</span><span class="p">):</span>
-    <span class="n">S</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">y</span>
-    <span class="n">dSdt</span> <span class="o">=</span> <span class="o">-</span><span class="n">β</span> <span class="o">*</span> <span class="n">S</span> <span class="o">*</span> <span class="n">I</span> <span class="o">/</span> <span class="n">N</span>
-    <span class="n">dIdt</span> <span class="o">=</span> <span class="n">β</span> <span class="o">*</span> <span class="n">S</span> <span class="o">*</span> <span class="n">I</span> <span class="o">/</span> <span class="n">N</span> <span class="o">-</span> <span class="n">γ</span> <span class="o">*</span> <span class="n">I</span>
-    <span class="n">dRdt</span> <span class="o">=</span> <span class="n">γ</span> <span class="o">*</span> <span class="n">I</span>
-    <span class="k">return</span> <span class="n">dSdt</span><span class="p">,</span> <span class="n">dIdt</span><span class="p">,</span> <span class="n">dRdt</span>
-</pre></div>
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-<p>We want start with an infected, $I_0=I(0)=1$ and a $R_0=R(0)=1.2$.
-Then, by using $D=4$ 
-$$\beta=R_0/D=0.3\,.$$</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">N</span> <span class="o">=</span> <span class="mi">1000</span>
-<span class="n">β</span> <span class="o">=</span> <span class="mf">0.32</span>  <span class="c1"># infected person infects 0.275 other person per day, such that R_0=1.2</span>
-<span class="n">D</span> <span class="o">=</span> <span class="mf">4.0</span> <span class="c1"># infections lasts four days</span>
-<span class="n">γ</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="o">/</span> <span class="n">D</span>
-
-<span class="n">S0</span><span class="p">,</span> <span class="n">I0</span><span class="p">,</span> <span class="n">R0</span> <span class="o">=</span> <span class="mi">999</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">1.2</span>  <span class="c1"># initial conditions: one infected, rest susceptible</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">199</span><span class="p">,</span> <span class="mi">200</span><span class="p">)</span> <span class="c1"># Grid of time points (in days)</span>
-<span class="n">y0</span> <span class="o">=</span> <span class="p">[</span><span class="n">S0</span><span class="p">,</span> <span class="n">I0</span><span class="p">,</span> <span class="n">R0</span><span class="p">]</span> <span class="c1"># Initial conditions vector</span>
-
-<span class="c1"># Integrate the SIR equations over the time grid, t.</span>
-<span class="n">ret</span> <span class="o">=</span> <span class="n">odeint</span><span class="p">(</span><span class="n">deriv</span><span class="p">,</span> <span class="n">y0</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">β</span><span class="p">,</span> <span class="n">γ</span><span class="p">))</span>
-<span class="n">S</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">ret</span><span class="o">.</span><span class="n">T</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
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-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">plotsir</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span><span class="n">yscale</span><span class="o">=</span><span class="s1">&#39;linear&#39;</span><span class="p">):</span>
-  <span class="n">f</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
-  <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Susceptible&#39;</span><span class="p">)</span>
-  <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="s1">&#39;y&#39;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Infected&#39;</span><span class="p">)</span>
-  <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="s1">&#39;g&#39;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Recovered&#39;</span><span class="p">)</span>
-  <span class="n">plt</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="n">yscale</span><span class="p">)</span>
-  <span class="c1">#plt.grid()</span>
-  <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Time (days)&#39;</span><span class="p">)</span>
-
-  <span class="n">ax</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">set_tick_params</span><span class="p">(</span><span class="n">length</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-  <span class="n">ax</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">set_tick_params</span><span class="p">(</span><span class="n">length</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-  <span class="c1">#ax.grid(b=True, which=&#39;major&#39;, c=&#39;w&#39;, lw=2, ls=&#39;-&#39;)</span>
-  <span class="n">legend</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
-  <span class="n">legend</span><span class="o">.</span><span class="n">get_frame</span><span class="p">()</span><span class="o">.</span><span class="n">set_alpha</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-  <span class="k">for</span> <span class="n">spine</span> <span class="ow">in</span> <span class="p">(</span><span class="s1">&#39;top&#39;</span><span class="p">,</span> <span class="s1">&#39;right&#39;</span><span class="p">,</span> <span class="s1">&#39;bottom&#39;</span><span class="p">,</span> <span class="s1">&#39;left&#39;</span><span class="p">):</span>
-      <span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="n">spine</span><span class="p">]</span><span class="o">.</span><span class="n">set_visible</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
-  <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">plotsir</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span><span class="n">yscale</span><span class="o">=</span><span class="s1">&#39;log&#39;</span><span class="p">)</span>
-</pre></div>
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-      });
-    });
-}else{
-    // require.js not available: dynamically load d3 & mpld3
-    mpld3_load_lib("https://d3js.org/d3.v5.js", function(){
-         mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.1.js", function(){
-
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-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
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-<p>Scaling:</p>
-
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-
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-<span class="n">sc</span>
-</pre></div>
-
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-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
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-<pre>50000.0</pre>
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-
-</div>
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-
-<div class="jb_cell">
-
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-<span class="s1">&#39;Nmax=</span><span class="si">{:g}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="mi">18</span><span class="o">*</span><span class="n">sc</span><span class="p">)</span>
-</pre></div>
-
-    </div>
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-</div>
-
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-       995.58735649, 995.01398138, 994.40037971, 993.74388596,
-       993.04168115, 992.29078738, 991.48806285, 990.63019731,
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-       985.38567616, 984.1176282 , 982.76633648, 981.32708715,
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-       938.34516187, 934.43714656, 930.35035052, 926.0830852 ,
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-       879.70482288, 873.7370976 , 867.63248712, 861.40128456,
-       855.05474889, 848.60502949, 842.06507572, 835.4485329 ,
-       828.76962599, 822.04303349, 815.28375391, 808.50696744,
-       801.72789565, 794.96166199, 788.22315562, 781.52690148,
-       774.88693884, 768.31671044, 761.82896352, 755.43566464,
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-       725.24233147, 719.61618079, 714.14224588, 708.82489264,
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-       633.37900391, 631.28107722, 629.28791023, 627.39519049,
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-       614.21379162, 613.10919302, 612.06428675, 611.07608661,
-       610.14172888, 609.25847031, 608.42368604, 607.63486685,
-       606.88961643, 606.18564825, 605.52078215, 604.89294111,
-       604.30014788, 603.74052121, 603.21227228, 602.71370133,
-       602.24319382, 601.79921721, 601.38031727, 600.98511483,
-       600.61230232, 600.26064064, 599.92895592, 599.61613654,
-       599.32113008, 599.04294053, 598.78062551, 598.53329361,
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-       597.49343728, 597.31983085, 597.15619372, 597.00195898,
-       596.85659136, 596.71958547, 596.59046425, 596.46877743,
-       596.35410002, 596.24603102, 596.14419204, 596.04822608,
-       595.95779637, 595.87258522, 595.79229299, 595.71663706,
-       595.64535091, 595.5781832 , 595.51489694, 595.45526866,
-       595.39908766, 595.3461553 , 595.29628431, 595.24929815,
-       595.2050304 , 595.1633242 , 595.12403167, 595.08701346,
-       595.05213821, 595.01928211, 594.98832849, 594.95916738,
-       594.93169517, 594.90581418, 594.88143239, 594.85846309,
-       594.83682455, 594.81643977, 594.79723619, 594.77914545,
-       594.76210312, 594.7460485 , 594.73092441, 594.71667696,
-       594.70325538, 594.69061183, 594.67870124, 594.66748115,
-       594.65691157, 594.64695479, 594.6375753 , 594.62873966,
-       594.62041635, 594.61257568, 594.60518966, 594.59823195])</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="mf">5E4</span><span class="p">,</span><span class="mi">5</span><span class="o">*</span><span class="mf">5E4</span>
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----
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-title: |-
-  Optimization
-pagenum: 8
-prev_page:
-  url: /features/algorithms-convergence.html
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-  url: /features/one-variable-equations.html
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-    <main class="jupyter-page">
-    <div id="page-info"><div id="page-title">Optimization</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/numba.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
-<h1 id="Numba">Numba<a class="anchor-link" href="#Numba"> </a></h1><p>Compile numerical functions for faster running</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">time</span>
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">from</span> <span class="nn">numba</span> <span class="kn">import</span> <span class="n">jit</span>
-<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-</pre></div>
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-<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
-<span class="ansi-red-fg">ImportError</span>                               Traceback (most recent call last)
-<span class="ansi-green-fg">&lt;ipython-input-2-1194e744ea9c&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span><span class="ansi-blue-fg">()</span>
-<span class="ansi-green-intense-fg ansi-bold">      1</span> <span class="ansi-green-fg">import</span> time
-<span class="ansi-green-intense-fg ansi-bold">      2</span> <span class="ansi-green-fg">import</span> numpy <span class="ansi-green-fg">as</span> np
-<span class="ansi-green-fg">----&gt; 3</span><span class="ansi-red-fg"> </span><span class="ansi-green-fg">from</span> numba <span class="ansi-green-fg">import</span> jit
-<span class="ansi-green-intense-fg ansi-bold">      4</span> <span class="ansi-green-fg">import</span> pandas <span class="ansi-green-fg">as</span> pd
-
-<span class="ansi-red-fg">ImportError</span>: No module named &#39;numba&#39;</pre>
-</div>
-</div>
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-
-</div>
-</div>
-
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-
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Example of the problem: integer summation</p>
-
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-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">st</span><span class="o">=</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
-<span class="n">imax</span><span class="o">=</span><span class="mi">10000</span>
-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">imax</span><span class="p">):</span>
-    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">imax</span><span class="p">):</span>
-        <span class="n">a</span><span class="o">=</span><span class="n">i</span><span class="o">+</span><span class="n">j</span>
-
-<span class="nb">print</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">st</span><span class="p">)</span>
-</pre></div>
-
-    </div>
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-<pre>9.316529273986816
-</pre>
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-</div>
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-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">suma</span><span class="p">(</span><span class="n">imax</span><span class="p">):</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">imax</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">imax</span><span class="p">):</span>
-            <span class="n">a</span><span class="o">=</span><span class="n">i</span><span class="o">+</span><span class="n">j</span>
-    <span class="k">return</span> <span class="n">a</span>
-</pre></div>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">st</span><span class="o">=</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
-<span class="n">a</span><span class="o">=</span><span class="n">suma</span><span class="p">(</span><span class="n">imax</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">st</span><span class="p">)</span>
-</pre></div>
-
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-<pre>4.799173831939697
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Precompile the function with <code>@jit</code> of <code>numba</code></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class="inner_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># jit decorator tells Numba to compile this function.</span>
-<span class="c1"># The argument types will be inferred by Numba when function is called.</span>
-<span class="nd">@jit</span>
-<span class="k">def</span> <span class="nf">suma</span><span class="p">(</span><span class="n">imax</span><span class="p">):</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">imax</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">imax</span><span class="p">):</span>
-            <span class="n">a</span><span class="o">=</span><span class="n">i</span><span class="o">+</span><span class="n">j</span>
-    <span class="k">return</span> <span class="n">a</span>
-</pre></div>
-
-    </div>
-</div>
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-
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">st</span><span class="o">=</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
-<span class="n">a</span><span class="o">=</span><span class="n">suma</span><span class="p">(</span><span class="n">imax</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">st</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<pre>0.09396576881408691
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">()</span><span class="c1"># []#0</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
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-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>A more complicated example with arrays</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-<span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">a</span><span class="p">,</span><span class="n">a</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([0, 0, 0, 0])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># jit decorator tells Numba to compile this function.</span>
-<span class="c1"># The argument types will be inferred by Numba when function is called.</span>
-<span class="nd">@jit</span><span class="c1">#(nopython=True)</span>
-<span class="k">def</span> <span class="nf">suma</span><span class="p">(</span><span class="n">imax</span><span class="p">,</span><span class="n">a</span><span class="p">):</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">imax</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">imax</span><span class="p">):</span>
-            <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">=</span><span class="n">i</span><span class="o">+</span><span class="n">j</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">imax</span><span class="o">=</span><span class="mi">10000</span>
-<span class="n">s</span><span class="o">=</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
-<span class="n">a</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">imax</span><span class="p">)</span>
-<span class="n">a</span><span class="o">=</span><span class="n">suma</span><span class="p">(</span><span class="n">imax</span><span class="p">,</span><span class="n">a</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">s</span><span class="p">)</span>        
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>0.00023245811462402344
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span><span class="o">=</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
-<span class="n">a</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">imax</span><span class="p">)</span>
-<span class="n">a</span><span class="o">=</span><span class="n">suma</span><span class="p">(</span><span class="n">imax</span><span class="p">,</span><span class="n">a</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">s</span><span class="p">)</span>        
-</pre></div>
-
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-<pre>0.0002434253692626953
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span><span class="o">=</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
-<span class="n">a</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">imax</span><span class="p">)</span>
-<span class="n">a</span><span class="o">=</span><span class="n">suma</span><span class="p">(</span><span class="n">imax</span><span class="p">,</span><span class="n">a</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">s</span><span class="p">)</span>        
-</pre></div>
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-<pre>0.0002474784851074219
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">&gt;&gt;&gt;</span> <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">)</span>
-<span class="o">&gt;&gt;&gt;</span> <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">)</span>
-<span class="o">&gt;&gt;&gt;</span> <span class="n">xx</span><span class="p">,</span> <span class="n">yy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">sparse</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-<span class="o">&gt;&gt;&gt;</span> <span class="n">z</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">xx</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">yy</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">xx</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">yy</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="c1">#&gt;&gt;&gt; h = plt.contourf(x,y,z)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">imax</span>
-</pre></div>
-
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-<pre>10000</pre>
-</div>
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-</div>
-</div>
-</div>
-</div>
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-</div>
-</div>
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-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">&gt;&gt;&gt;</span> <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">imax</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="o">&gt;&gt;&gt;</span> <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">imax</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="o">&gt;&gt;&gt;</span> <span class="n">xx</span><span class="p">,</span> <span class="n">yy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">sparse</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
-<span class="o">&gt;&gt;&gt;</span> <span class="n">z</span> <span class="o">=</span> <span class="n">xx</span><span class="o">+</span><span class="n">yy</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">xx</span>
-</pre></div>
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-<pre>array([[   0,    1,    2, ..., 9997, 9998, 9999],
-       [   0,    1,    2, ..., 9997, 9998, 9999],
-       [   0,    1,    2, ..., 9997, 9998, 9999],
-       ..., 
-       [   0,    1,    2, ..., 9997, 9998, 9999],
-       [   0,    1,    2, ..., 9997, 9998, 9999],
-       [   0,    1,    2, ..., 9997, 9998, 9999]])</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">numba</span> <span class="kn">import</span> <span class="n">jit</span>
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-
-<span class="nd">@jit</span><span class="p">(</span><span class="n">nopython</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="c1">#return np.array([ [ [x,y]  for x in range(n) ] for y in range(n) ])</span>
-    <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>  <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="p">]</span> <span class="p">)</span>
-</pre></div>
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-    </div>
-</div>
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-<div class="cell border-box-sizing code_cell rendered">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span><span class="o">=</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
-<span class="n">a</span><span class="o">=</span><span class="n">f</span><span class="p">(</span><span class="mi">10000</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">s</span><span class="p">)</span>        
-</pre></div>
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-<pre>
-<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
-<span class="ansi-red-fg">NotImplementedError</span>                       Traceback (most recent call last)
-<span class="ansi-green-fg">&lt;ipython-input-21-ad659232eeb3&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span><span class="ansi-blue-fg">()</span>
-<span class="ansi-green-intense-fg ansi-bold">      1</span> s<span class="ansi-blue-fg">=</span>time<span class="ansi-blue-fg">.</span>time<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-fg">----&gt; 2</span><span class="ansi-red-fg"> </span>a<span class="ansi-blue-fg">=</span>f<span class="ansi-blue-fg">(</span><span class="ansi-cyan-fg">10000</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">      3</span> print<span class="ansi-blue-fg">(</span>time<span class="ansi-blue-fg">.</span>time<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">-</span>s<span class="ansi-blue-fg">)</span>
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/dispatcher.py</span> in <span class="ansi-cyan-fg">_compile_for_args</span><span class="ansi-blue-fg">(self, *args, **kws)</span>
-<span class="ansi-green-intense-fg ansi-bold">    305</span>                 argtypes<span class="ansi-blue-fg">.</span>append<span class="ansi-blue-fg">(</span>self<span class="ansi-blue-fg">.</span>typeof_pyval<span class="ansi-blue-fg">(</span>a<span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    306</span>         <span class="ansi-green-fg">try</span><span class="ansi-blue-fg">:</span>
-<span class="ansi-green-fg">--&gt; 307</span><span class="ansi-red-fg">             </span><span class="ansi-green-fg">return</span> self<span class="ansi-blue-fg">.</span>compile<span class="ansi-blue-fg">(</span>tuple<span class="ansi-blue-fg">(</span>argtypes<span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    308</span>         <span class="ansi-green-fg">except</span> errors<span class="ansi-blue-fg">.</span>TypingError <span class="ansi-green-fg">as</span> e<span class="ansi-blue-fg">:</span>
-<span class="ansi-green-intense-fg ansi-bold">    309</span>             <span class="ansi-red-fg"># Intercept typing error that may be due to an argument</span>
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/dispatcher.py</span> in <span class="ansi-cyan-fg">compile</span><span class="ansi-blue-fg">(self, sig)</span>
-<span class="ansi-green-intense-fg ansi-bold">    577</span> 
-<span class="ansi-green-intense-fg ansi-bold">    578</span>                 self<span class="ansi-blue-fg">.</span>_cache_misses<span class="ansi-blue-fg">[</span>sig<span class="ansi-blue-fg">]</span> <span class="ansi-blue-fg">+=</span> <span class="ansi-cyan-fg">1</span>
-<span class="ansi-green-fg">--&gt; 579</span><span class="ansi-red-fg">                 </span>cres <span class="ansi-blue-fg">=</span> self<span class="ansi-blue-fg">.</span>_compiler<span class="ansi-blue-fg">.</span>compile<span class="ansi-blue-fg">(</span>args<span class="ansi-blue-fg">,</span> return_type<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    580</span>                 self<span class="ansi-blue-fg">.</span>add_overload<span class="ansi-blue-fg">(</span>cres<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    581</span>                 self<span class="ansi-blue-fg">.</span>_cache<span class="ansi-blue-fg">.</span>save_overload<span class="ansi-blue-fg">(</span>sig<span class="ansi-blue-fg">,</span> cres<span class="ansi-blue-fg">)</span>
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/dispatcher.py</span> in <span class="ansi-cyan-fg">compile</span><span class="ansi-blue-fg">(self, args, return_type)</span>
-<span class="ansi-green-intense-fg ansi-bold">     78</span>                                       impl<span class="ansi-blue-fg">,</span>
-<span class="ansi-green-intense-fg ansi-bold">     79</span>                                       args<span class="ansi-blue-fg">=</span>args<span class="ansi-blue-fg">,</span> return_type<span class="ansi-blue-fg">=</span>return_type<span class="ansi-blue-fg">,</span>
-<span class="ansi-green-fg">---&gt; 80</span><span class="ansi-red-fg">                                       flags=flags, locals=self.locals)
-</span><span class="ansi-green-intense-fg ansi-bold">     81</span>         <span class="ansi-red-fg"># Check typing error if object mode is used</span>
-<span class="ansi-green-intense-fg ansi-bold">     82</span>         <span class="ansi-green-fg">if</span> cres<span class="ansi-blue-fg">.</span>typing_error <span class="ansi-green-fg">is</span> <span class="ansi-green-fg">not</span> <span class="ansi-green-fg">None</span> <span class="ansi-green-fg">and</span> <span class="ansi-green-fg">not</span> flags<span class="ansi-blue-fg">.</span>enable_pyobject<span class="ansi-blue-fg">:</span>
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/compiler.py</span> in <span class="ansi-cyan-fg">compile_extra</span><span class="ansi-blue-fg">(typingctx, targetctx, func, args, return_type, flags, locals, library)</span>
-<span class="ansi-green-intense-fg ansi-bold">    777</span>     pipeline = Pipeline(typingctx, targetctx, library,
-<span class="ansi-green-intense-fg ansi-bold">    778</span>                         args, return_type, flags, locals)
-<span class="ansi-green-fg">--&gt; 779</span><span class="ansi-red-fg">     </span><span class="ansi-green-fg">return</span> pipeline<span class="ansi-blue-fg">.</span>compile_extra<span class="ansi-blue-fg">(</span>func<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    780</span> 
-<span class="ansi-green-intense-fg ansi-bold">    781</span> 
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/compiler.py</span> in <span class="ansi-cyan-fg">compile_extra</span><span class="ansi-blue-fg">(self, func)</span>
-<span class="ansi-green-intense-fg ansi-bold">    360</span>         self<span class="ansi-blue-fg">.</span>lifted <span class="ansi-blue-fg">=</span> <span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    361</span>         self<span class="ansi-blue-fg">.</span>lifted_from <span class="ansi-blue-fg">=</span> <span class="ansi-green-fg">None</span>
-<span class="ansi-green-fg">--&gt; 362</span><span class="ansi-red-fg">         </span><span class="ansi-green-fg">return</span> self<span class="ansi-blue-fg">.</span>_compile_bytecode<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    363</span> 
-<span class="ansi-green-intense-fg ansi-bold">    364</span>     <span class="ansi-green-fg">def</span> compile_ir<span class="ansi-blue-fg">(</span>self<span class="ansi-blue-fg">,</span> func_ir<span class="ansi-blue-fg">,</span> lifted<span class="ansi-blue-fg">=</span><span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">,</span> lifted_from<span class="ansi-blue-fg">=</span><span class="ansi-green-fg">None</span><span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">:</span>
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/compiler.py</span> in <span class="ansi-cyan-fg">_compile_bytecode</span><span class="ansi-blue-fg">(self)</span>
-<span class="ansi-green-intense-fg ansi-bold">    736</span>         &#34;&#34;&#34;
-<span class="ansi-green-intense-fg ansi-bold">    737</span>         <span class="ansi-green-fg">assert</span> self<span class="ansi-blue-fg">.</span>func_ir <span class="ansi-green-fg">is</span> <span class="ansi-green-fg">None</span>
-<span class="ansi-green-fg">--&gt; 738</span><span class="ansi-red-fg">         </span><span class="ansi-green-fg">return</span> self<span class="ansi-blue-fg">.</span>_compile_core<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    739</span> 
-<span class="ansi-green-intense-fg ansi-bold">    740</span>     <span class="ansi-green-fg">def</span> _compile_ir<span class="ansi-blue-fg">(</span>self<span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">:</span>
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/compiler.py</span> in <span class="ansi-cyan-fg">_compile_core</span><span class="ansi-blue-fg">(self)</span>
-<span class="ansi-green-intense-fg ansi-bold">    723</span> 
-<span class="ansi-green-intense-fg ansi-bold">    724</span>         pm<span class="ansi-blue-fg">.</span>finalize<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-fg">--&gt; 725</span><span class="ansi-red-fg">         </span>res <span class="ansi-blue-fg">=</span> pm<span class="ansi-blue-fg">.</span>run<span class="ansi-blue-fg">(</span>self<span class="ansi-blue-fg">.</span>status<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    726</span>         <span class="ansi-green-fg">if</span> res <span class="ansi-green-fg">is</span> <span class="ansi-green-fg">not</span> <span class="ansi-green-fg">None</span><span class="ansi-blue-fg">:</span>
-<span class="ansi-green-intense-fg ansi-bold">    727</span>             <span class="ansi-red-fg"># Early pipeline completion</span>
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/compiler.py</span> in <span class="ansi-cyan-fg">run</span><span class="ansi-blue-fg">(self, status)</span>
-<span class="ansi-green-intense-fg ansi-bold">    246</span>                     <span class="ansi-red-fg"># No more fallback pipelines?</span>
-<span class="ansi-green-intense-fg ansi-bold">    247</span>                     <span class="ansi-green-fg">if</span> is_final_pipeline<span class="ansi-blue-fg">:</span>
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-<span class="ansi-green-intense-fg ansi-bold">    249</span>                     <span class="ansi-red-fg"># Go to next fallback pipeline</span>
-<span class="ansi-green-intense-fg ansi-bold">    250</span>                     <span class="ansi-green-fg">else</span><span class="ansi-blue-fg">:</span>
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-<span class="ansi-green-intense-fg ansi-bold">    241</span>                 <span class="ansi-green-fg">except</span> _EarlyPipelineCompletion <span class="ansi-green-fg">as</span> e<span class="ansi-blue-fg">:</span>
-<span class="ansi-green-intense-fg ansi-bold">    242</span>                     <span class="ansi-green-fg">return</span> e<span class="ansi-blue-fg">.</span>result
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-<span class="ansi-green-intense-fg ansi-bold">    611</span>             self<span class="ansi-blue-fg">.</span>library<span class="ansi-blue-fg">.</span>enable_object_caching<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">)</span>
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-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/compiler.py</span> in <span class="ansi-cyan-fg">backend_nopython_mode</span><span class="ansi-blue-fg">(self)</span>
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-<span class="ansi-green-intense-fg ansi-bold">    599</span>                 self<span class="ansi-blue-fg">.</span>calltypes<span class="ansi-blue-fg">,</span>
-<span class="ansi-green-fg">--&gt; 600</span><span class="ansi-red-fg">                 self.flags)
-</span><span class="ansi-green-intense-fg ansi-bold">    601</span> 
-<span class="ansi-green-intense-fg ansi-bold">    602</span>     <span class="ansi-green-fg">def</span> _backend<span class="ansi-blue-fg">(</span>self<span class="ansi-blue-fg">,</span> lowerfn<span class="ansi-blue-fg">,</span> objectmode<span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">:</span>
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/compiler.py</span> in <span class="ansi-cyan-fg">native_lowering_stage</span><span class="ansi-blue-fg">(targetctx, library, interp, typemap, restype, calltypes, flags)</span>
-<span class="ansi-green-intense-fg ansi-bold">    898</span>     lower<span class="ansi-blue-fg">.</span>lower<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">)</span>
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-<span class="ansi-green-intense-fg ansi-bold">    901</span>     env <span class="ansi-blue-fg">=</span> lower<span class="ansi-blue-fg">.</span>env
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-<span class="ansi-green-intense-fg ansi-bold">    227</span>         self.context.create_cpython_wrapper(self.library, self.fndesc,
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-</span><span class="ansi-green-intense-fg ansi-bold">    230</span> 
-<span class="ansi-green-intense-fg ansi-bold">    231</span>     <span class="ansi-green-fg">def</span> setup_function<span class="ansi-blue-fg">(</span>self<span class="ansi-blue-fg">,</span> fndesc<span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">:</span>
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-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/targets/cpu.py</span> in <span class="ansi-cyan-fg">create_cpython_wrapper</span><span class="ansi-blue-fg">(self, library, fndesc, env, call_helper, release_gil)</span>
-<span class="ansi-green-intense-fg ansi-bold">    147</span>                                 fndesc<span class="ansi-blue-fg">,</span> env<span class="ansi-blue-fg">,</span> call_helper<span class="ansi-blue-fg">=</span>call_helper<span class="ansi-blue-fg">,</span>
-<span class="ansi-green-intense-fg ansi-bold">    148</span>                                 release_gil=release_gil)
-<span class="ansi-green-fg">--&gt; 149</span><span class="ansi-red-fg">         </span>builder<span class="ansi-blue-fg">.</span>build<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    150</span>         library<span class="ansi-blue-fg">.</span>add_ir_module<span class="ansi-blue-fg">(</span>wrapper_module<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    151</span> 
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/callwrapper.py</span> in <span class="ansi-cyan-fg">build</span><span class="ansi-blue-fg">(self)</span>
-<span class="ansi-green-intense-fg ansi-bold">    120</span> 
-<span class="ansi-green-intense-fg ansi-bold">    121</span>         api <span class="ansi-blue-fg">=</span> self<span class="ansi-blue-fg">.</span>context<span class="ansi-blue-fg">.</span>get_python_api<span class="ansi-blue-fg">(</span>builder<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-fg">--&gt; 122</span><span class="ansi-red-fg">         </span>self<span class="ansi-blue-fg">.</span>build_wrapper<span class="ansi-blue-fg">(</span>api<span class="ansi-blue-fg">,</span> builder<span class="ansi-blue-fg">,</span> closure<span class="ansi-blue-fg">,</span> args<span class="ansi-blue-fg">,</span> kws<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    123</span> 
-<span class="ansi-green-intense-fg ansi-bold">    124</span>         <span class="ansi-green-fg">return</span> wrapper<span class="ansi-blue-fg">,</span> api
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/callwrapper.py</span> in <span class="ansi-cyan-fg">build_wrapper</span><span class="ansi-blue-fg">(self, api, builder, closure, args, kws)</span>
-<span class="ansi-green-intense-fg ansi-bold">    174</span> 
-<span class="ansi-green-intense-fg ansi-bold">    175</span>             retty <span class="ansi-blue-fg">=</span> self<span class="ansi-blue-fg">.</span>_simplified_return_type<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-fg">--&gt; 176</span><span class="ansi-red-fg">             </span>obj <span class="ansi-blue-fg">=</span> api<span class="ansi-blue-fg">.</span>from_native_return<span class="ansi-blue-fg">(</span>retty<span class="ansi-blue-fg">,</span> retval<span class="ansi-blue-fg">,</span> env_manager<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    177</span>             builder<span class="ansi-blue-fg">.</span>ret<span class="ansi-blue-fg">(</span>obj<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    178</span> 
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/pythonapi.py</span> in <span class="ansi-cyan-fg">from_native_return</span><span class="ansi-blue-fg">(self, typ, val, env_manager)</span>
-<span class="ansi-green-intense-fg ansi-bold">   1323</span>                                                     <span class="ansi-blue-fg">&#34;prevented the return of &#34;</span><span class="ansi-red-fg"> </span><span class="ansi-red-fg">\</span>
-<span class="ansi-green-intense-fg ansi-bold">   1324</span>                                                     <span class="ansi-blue-fg">&#34;optional value&#34;</span>
-<span class="ansi-green-fg">-&gt; 1325</span><span class="ansi-red-fg">         </span>out <span class="ansi-blue-fg">=</span> self<span class="ansi-blue-fg">.</span>from_native_value<span class="ansi-blue-fg">(</span>typ<span class="ansi-blue-fg">,</span> val<span class="ansi-blue-fg">,</span> env_manager<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">   1326</span>         <span class="ansi-green-fg">return</span> out
-<span class="ansi-green-intense-fg ansi-bold">   1327</span> 
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/pythonapi.py</span> in <span class="ansi-cyan-fg">from_native_value</span><span class="ansi-blue-fg">(self, typ, val, env_manager)</span>
-<span class="ansi-green-intense-fg ansi-bold">   1337</span> 
-<span class="ansi-green-intense-fg ansi-bold">   1338</span>         c <span class="ansi-blue-fg">=</span> _BoxContext<span class="ansi-blue-fg">(</span>self<span class="ansi-blue-fg">.</span>context<span class="ansi-blue-fg">,</span> self<span class="ansi-blue-fg">.</span>builder<span class="ansi-blue-fg">,</span> self<span class="ansi-blue-fg">,</span> env_manager<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-fg">-&gt; 1339</span><span class="ansi-red-fg">         </span><span class="ansi-green-fg">return</span> impl<span class="ansi-blue-fg">(</span>typ<span class="ansi-blue-fg">,</span> val<span class="ansi-blue-fg">,</span> c<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">   1340</span> 
-<span class="ansi-green-intense-fg ansi-bold">   1341</span>     <span class="ansi-green-fg">def</span> reflect_native_value<span class="ansi-blue-fg">(</span>self<span class="ansi-blue-fg">,</span> typ<span class="ansi-blue-fg">,</span> val<span class="ansi-blue-fg">,</span> env_manager<span class="ansi-blue-fg">=</span><span class="ansi-green-fg">None</span><span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">:</span>
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/targets/boxing.py</span> in <span class="ansi-cyan-fg">box_array</span><span class="ansi-blue-fg">(typ, val, c)</span>
-<span class="ansi-green-intense-fg ansi-bold">    307</span>     nativeary <span class="ansi-blue-fg">=</span> nativearycls<span class="ansi-blue-fg">(</span>c<span class="ansi-blue-fg">.</span>context<span class="ansi-blue-fg">,</span> c<span class="ansi-blue-fg">.</span>builder<span class="ansi-blue-fg">,</span> value<span class="ansi-blue-fg">=</span>val<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    308</span>     <span class="ansi-green-fg">if</span> c<span class="ansi-blue-fg">.</span>context<span class="ansi-blue-fg">.</span>enable_nrt<span class="ansi-blue-fg">:</span>
-<span class="ansi-green-fg">--&gt; 309</span><span class="ansi-red-fg">         </span>np_dtype <span class="ansi-blue-fg">=</span> numpy_support<span class="ansi-blue-fg">.</span>as_dtype<span class="ansi-blue-fg">(</span>typ<span class="ansi-blue-fg">.</span>dtype<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    310</span>         dtypeptr <span class="ansi-blue-fg">=</span> c<span class="ansi-blue-fg">.</span>env_manager<span class="ansi-blue-fg">.</span>read_const<span class="ansi-blue-fg">(</span>c<span class="ansi-blue-fg">.</span>env_manager<span class="ansi-blue-fg">.</span>add_const<span class="ansi-blue-fg">(</span>np_dtype<span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    311</span>         <span class="ansi-red-fg"># Steals NRT ref</span>
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/numba/numpy_support.py</span> in <span class="ansi-cyan-fg">as_dtype</span><span class="ansi-blue-fg">(nbtype)</span>
-<span class="ansi-green-intense-fg ansi-bold">    134</span>         <span class="ansi-green-fg">return</span> as_dtype<span class="ansi-blue-fg">(</span>nbtype<span class="ansi-blue-fg">.</span>dtype<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    135</span>     raise NotImplementedError(&#34;%r cannot be represented as a Numpy dtype&#34;
-<span class="ansi-green-fg">--&gt; 136</span><span class="ansi-red-fg">                               % (nbtype,))
-</span><span class="ansi-green-intense-fg ansi-bold">    137</span> 
-<span class="ansi-green-intense-fg ansi-bold">    138</span> 
-
-<span class="ansi-red-fg">NotImplementedError</span>: Failed at nopython (nopython mode backend)
-(int64 x 2) cannot be represented as a Numpy dtype</pre>
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-
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-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">(</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">a</span><span class="p">))</span>
-</pre></div>
-
-    </div>
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-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">n_i</span><span class="o">=</span><span class="mi">101</span>
-<span class="n">n_i</span><span class="o">%</span><span class="k">10</span>==0
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-    </div>
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-<h2 id="Numerical-approximation-of-integrals">Numerical approximation of integrals<a class="anchor-link" href="#Numerical-approximation-of-integrals"> </a></h2>
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-<p>The methods for numerical approximation that we will study here rely in the idea of approximate 
-$$\int_a^b f(x)\operatorname{d}x\,,$$
-by using some polinomial $p$, such that
-$$\int_a^b f(x)\operatorname{d}x\approx \int_a^b P_n(x)\operatorname{d}x\,,$$
-The integral of a polynomial can be calculated analytically because we know its antiderivative.</p>
-<p>Since, as we seen in <a href="interpolation.ipynb#Lagrange-Polynomial">Lagrange-Polynomial</a>, the next polynomial of $n$th-degree 
-$$P_n(x) = \sum_{i=0}^n f(x_i)L_{n,i}(x) = \sum_{i=0}^n y_iL_{n,i}(x)\,,$$
-where 
-$$L_{n,i}(x) = \prod_{\begin{smallmatrix}m=0\\ m\neq i\end{smallmatrix}}^n \frac{x-x_m}{x_i-x_m} =\frac{(x-x_0)}{(x_i-x_0)}\frac{(x-x_1)}{(x_i-x_1)}\cdots \frac{(x-x_{i-1})}{(x_i-x_{i-1})}\underbrace{\frac{}{}}_{m\ne i}
-\frac{(x-x_{i+1})}{(x_i-x_{i+1})} \cdots \frac{(x-x_{n-1})}{(x_i-x_{n-1})}\frac{(x-x_n)}{(x_i-x_n)}  $$
-Given a well-behaved function $f(x)$, we have then</p>
-$$f(x) \approx \sum_{k=0}^n f(x_k)L_{n,k}(x)$$\begin{align}
-%$$ trick notebook cell
-\int_a^b f(x)dx =&amp; \int_a^b\sum_{k=0}^n f(x_k)L_{n,k}(x)dx \\
-=&amp;\sum_{k=0}^n f(x_k) \int_a^b L_{n,k}(x)dx \\
-=&amp;\sum_{k=0}^n f(x_k) \omega_k \,
-%$$
-\end{align}<p><!-- http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture18.pdf -->
-where $\omega_k$ is a <em>weight</em> applied to each function value:
-$$\omega_k = \int_a^b L_{n,k}(x) dx= \int_a^b\prod_{j=0,\ j\neq k}^{n}\frac{(x-x_j)}{(x_k-x_j)}dx$$
-Note that, since Lagrange polynomials do not depend on the function, we can calculate the weights $\omega_i$ using only the nodes $[x_0,x_1,\ldots x_n]$</p>
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-<h3 id="Error-calculation">Error calculation<a class="anchor-link" href="#Error-calculation"> </a></h3><p>Including the error, the previous function $f(x)$ can be written as</p>
-$$f(x) = \sum_{k=0}^n f(x_k)L_{n,k}(x) + \frac{(x-x_0)(x-x_1)\cdots(x-x_n)}{(n+1)!}f^{(n+1)}(\xi(x))$$<p>with $L_{n,k}(x)$ the lagrange basis functions. Integrating $f(x)$ over $[a,b]$, we obtain the next expression:</p>
-$$\int_a^b f(x)dx = \int_a^b\sum_{k=0}^n f(x_k)L_{n,k}(x)dx + \int_a^b\frac{(x-x_0)(x-x_1)\cdots(x-x_n)}{(n+1)!}f^{(n+1)}(\xi(x))dx$$<p>It is worth mentioning this expression is a number, unlike differentiation where we obtained a function.</p>
-<p>We can readily convert this expression in a weighted summation as</p>
-$$\int_a^b f(x)dx = \sum_{k=0}^n \omega_i\,f(x_k) + \frac{1}{(n+1)!}\int_a^bf^{(n+1)}(\xi(x)) \prod_{k=0}^{n}(x-x_k)dx$$<p>Finally, the quadrature formula or <strong>Newton-Cotes formula</strong> is given by the next expression:</p>
-$$\int_a^b f(x) dx = \sum \omega_i\, f(x_i) + E[f]$$<p>where the estimated error is</p>
-$$E[f] = \frac{1}{(n+1)!}\int_a^bf^{(n+1)}(\xi(x)) \prod_{k=0}^{n}(x-x_k)dx $$<p>Asumming besides intervals equally spaced such that $x_i = x_0 + i\times h$, the error formula becomes:</p>
-$$E[f] = \frac{h^{n+3}f^{n+2}(\xi)}{(n+1)!}\int_0^nt^2(t-1)\cdots(t-n) $$<p>if $n$ is even and</p>
-$$E[f] = \frac{h^{n+2}f^{n+1}(\xi)}{(n+1)!}\int_0^nt(t-1)\cdots(t-n) $$<p>if $n$ is odd.</p>
-<p>Below is a simple implementation of the quadrature formula or <strong>Newton-Cotes formula</strong> by using the implementation in <code>scipy</code> of the Interpolation polynomial in the Lagrangian form of order $n$ discussed in <a href="./interpolation.ipynb#Implementation-in-Scipy">Interpolation</a> from  <code>scipy interpolation</code></p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%</span><span class="k">pylab</span> inline
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">interpolate</span>
-<span class="kn">import</span> <span class="nn">scipy.integrate</span> <span class="k">as</span> <span class="nn">integrate</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">F</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span><span class="n">X</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Antiderivate approximantion with Lagrange Polynomial </span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="k">return</span> <span class="n">interpolate</span><span class="o">.</span><span class="n">lagrange</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="p">)</span><span class="o">.</span><span class="n">integ</span><span class="p">()</span>
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-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Antiderivate approximantion with Lagrange Polynomial </span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="n">antif</span><span class="o">=</span><span class="n">F</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span><span class="n">X</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">antif</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="o">-</span><span class="n">antif</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
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-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Implementation of Newton-Cotes formula for A SINGLE INTERVAL</span>
-<span class="sd">    f: Function to integrate in A SINGLE INTERVAL</span>
-<span class="sd">    X: nodes of the Lagrangian interpolation polynomial</span>
-<span class="sd">    xmin,xmax,ymin, ymax: size of the figure</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="n">Y</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span> <span class="n">X</span> <span class="p">)</span>
-    
-    <span class="c1">#X array</span>
-    <span class="n">Xarray</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">,</span> <span class="mi">1000</span> <span class="p">)</span>
-    <span class="c1">#X area</span>
-    <span class="n">Xarea</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">X</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1000</span> <span class="p">)</span>
-    <span class="c1">#F array</span>
-    <span class="n">Yarray</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span> <span class="n">Xarray</span> <span class="p">)</span>
-    
-    <span class="c1">#Lagrange polynomial</span>
-    <span class="n">Ln</span> <span class="o">=</span> <span class="n">interpolate</span><span class="o">.</span><span class="n">lagrange</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="p">)</span>
-    <span class="c1">#Interpolated array</span>
-    <span class="n">Parray</span> <span class="o">=</span> <span class="n">Ln</span><span class="p">(</span> <span class="n">Xarray</span> <span class="p">)</span>
-    <span class="c1">#Interpolated array for area</span>
-    <span class="n">Parea</span> <span class="o">=</span> <span class="n">Ln</span><span class="p">(</span> <span class="n">Xarea</span> <span class="p">)</span>
-    
-    <span class="c1">#Plotting</span>
-    <span class="k">if</span> <span class="n">fig</span><span class="o">==</span><span class="kc">None</span><span class="p">:</span>
-        <span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">)</span> <span class="p">)</span>
-    <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
-    <span class="c1">#Function</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">Xarray</span><span class="p">,</span> <span class="n">Yarray</span><span class="p">,</span> <span class="n">linewidth</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="n">color</span> <span class="o">=</span> <span class="s2">&quot;blue&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$f(x)$&quot;</span> <span class="p">)</span>
-    <span class="c1">#Points</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="s2">&quot;o&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;points&quot;</span><span class="p">,</span> <span class="n">zorder</span> <span class="o">=</span> <span class="mi">10</span> <span class="p">)</span>
-    <span class="c1">#Interpolator</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">Xarray</span><span class="p">,</span> <span class="n">Parray</span><span class="p">,</span> <span class="n">linewidth</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">color</span> <span class="o">=</span> <span class="s2">&quot;black&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_{</span><span class="si">%d</span><span class="s2">}(x)$&quot;</span><span class="o">%</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="p">)</span>
-    <span class="c1">#Area</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span> <span class="n">Xarea</span><span class="p">,</span> <span class="n">Parea</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;green&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span> <span class="p">)</span>
-    
-    <span class="c1">#Format</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span> <span class="s2">&quot;</span><span class="si">%d</span><span class="s2">-point Quadrature&quot;</span><span class="o">%</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">)),</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span> <span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span> <span class="p">(</span><span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">)</span> <span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span> <span class="s2">&quot;$x$&quot;</span> <span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span> <span class="s2">&quot;$y$&quot;</span> <span class="p">)</span>
-    <span class="k">if</span> <span class="n">leg</span><span class="p">:</span>
-        <span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span> <span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper left&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span> <span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-    
-    <span class="k">return</span> <span class="n">Quadrature</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span><span class="n">X</span><span class="p">)</span>
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-<h2 id="Trapezoidal-rule">Trapezoidal rule<a class="anchor-link" href="#Trapezoidal-rule"> </a></h2><p>Using the previous formula, it is easily to derivate a set of low-order approximations for integration. Asumming a function $f(x)$ and an interval $[x_0,x_1]$, the associated quadrature formula is that obtained from a first-order Lagrange polynomial $P_1(x)$ given by:</p>
-$$P_1(x) = \frac{(x-x_1)}{x_0-x_1}f(x_0) + \frac{(x-x_0)}{(x_1-x_0)}f(x_1)$$<p>Using this, it is readible to obtain the integrate:</p>
-$$\int_{x_0}^{x_1}f(x)dx = \frac{h}{2}[ f(x_0) + f(x_1) ]-\frac{h^3}{12}f^{''}(\xi)$$<p>with $\xi \in [x_0, x_1]$ and $h = x_1-x_0$.</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Function</span>
-<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="mi">1</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">x</span>
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-<span class="c1">#Quadrature with 2 points (Trapezoidal rule)</span>
-<span class="n">a</span><span class="o">=-</span><span class="mf">0.5</span>
-<span class="n">b</span><span class="o">=</span><span class="mf">1.5</span>
-<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">])</span>
-<span class="n">Q</span><span class="o">=</span><span class="n">PlotQuadrature</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">xmin</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">xmax</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">ymin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">ymax</span><span class="o">=</span><span class="mi">4</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Full=</span><span class="si">{}</span><span class="s1">, trapezoid=</span><span class="si">{}</span><span class="s1">, rectangle=</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> <span class="n">integrate</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span> <span class="p">,</span>  <span class="n">Quadrature</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">),</span><span class="n">X</span><span class="p">),</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">+</span><span class="n">f</span><span class="p">(</span><span class="n">a</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span> <span class="p">)</span> <span class="p">)</span>
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-<pre>Full=4.245647748216942, trapezoid=3.775154904633847, rectangle=3.7751549046338475
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">a</span><span class="p">),(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
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-<pre>(2.5403023058681398, 5.010007503399555)</pre>
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-<h3 id="Scipy-implementation"><a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.trapz.html#r78">Scipy implementation</a><a class="anchor-link" href="#Scipy-implementation"> </a></h3>
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-<div class=" highlight hl-ipython3"><pre><span></span>integrate.trapz<span class="o">?</span>
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-<pre><span class="ansi-red-fg">Signature:</span> integrate<span class="ansi-blue-fg">.</span>trapz<span class="ansi-blue-fg">(</span>y<span class="ansi-blue-fg">,</span> x<span class="ansi-blue-fg">=</span><span class="ansi-green-fg">None</span><span class="ansi-blue-fg">,</span> dx<span class="ansi-blue-fg">=</span><span class="ansi-cyan-fg">1.0</span><span class="ansi-blue-fg">,</span> axis<span class="ansi-blue-fg">=</span><span class="ansi-blue-fg">-</span><span class="ansi-cyan-fg">1</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-red-fg">Docstring:</span>
-Integrate along the given axis using the composite trapezoidal rule.
-
-Integrate `y` (`x`) along given axis.
-
-Parameters
-----------
-y : array_like
-    Input array to integrate.
-x : array_like, optional
-    The sample points corresponding to the `y` values. If `x` is None,
-    the sample points are assumed to be evenly spaced `dx` apart. The
-    default is None.
-dx : scalar, optional
-    The spacing between sample points when `x` is None. The default is 1.
-axis : int, optional
-    The axis along which to integrate.
-
-Returns
--------
-trapz : float
-    Definite integral as approximated by trapezoidal rule.
-
-See Also
---------
-sum, cumsum
-
-Notes
------
-Image [2]_ illustrates trapezoidal rule -- y-axis locations of points
-will be taken from `y` array, by default x-axis distances between
-points will be 1.0, alternatively they can be provided with `x` array
-or with `dx` scalar.  Return value will be equal to combined area under
-the red lines.
-
-
-References
-----------
-.. [1] Wikipedia page: http://en.wikipedia.org/wiki/Trapezoidal_rule
-
-.. [2] Illustration image:
-       http://en.wikipedia.org/wiki/File:Composite_trapezoidal_rule_illustration.png
-
-Examples
---------
-&gt;&gt;&gt; np.trapz([1,2,3])
-4.0
-&gt;&gt;&gt; np.trapz([1,2,3], x=[4,6,8])
-8.0
-&gt;&gt;&gt; np.trapz([1,2,3], dx=2)
-8.0
-&gt;&gt;&gt; a = np.arange(6).reshape(2, 3)
-&gt;&gt;&gt; a
-array([[0, 1, 2],
-       [3, 4, 5]])
-&gt;&gt;&gt; np.trapz(a, axis=0)
-array([ 1.5,  2.5,  3.5])
-&gt;&gt;&gt; np.trapz(a, axis=1)
-array([ 2.,  8.])
-<span class="ansi-red-fg">File:</span>      /usr/local/lib/python3.5/dist-packages/numpy/lib/function_base.py
-<span class="ansi-red-fg">Type:</span>      function
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-<p>In fact, as expected</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Y</span><span class="o">=</span><span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="n">integrate</span><span class="o">.</span><span class="n">trapz</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">),</span><span class="n">X</span><span class="p">)</span>
-</pre></div>
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-<pre>3.7751549046338475</pre>
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-<p><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/4/42/Composite_trapezoidal_rule_illustration.png/1200px-Composite_trapezoidal_rule_illustration.png" height="400" width="400"></p>
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-<p>For a better approximation we can use more intervals</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">X</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1000</span> <span class="p">)</span>
-<span class="n">integrate</span><span class="o">.</span><span class="n">trapz</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">x</span><span class="p">)</span>
-</pre></div>
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-<pre>4.245647420030478</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">X</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</pre></div>
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-<pre>-0.5</pre>
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-<h2 id="Simpson's-rule">Simpson's rule<a class="anchor-link" href="#Simpson's-rule"> </a></h2>
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-<p>A slightly better approximation to integration is the Simpson's rule. For this, assume a function $f(x)$ and an interval $[x_0,x_2]$, with a intermediate point $x_1$. The associate second-order Lagrange polynomial is given by: See previous <a href="./interpolation.ipynb#Exercise-interpolation">exercise</a></p>
-$$P_2(x) = \frac{(x-x_1)(x-x_2)}{(x_0-x_1)(x_0-x_2)}f(x_0) + \frac{(x-x_0)(x-x_2)}{(x_1-x_0)(x_1-x_2)}f(x_1) + \frac{(x-x_0)(x-x_1)}{(x_2-x_0)(x_2-x_1)}f(x_2)$$<p>The final expression is then:</p>
-$$\int_{x_0}^{x_2} f(x)dx = \frac{h}{3}[ f(x_0)+4f(x_1)+f(x_2) ]-\frac{h^5}{90}f^{(4)}(\xi)$$
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-<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="mi">1</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">x</span>
-
-<span class="c1">#Quadrature with 3 points (Simpson&#39;s rule)</span>
-<span class="n">a</span><span class="o">=-</span><span class="mf">0.5</span>
-<span class="n">b</span><span class="o">=</span><span class="mf">1.5</span>
-<span class="n">x0</span><span class="o">=</span><span class="mf">0.5</span>
-<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">a</span><span class="p">,</span><span class="n">x0</span><span class="p">,</span><span class="n">b</span><span class="p">])</span>
-<span class="n">Q</span><span class="o">=</span><span class="n">PlotQuadrature</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">xmin</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">xmax</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">ymin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">ymax</span><span class="o">=</span><span class="mi">4</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Full=</span><span class="si">{}</span><span class="s1">, P2=</span><span class="si">{}</span><span class="s1">, rectangle=</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> <span class="n">integrate</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span> <span class="p">,</span>  <span class="n">Quadrature</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">),</span><span class="n">X</span><span class="p">),</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">+</span><span class="n">f</span><span class="p">(</span><span class="n">a</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span> <span class="p">)</span> <span class="p">)</span>
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-<pre>Full=4.245647748216942, P2=4.285253172123376, rectangle=3.7751549046338475
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-<h3 id="Scipy-implementation">Scipy implementation<a class="anchor-link" href="#Scipy-implementation"> </a></h3><p>Improves with number of points:
-<img src="https://upload.wikimedia.org/wikipedia/en/thumb/6/67/Simpsonsrule2.gif/220px-Simpsonsrule2.gif" alt="Simpons"></p>
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-<div class=" highlight hl-ipython3"><pre><span></span>integrate.simps<span class="o">?</span>
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-<pre><span class="ansi-red-fg">Signature:</span> integrate<span class="ansi-blue-fg">.</span>simps<span class="ansi-blue-fg">(</span>y<span class="ansi-blue-fg">,</span> x<span class="ansi-blue-fg">=</span><span class="ansi-green-fg">None</span><span class="ansi-blue-fg">,</span> dx<span class="ansi-blue-fg">=</span><span class="ansi-cyan-fg">1</span><span class="ansi-blue-fg">,</span> axis<span class="ansi-blue-fg">=</span><span class="ansi-blue-fg">-</span><span class="ansi-cyan-fg">1</span><span class="ansi-blue-fg">,</span> even<span class="ansi-blue-fg">=</span><span class="ansi-blue-fg">&#39;avg&#39;</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-red-fg">Docstring:</span>
-Integrate y(x) using samples along the given axis and the composite
-Simpson&#39;s rule.  If x is None, spacing of dx is assumed.
-
-If there are an even number of samples, N, then there are an odd
-number of intervals (N-1), but Simpson&#39;s rule requires an even number
-of intervals.  The parameter &#39;even&#39; controls how this is handled.
-
-Parameters
-----------
-y : array_like
-    Array to be integrated.
-x : array_like, optional
-    If given, the points at which `y` is sampled.
-dx : int, optional
-    Spacing of integration points along axis of `y`. Only used when
-    `x` is None. Default is 1.
-axis : int, optional
-    Axis along which to integrate. Default is the last axis.
-even : str {&#39;avg&#39;, &#39;first&#39;, &#39;last&#39;}, optional
-    &#39;avg&#39; : Average two results:1) use the first N-2 intervals with
-              a trapezoidal rule on the last interval and 2) use the last
-              N-2 intervals with a trapezoidal rule on the first interval.
-
-    &#39;first&#39; : Use Simpson&#39;s rule for the first N-2 intervals with
-            a trapezoidal rule on the last interval.
-
-    &#39;last&#39; : Use Simpson&#39;s rule for the last N-2 intervals with a
-           trapezoidal rule on the first interval.
-
-See Also
---------
-quad: adaptive quadrature using QUADPACK
-romberg: adaptive Romberg quadrature
-quadrature: adaptive Gaussian quadrature
-fixed_quad: fixed-order Gaussian quadrature
-dblquad: double integrals
-tplquad: triple integrals
-romb: integrators for sampled data
-cumtrapz: cumulative integration for sampled data
-ode: ODE integrators
-odeint: ODE integrators
-
-Notes
------
-For an odd number of samples that are equally spaced the result is
-exact if the function is a polynomial of order 3 or less.  If
-the samples are not equally spaced, then the result is exact only
-if the function is a polynomial of order 2 or less.
-<span class="ansi-red-fg">File:</span>      /usr/local/lib/python3.5/dist-packages/scipy/integrate/quadrature.py
-<span class="ansi-red-fg">Type:</span>      function
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">integrate</span><span class="o">.</span><span class="n">simps</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">),</span><span class="n">X</span><span class="p">)</span>
-</pre></div>
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-<pre>4.285253172123376</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">X</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">100000</span> <span class="p">)</span>
-<span class="n">integrate</span><span class="o">.</span><span class="n">trapz</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">x</span><span class="p">),</span><span class="n">integrate</span><span class="o">.</span><span class="n">simps</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">x</span><span class="p">)</span>
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-<pre>(4.245647748184187, 4.245647748216941)</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="mf">4.245647748216942</span>
-</pre></div>
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-<pre>4.245647748216942</pre>
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-<p><strong>Activity</strong>: Implement the Simpson method <a href="https://beta.deepnote.com/project/52b463ef-f58d-4842-be84-cc155faa49b9#">here</a> or <a href="https://repl.it/@restrepo/integration">here</a> by generalizing the previous <code>Quadrature</code> function to one that accepts lists of any size. Compares with the <code>Scipy</code> implementation for the next integral with a linspace of 100 points.</p>
-<p><em>Hint</em>: Note that the intervals for each 3 points can be obtained by slicing in steps of 2:</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">X</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">12</span><span class="p">]</span>
-<span class="n">i</span><span class="o">=</span><span class="mi">0</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span><span class="o">+</span><span class="mi">3</span><span class="p">])</span>
-<span class="n">i</span><span class="o">=</span><span class="n">i</span><span class="o">+</span><span class="mi">2</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span><span class="o">+</span><span class="mi">3</span><span class="p">])</span>
-<span class="n">i</span><span class="o">=</span><span class="n">i</span><span class="o">+</span><span class="mi">2</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span><span class="o">+</span><span class="mi">3</span><span class="p">])</span>
-<span class="n">i</span><span class="o">=</span><span class="n">i</span><span class="o">+</span><span class="mi">2</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span><span class="o">+</span><span class="mi">3</span><span class="p">])</span>
-</pre></div>
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-<pre>[1, 2, 3]
-[3, 4, 5]
-[5, 6, 7]
-[7, 8, 9]
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-<h2 id="-----Activity-"><span style="color:red">     Activity </span><a class="anchor-link" href="#-----Activity-"> </a></h2>
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-<ul>
-<li>Using the trapezoidal and the Simpson's rules, determine the value of the integral (4.24565)</li>
-</ul>
-$$ \int_{-0.5}^{1.5}(1+\cos^2x + x)dx $$<ul>
-<li>Take the previous routine Quadrature and the above function and explore high-order quadratures. What happends when you increase  the number of points?</li>
-</ul>
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-<p>N=20
-def f(x):
-    return 1+np.cos(x)**2+x</p>
-<h1 id="Quadrature-with-N-points-(Simpson's-rule)">Quadrature with N points (Simpson's rule)<a class="anchor-link" href="#Quadrature-with-N-points-(Simpson's-rule)"> </a></h1><p>X = np.array(np.linspace(-0.5,1.5,N))
-Ln=Quadrature( f, X, xmin=-1, xmax=2, ymin=0, ymax=4 )
-integrate.simps(Ln(X),X)
-print(poly1d(Ln))</p>
-
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-<h2 id="-----Activity-"><span style="color:red">     Activity </span><a class="anchor-link" href="#-----Activity-"> </a></h2>
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-<p>Approximate the following integrals using formulas Trapezoidal and Simpson rules. Are the accuracies of
-the approximations consistent with the error formulas?</p>
-\begin{eqnarray*}
-&amp;\int_{0}^{0.1}&amp;\sqrt{1+ x}dx \\
-&amp;\int_{0}^{\pi/2}&amp;(\sin x)^2dx\\ 
-&amp;\int_{1.1}^{1.5}&amp;e^xdx 
-\end{eqnarray*}
-</div>
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-<h1 id="Composite-Numerical-Integration">Composite Numerical Integration<a class="anchor-link" href="#Composite-Numerical-Integration"> </a></h1>
-</div>
-</div>
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-<p>Although above-described methods are good enough when we want to integrate along small intervals, larger intervals would require more sampling points, where the resulting Lagrange interpolant will be a high-order polynomial. These interpolant polynomials exihibit usually an oscillatory behaviour (best known as <a href="http://en.wikipedia.org/wiki/Runge%27s_phenomenon">Runge's phenomenon</a>), being more inaccurate as we increase $n$.</p>
-<p>An elegant and computationally inexpensive solution to this problem is a <em>piecewise</em> approach, where low-order Newton-Cotes formula (like trapezoidal and Simpson's rules) are applied over subdivided intervals. This methods are already implemented in the previous <code>scipy</code> trapezoidal and Simpsons implementations. An internal implementation is given in the <a href="./Appendix.ipynb#composite-numerical integration">Appendix of integration</a>, the <em>Composite Trapezoidal rule</em> given by:</p>
-$$ \int_a^b f(x) dx = \frac{h}{2}\left[ f(a) + 2\sum_{j=1}^{N-1}f(x_j) + f(b) \right] - \frac{b-a}{12}h^2 f^{''}(\mu)$$<p>and the <em>Composite Simpson's rule</em> given by</p>
-$$ \int_a^bf(x)dx = \frac{h}{3}\left[ f(a) +2 \sum_{j=1}^{(n/2)-1}f(x_{2j})+4\sum_{j=1}^{n/2}f(x_{2j-1})+f(b) \right] - \frac{b-a}{180}h^4f^{(4)}(\mu)$$
-</div>
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-<h2 id="Composite-trapezoidal-rule">Composite trapezoidal rule<a class="anchor-link" href="#Composite-trapezoidal-rule"> </a></h2>
-</div>
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-<p>This formula is obtained when we subdivide the integration interval $[a,b]$ within sets of two points, such that we can apply the previous Trapezoidal rule to each one.</p>
-<p>Let $f(x)$ be a well behaved function ($f\in C^2[a,b]$), defining the interval space as $h = (b-a)/N$, where N is the number of intervals we take, the <strong>Composite Trapezoidal rule</strong> is given by:</p>
-$$ \int_a^b f(x) dx = \frac{h}{2}\left[ f(a) + 2\sum_{j=1}^{N-1}f(x_j) + f(b) \right] - \frac{b-a}{12}h^2 f^{''}(\mu)$$<p>for some value $\mu$ in $(a,b)$.</p>
-
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-<p>Determine the value of the integral (4.24565)</p>
-$$ \int_{-0.5}^{1.5}(1+\cos^2x + x)dx $$
-</div>
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-<h2 id="-----Activity-"><span style="color:red">     Activity </span><a class="anchor-link" href="#-----Activity-"> </a></h2>
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-<p>An experiment has measured $dN(t)/dt$, the number of particles entering a counter, per unit time, as a function of time. Your problem is to integrate this spectrum to obtain the number of particles $N(1)$ that entered the counter
-in the first second</p>
-$$ N(1) = \int_0^1 \frac{dN}{dt} dt$$<p>For the problem it is assumed exponential decay so that there actually is an analytic answer.</p>
-$$ \frac{dN}{dt} = e^{-t} $$<p>Compare the relative error for the composite trapezoid and Simpson rules. Try different values of N. Make a logarithmic plot of N vs Error.</p>
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-<h1 id="Adaptive-Quadrature-Methods">Adaptive Quadrature Methods<a class="anchor-link" href="#Adaptive-Quadrature-Methods"> </a></h1>
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-<p>Calculating the integrate of the function $f(x) = e^{-3x}\sin(4x)$ within the interval $[0,4]$, we obtain:</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Function</span>
-<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-
-<span class="c1">#Plotting</span>
-<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">200</span> <span class="p">)</span>
-<span class="n">Y</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">14</span><span class="p">,</span><span class="mi">7</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;blue&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">3</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;blue&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span> <span class="mi">0</span><span class="p">,</span><span class="mi">4</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-</pre></div>
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-<p>Using composite numerical integration is not completely adequate for this problem as the function exhibits different behaviours for differente intervals. For the interval $[0,2]$ the function varies noticeably, requiring a rather small integration interval $h$. However, for the interval $[2,4]$ variations are not considerable and low-order composite integration is enough. This lays a pathological situation where simple composite methods are not efficient. In order to remedy this, we introduce an adaptive quadrature methods, where the integration step $h$ can vary according to the interval. The main advantage of this is a controlable precision of the result.</p>
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-<h2 id="Simpson's-adaptive-quadrature">Simpson's adaptive quadrature<a class="anchor-link" href="#Simpson's-adaptive-quadrature"> </a></h2>
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-<p>Although adaptive quadrature can be readily applied to any quadrature method, we illustrate the method with  the Simpson's adaptive quadrature.</p>
-<p>Let's assume a function $f(x)$. We want to compute the integral within the interval $[a,b]$. Using a simple Simpson's quadrature, we obtain:</p>
-$$\int_a^bf(x)dx = S(a,b) - \frac{h^5}{90}f^{(4)}(\xi)$$<p>where we introduce the notation:</p>
-$$S(a,b) = \frac{h}{3}\left[ f(a) + 4f(a+h) + f(b) \right]$$<p>and $h$ is simply $h = (b-a)/2$.</p>
-<p><img src="https://raw.githubusercontent.com/sbustamante/ComputationalMethods/bb0f137366b3d2bbfdd33425ad78be4003716703/material/figures/adaptive_quadrature.png" alt=""></p>
-<p>Now, instead of using an unique Simpson's quadrature, we implement two, by adding a new point in the midle of the interval $[a,b]$, $(a+b)/2$, yielding:</p>
-$$\int_a^bf(x)dx = S\left(a,\frac{a+b}{2}\right) + S\left(\frac{a+b}{2},b\right) - \frac{1}{16}\left(\frac{h^5}{90}\right)f^{(4)}(\xi)$$<p>For this expression, we reasonably assume an equal fourth-order derivative $f^{(4)}(\xi) = f^{(4)}(\xi_1) = f^{(4)}(\xi_2) $, where $\xi_1$ is the estimative for the first subtinterval (i.e. $\xi_1\in[a,(a+b)/2]$), and $\xi_2$ for the second one (i.e. $\xi_1\in[(a+b)/2, b]$).</p>
-<p>As both expressions can approximate the real value of the integrate, we can equal them, obtaining:</p>
-$$
-\int_a^bf(x)dx \begin{cases} \sim S(a,b) - \frac{h^5}{90}f^{(4)}(\xi)\\ 
-\approx S\left(a,\frac{a+b}{2}\right) + S\left(\frac{a+b}{2},b\right) - \frac{1}{16}\left(\frac{h^5}{90}\right)f^{(4)}(\xi)\\
-\end{cases}
-$$<p>which leads us to a simple way to estimate the error without knowing the fourth-order derivative, i.e.</p>
-$$\frac{h^5}{90}f^{(4)}(\xi) = \frac{16}{15}\left| S(a,b) - S\left(a,\frac{a+b}{2}\right) - S\left(\frac{a+b}{2},b\right) \right|$$<p>If we fix a precision $\epsilon$, such that the obtained error for the second iteration is smaller</p>
-$$\frac{1}{16}\frac{h^5}{90}f^{(4)}(\xi) &lt; \epsilon $$<p>it implies:</p>
-$$\left| S(a,b) - S\left(a,\frac{a+b}{2}\right) - S\left(\frac{a+b}{2},b\right) \right|&lt; 15 \epsilon$$<p>and</p>
-$$\left| \int_a^bf(x) dx- S\left(a,\frac{a+b}{2}\right) - S\left(\frac{a+b}{2},b\right) \right|&lt; \epsilon$$<p>The second iteration is then $15$ times more precise than the first one.</p>
-
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-<h2 id="Steps-Simpson's-adaptive-quadrature">Steps Simpson's adaptive quadrature<a class="anchor-link" href="#Steps-Simpson's-adaptive-quadrature"> </a></h2>
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-<p><strong>1.</strong> Give the function $f(x)$ to be integrated, the inverval $[a,b]$ and set a desired precision $\epsilon$.</p>
-<p><strong>2.</strong> Compute the next Simpsons's quadratures:</p>
-$$ S(a,b),\ S\left(a,\frac{a+b}{2}\right),\ S\left(\frac{a+b}{2},b\right) $$<p><strong>3.</strong> If</p>
-$$\frac{1}{15}\left| S(a,b) - S\left(a,\frac{a+b}{2}\right) - S\left(\frac{a+b}{2},b\right) \right|&lt;\epsilon$$<p>then the integration is ready and is given by:</p>
-$$\int_a^bf(x) dx \approx S\left(a,\frac{a+b}{2}\right) + S\left(\frac{a+b}{2},b\right) $$<p>within the given precision.</p>
-<p><strong>4.</strong> If the previous step is not fulfilled, repeat from step <strong>2</strong> using as new intervals $[a,(a+b)/2]$ and $[(a+b)/2,b]$ and a new precision $\epsilon_1 = \epsilon/2$. Repeating until step 3 is fulfilled for all the subintervals.</p>
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-  Numerical integration
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-
-    <main class="jupyter-page">
-    <div id="page-info"><div id="page-title">Numerical integration</div>
-</div>
-    <div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/numerical-calculus-integration.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<h1 id="Numerical-Calculus.-Integration">Numerical Calculus. Integration<a class="anchor-link" href="#Numerical-Calculus.-Integration"> </a></h1><p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/numerical-calculus-integration.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<p>Throughout this section and the next ones, we shall cover the topic of numerical calculus. Calculus has been identified since ancient times as a powerful toolkit for analysing and handling geometrical problems. Since differential calculus was developed by Newton and Leibniz (in its actual notation), many different applications have been found, at the point that most of the current science is founded on it (e.g. differential and integral equations). Due to the ever increasing complexity of analytical expressions used in physics and astronomy, their usage becomes more and more impractical, and numerical approaches are more than necessary when one wants to go deeper. This issue has been identified since long ago and many numerical techniques have been developed. We shall cover only the most basic schemes, but also providing a basis for more formal approaches.</p>
-<p>From <a href="http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture18.pdf">7 (Chapter 18)</a></p>
-<blockquote><p>We can only find anti-derivatives for a very small number of functions, such as those functions that are popular as problems in mathematics textbooks. Outside of math classes, these functions are rare</p>
-</blockquote>
-
-</div>
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-<hr>
-<ul>
-<li><a href="./numerical-calculus.ipynb">Numerical Differentiation</a> </li>
-<li><a href="#Numerical-Integration">Numerical Integration</a><ul>
-<li><a href="#Numerical-quadrature">Numerical quadrature</a></li>
-<li><a href="#Trapezoidal-rule">Trapezoidal rule</a></li>
-<li><a href="#Simpson&#39;s-rule">Simpson's rule</a></li>
-</ul>
-</li>
-<li><a href="#Composite-Numerical-Integration">Composite Numerical Integration</a><ul>
-<li><a href="#Composite-trapezoidal-rule">Composite trapezoidal rule</a></li>
-<li><a href="#Composite-Simpson&#39;s-rule">Composite Simpson's rule</a></li>
-</ul>
-</li>
-<li><a href="#Adaptive-Quadrature-Methods">Adaptive Quadrature Methods</a><ul>
-<li><a href="#Simpson&#39;s-adaptive-quadrature">Simpson's adaptive quadrature</a></li>
-<li><a href="#Steps-Simpson&#39;s-adaptive-quadrature">Steps Simpson's adaptive quadrature</a></li>
-</ul>
-</li>
-<li><a href="#Improper-Integrals">Improper Integrals</a><ul>
-<li><a href="#Left-endpoint-singularity">Left endpoint singularity</a></li>
-<li><a href="#Right-endpoint-singularity">Right endpoint singularity</a></li>
-<li><a href="#Infinite-singularity">Infinite singularity</a></li>
-</ul>
-</li>
-</ul>
-<hr>
-
-</div>
-</div>
-</div>
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-<h2 id="Bibliograpy">Bibliograpy<a class="anchor-link" href="#Bibliograpy"> </a></h2><ul>
-<li><a href="https://github.com/restrepo/Calculus">https://github.com/restrepo/Calculus</a></li>
-<li>Thomas J. Sargent John Stachurski, <a href="https://python-programming.quantecon.org/scipy.html">Python Programming for Quantitative Economics GitHub</a></li>
-</ul>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%</span><span class="k">pylab</span> inline
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">import</span> <span class="nn">scipy.interpolate</span> <span class="k">as</span> <span class="nn">interpolate</span>
-<span class="kn">import</span> <span class="nn">scipy.integrate</span> <span class="k">as</span> <span class="nn">integrate</span>
-</pre></div>
-
-    </div>
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-<pre>Populating the interactive namespace from numpy and matplotlib
-</pre>
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-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h1 id="Numerical-Integration">Numerical Integration<a class="anchor-link" href="#Numerical-Integration"> </a></h1>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Integration is the second fundamental concept of calculus (along with differentiation). Numerical approaches are generally more useful here than in differentiation as the antiderivative procedure (analytically) is often much more complex, or even not possible. In this section we will cover some basic schemes, including numerical quadratures.</p>
-<p>Geometrically, integration can be understood as the area below a funtion within a given interval. Formally, given a function $f(x)$ such that $f\in C^{1}[a,b]$, the antiderivative is defined as</p>
-$$F(x) = \int f(x) dx$$<p>valid for all $x$ in $[a,b]$. However, a more useful expression is a definite integral, where the antiderivative is evaluated within some interval, i.e.</p>
-$$F(b) - F(a) = \int_{a}^{b} f(x) dx$$<p>This procedure can be formally thought as a generalization of discrete weighted summation. This idea will be exploited below and will lead us to some first approximations to integration.</p>
-<p>See pag. 66 of <a href="https://docs.google.com/viewer?a=v&amp;pid=sites&amp;srcid=ZmlzaWNhLnVkZWEuZWR1LmNvfGNvbXB1dGFkb3Jlcy1lbi1maXNpY2EtMjAxMC0yfGd4OjFmMDJjN2NlOTc3YWE0YTQ">PDF</a></p>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Numpy-abstractions">Numpy abstractions<a class="anchor-link" href="#Numpy-abstractions"> </a></h2><p>The programming paradigm with Numpy are the abstractions, where the algorithms are written in terms of operations with arrays, avoiding the use of  programming loops. Low level operations between arrays are implemented in C/Fortran within Numpy. Therefore, a code with abstractions is automatically optimized.</p>
-<p>Consider for example the numerical integration with the trapezoidal method</p>
-<p><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/2/26/Integration_rectangle.svg/300px-Integration_rectangle.svg.png" alt="Credit Wikipedia"> <small>[See Wikipedia](https://en.wikipedia.org/wiki/Numerical_integration#Methods_for_one-dimensional_integrals)</small></p>
-<p>Wehere for each internval between $x_i$ and $x_{i+1}$, the average height of the function 
-$$\frac{f(x_i)+f(x_{i-1})}{2}\,,$$
-is used. Therefore
-\begin{align*}
-  \int_a^b f(x)d x\approx \frac{1}{2}\sum_{i=1}^n (x_i-x_{i-1})(f(x_i)+f(x_{i-1}))\,.
-\end{align*}</p>
-<p>To implement the correspponding abstractions in Numpy it is convenient to define the following vectors
-\begin{align*}
-  \Delta \mathbf{X}&amp;=(x_1-x_0,x_2-x_1,\cdots,x_n-x_{n-1})\\ 
-   \langle\mathbf{F}\rangle&amp;=\frac{1}{2}(f(x_1)+f(x_0),f(x_2)+f(x_1),\cdots,f(x_n)+f(x_{n-1}))\,.
-\end{align*}
-The numerical approximation of the integral with the trapezoidal method is then just the dot product between the two previous vectors
-\begin{align*}
-   \int_a^b f(x)d x\approx \Delta \mathbf{X}\cdot \langle\mathbf{F}\rangle
-\end{align*}</p>
-<p>We next present the code with different levels of abstractions. The faster is the first one with the direct implementation of the dot product.</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">integracion</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">n</span><span class="o">=</span><span class="mi">4</span><span class="p">,</span><span class="n">test</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;Trapezoidal method for the numerical integration  de f entre a y b</span>
-<span class="sd">         con n intervalos</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-    <span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-    <span class="n">y</span><span class="o">=</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">test</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span><span class="o">-</span><span class="n">x</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]),(</span><span class="n">y</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span><span class="o">+</span><span class="n">y</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]))</span>
-    <span class="k">elif</span> <span class="n">test</span> <span class="o">==</span><span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mf">0.5</span><span class="o">*</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span><span class="o">-</span><span class="n">x</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="o">*</span><span class="p">(</span><span class="n">y</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span><span class="o">+</span><span class="n">y</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]))</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
-</pre></div>
-
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-<h2 id="Implementation-in-scipy">Implementation in scipy<a class="anchor-link" href="#Implementation-in-scipy"> </a></h2>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<div class="highlight"><pre><span></span><span class="n">integrate</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="n">func</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">args</span><span class="o">=</span><span class="p">(),</span><span class="n">full_output</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span><span class="o">...</span><span class="p">)</span>
-</pre></div>
-<p><strong>Parameters</strong>:
-<code>python</code></p>
-<p>See <code>integrate.quad?</code> for further details</p>
-
-</div>
-</div>
-</div>
-</div>
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-<h3 id="Usage-example">Usage example<a class="anchor-link" href="#Usage-example"> </a></h3>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f</span><span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span> <span class="p">(</span> <span class="n">x</span> <span class="p">)</span><span class="o">/</span> <span class="n">x</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Note that:</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-</pre></div>
-
-    </div>
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-<div class="output_wrapper">
-<div class="output">
-
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-<div class="output_subarea output_stream output_stderr output_text">
-<pre>/usr/local/lib/python3.5/dist-packages/ipykernel_launcher.py:1: RuntimeWarning: invalid value encountered in double_scalars
-  &#34;&#34;&#34;Entry point for launching an IPython kernel.
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>nan</pre>
-</div>
-
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-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Even so, the full tuple output with result and numerical error is obtained from</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">integrate</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
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-<div class="output_text output_subarea output_execute_result">
-<pre>(0.9460830703671831, 1.0503632079297089e-14)</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">integrate</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>0.9460830703671831</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">time</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span><span class="o">=</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">integrate</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">s</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>(0.9460830703671831, 1.0503632079297089e-14)
-0.00041174888610839844
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Force output to have only the result by asking for the entry 0 of the output tuple, note the <code>[0]</code> at the end:</p>
-
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">integrate</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span> <span class="p">(</span> <span class="n">x</span> <span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-</pre></div>
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-<pre>(0.9460830703671831, 1.0503632079297089e-14)</pre>
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-<p>Avoid singularities in custom solutions</p>
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-</pre></div>
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-<pre>0.9450787809533919</pre>
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-<p>Sinintegrate: 
-$$\operatorname{Si}(t)=\int_0^t \frac{\sin x}{x}\, \operatorname{d}x$$</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">import</span> <span class="nn">scipy.integrate</span> <span class="k">as</span> <span class="nn">integrate</span>
-
-<span class="n">Sifloat</span><span class="o">=</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">integrate</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">t</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-<span class="k">def</span> <span class="nf">Si</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">nn</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">f</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="n">Sifloat</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
-        <span class="n">f</span><span class="o">=</span><span class="n">Sifloat</span>
-    <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Si</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
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-<pre>0.9460830703671831</pre>
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-<pre>array([0.94608307, 1.60541298])</pre>
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-<p>When you write <code>LaTeX</code>  in  <code>matplotlib</code>, you must be sure that the backslash, '\', is properly interpreted  as the key for <code>LaTeX</code> commands. This is accomplished  by writting an <code>r</code> before the string declaration. Note the different behaviour of the backslash in the next two cells of code. In the first one <code>\t</code> is intepreted as the keyboard key <code>&lt;TAB&gt;</code> and <code>\b</code> as an empty space, while in the second, with the prefix <code>r</code>, it is just a pair of <code>LaTeX</code> commands</p>
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-<pre>$	aeta$
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-<span class="n">tt</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">20</span><span class="p">,</span><span class="mi">200</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">semilogy</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">Si</span><span class="p">(</span><span class="n">t</span><span class="p">),</span><span class="s1">&#39;r-&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">semilogy</span><span class="p">(</span><span class="n">tt</span><span class="p">,</span><span class="n">Si</span><span class="p">(</span><span class="n">tt</span><span class="p">),</span><span class="s1">&#39;b-&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;$t$&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;${\rm Si}(t)=\int_0^t {\sin\, x}/</span><span class="si">{x}</span><span class="s1">\,{\rm d}\,x$&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
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->
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-<h2 id="---Activity-"><span style="color:red">   Activity </span><a class="anchor-link" href="#---Activity-"> </a></h2>
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-<h3 id="Models-of-Universe">Models of Universe<a class="anchor-link" href="#Models-of-Universe"> </a></h3><p>From the Friedmann equations can be found the dynamics of the Universe, i.e., the evolution of the expansion with time that depends on the content of matter and energy of the Universe. Before introducing the general expression, there are several quatities that need to be defined.</p>
-<p>It is convenient to express the density in terms of a critical density $\rho_c$ given by</p>
-\begin{equation}
-\rho_c = 3H_0^2/8\pi G
-\end{equation}<p>where $H_o$ is the Hubble constant. The critical density is the density needed in order the Universe to be flat. To obtained it, it is neccesary to make the curvature of the universe $\kappa = 0$. The critical density is one value per
-time and the geometry of the universe depends on this value, or equally on $\kappa$. For a universe with $\kappa&lt;0$ it would ocurre a big crunch(closed universe) and for a $\kappa&gt;0$ there would be an open universe.</p>
-<p>Now, it can also be defined a density parameter, $\Omega$, a normalized density</p>
-\begin{equation}
-\Omega_{i,0} = \rho_{i,0}/\rho_{crit}
-\end{equation}<p>where $\rho_{i,0}$ is the actual density($z=0$) for the component $i$. Then, it can be found the next expression</p>
-\begin{equation}
-\frac{H^2(t)}{H_{0}^{2}} = (1-\Omega_0)(1+z)^2 + \Omega_{m,0}(1+z^3)+ \Omega_{r,0}(1+z)^4 + \Omega_{\Lambda,0}
-\end{equation}<p>where $\Omega_{m,0}$, $\Omega_{r,0}$ and  $\Omega_{\Lambda,0}$ are the matter, radiation and vacuum density parameters. And $\Omega_0$ is the total density including the vacuum energy.</p>
-<p>This expression can also be written in terms of the expansion or scale factor, $a$, by using:</p>
-$$1+z = 1/a$$<p>For the next universe models, plot time($H_{0}^{-1}$ units) vs the scale factor:</p>
-<p>-Einstein-de Sitter Universe: Flat space, null vacuum energy and dominated by matter</p>
-\begin{equation}
-t = H_0^{-1} \int_0^{a'} a^{1/2}da
-\end{equation}<p>-Radiation dominated universe: All other components are not contributing</p>
-$$
-t = H_0^{-1} \int_0^{a'} \frac{a}{[\Omega_{r,0}+a^2(1-\Omega_{r,0})]^{1/2}}da
-$$<p>-WMAP9 Universe</p>
-\begin{equation}
-t = H_0^{-1} \int_0^{a'} \left[(1-\Omega_{0})+ \Omega_{M,0}a^{-1} + \Omega_{R,0}a^{-2} +\Omega_{\Lambda,0}a^2\right]^{-1/2} da
-\end{equation}<p>You can take the cosmological parameters  from the link</p>
-<p><a href="http://lambda.gsfc.nasa.gov/product/map/dr5/params/lcdm_wmap9.cfm">http://lambda.gsfc.nasa.gov/product/map/dr5/params/lcdm_wmap9.cfm</a> or use these ones: $\Omega_M = 0.266$,
-$\Omega_R = 8.24\times 10^{-5}$ and $\Omega_\Lambda = 0.734$.</p>
-<p>Use composite Simpson's rule to integrate and compare it with the analytical expression in case you can get it. 
-The superior limit in the integral corresponds to the actual redshift $z=0$. What is happening to our universe?</p>
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-<h2 id="--Activity-"><span style="color:red">  Activity </span><a class="anchor-link" href="#--Activity-"> </a></h2><ul>
-<li>Using the Simpson's adaptive quadrature determine the value of the next integral with a precision of float32.</li>
-</ul>
-$$\int_0^4 e^{-3x}\sin(4x)dx$$<h2 id="--Activity-"><span style="color:red">  Activity </span><a class="anchor-link" href="#--Activity-"> </a></h2><p>Fresnel integrals are commonly used in the study of light difraction at a rectangular aperture, they are given by:</p>
-$$c(t) = \int_0^t\cos\left(\frac{\pi}{2}\omega^2\right)d\omega$$$$s(t) = \int_0^t\sin\left(\frac{\pi}{2}\omega^2\right)d\omega$$<p>These integrals cannot be solved using analitical methods. Using the previous routine for adaptive quadrature, compute the integrals with a precision of $\epsilon=10^{-4}$ for values of $t=0.1,0.2,0.3,\cdots 1.0$. Create two arrays with those values and then make a plot of $c(t)$ vs $s(t)$. The resulting figure is called Euler spiral, that is a member of a family of curves called <a href="http://en.wikipedia.org/wiki/Vertical_loop">Clothoid loops</a>.</p>
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-<h1 id="Improper-Integrals">Improper Integrals<a class="anchor-link" href="#Improper-Integrals"> </a></h1>
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-<p>Improper integral can be processed with the <code>quad</code> function of <code>scipy.integrate</code>, or other integration functions, by using the infinity implementation of <code>numpy</code>, (which is the same of the <code>math</code> module, see below)</p>
-<ul>
-<li>Positive infinity ($+\infty$): <code>np.inf</code> and several aliases</li>
-<li>Negagite infinity ($-\infty$): <code>np.NINF</code>
-For details of the implementation and the aliases <a href="https://stackoverflow.com/a/42315752/2268280">see here</a></li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
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-<p>In Python the $\infty$ can be defined as</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">float</span><span class="p">(</span><span class="s1">&#39;inf&#39;</span><span class="p">)</span>
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-<p>It is already implemented in numpy</p>
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-<pre>True</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="mi">24343555355567</span><span class="o">&lt;</span><span class="n">np</span><span class="o">.</span><span class="n">inf</span>
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-<p>There are also function to check for infinity numbers:</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">isfinite</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
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-<pre>True</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">isfinite</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">inf</span><span class="p">)</span>
-</pre></div>
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-<pre>False</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">isfinite</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span><span class="n">np</span><span class="o">.</span><span class="n">isinf</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">inf</span><span class="p">)</span>
-</pre></div>
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-<pre>(True, True)</pre>
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-<p>So it is possible to evaluate improper integrals like</p>
-$$
-\int_1^\infty x^{-2}\,\operatorname{d}x=1\,.
-$$
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">integrate</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">inf</span><span class="p">)</span>
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-<pre>(1.0, 1.1102230246251565e-14)</pre>
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-<p>or
-$$
-\int_{-\infty}^\infty \operatorname{e}^{x^{-2}}\,\operatorname{d}x=\sqrt{\pi},.
-$$</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">integrate</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span><span class="n">np</span><span class="o">.</span><span class="n">NINF</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">inf</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span><span class="o">==</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span>
-</pre></div>
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-<pre>True</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">integrate</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span><span class="mi">0</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">inf</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span><span class="o">==</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-</pre></div>
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-<pre>True</pre>
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-<p>Although the previous integration methods can be applied in almost every situation, improper integrals pose a challenger to numerical methods as they involve indeterminations and infinite intervals. Next, we shall cover some tricks to rewrite improper integrals in terms of simple ones. See for example <a href="https://stackoverflow.com/q/30769913/2268280">here</a></p>
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-<h2 id="Left-endpoint-singularity">Left endpoint singularity<a class="anchor-link" href="#Left-endpoint-singularity"> </a></h2>
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-<p>Assuming a function $f(x)$ such that it can be rewritten as</p>
-$$ f(x) = \frac{g(x)}{(x-a)^p} $$<p>the integral over an interval $[a,b]$ converges only and only if $0&lt;p&lt;1$.</p>
-<p>Using Simpson's composite rule, it is possible to compute the fourth-order Taylor polynomial of the function $g(x)$ at the point $a$, obtaining</p>
-$$ P_4(x) = g(a) + g^{'}(a)(x-a)+\frac{g^{''}(a)}{2!}(x-a)^2 +\frac{g^{''}(a)}{3!}(x-a)^3+ \frac{g^{(4)}(a)}{4!}(x-a)^4$$<p>The integral can be then calculated as</p>
-$$\int_a^b f(x)dx = \int_a^b\frac{g(x)-P_4(x)}{(x-a)^p}dx + \int_a^b\frac{P_4(x)}{(x-a)^p}dx$$<p>The second term is an integral of a polynimal, which can be easily integrated using analytical methods.</p>
-<p>The first term is no longer pathologic as there is not any indetermination, and it can be determined using composite Simpson's rule</p>
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-<p>This is the contrary case, where the indetermination is present in the extreme $b$ of the integration interval $[a,b]$. For this problem is enough to make a variable substitution</p>
-$$ z=-x \  \ \ \ \ \  dz=-dx$$<p>With this, the right endpoint singularity becomes a left endpoint singularity and the previous method can be directly applied.</p>
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-<h2 id="Infinite-singularity">Infinite singularity<a class="anchor-link" href="#Infinite-singularity"> </a></h2>
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-<p>Finally, infinite singularities are those where the integration domain is infinite, i.e.</p>
-$$\int_a^\infty f(x)dx$$<p>this type of integrals can be easily turned into a left endpoint singularity jus making the next variable substitution</p>
-$$t = x^{-1}\ \ \ \ \ dt = -x^{-2}dx$$<p>yielding</p>
-$$ \int_a^\infty f(x)dx = \int_0^{1/a}t^{-2}f\left(\frac{1}{t}\right)dt $$
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-<h2 id="-Activity"><span style="color:red"> Activity</span><a class="anchor-link" href="#-Activity"> </a></h2><p>Error function is a special and non-elementary function that is widely used in probability, statistics and diffussion processes.
-It is defined through the integral:</p>
-$$\mbox{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt$$<p>Using the substitution $u=t^2$ it is possible to use the previous methods for impropers integrals in order to evaluate the error function. Create a routine called <code>ErrorFunction</code> that, given a value of $x$, return the respective value of the integral.</p>
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-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/numerical-calculus.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<h1 id="Numerical-Calculus">Numerical Calculus<a class="anchor-link" href="#Numerical-Calculus"> </a></h1><p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/numerical-calculus.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<p>Throughout this section and the next ones, we shall cover the topic of numerical calculus. Calculus has been identified since ancient times as a powerful toolkit for analysing and handling geometrical problems. Since differential calculus was developed by Newton and Leibniz (in its actual notation), many different applications have been found, at the point that most of the current science is founded on it (e.g. differential and integral equations). Due to the ever increasing complexity of analytical expressions used in physics and astronomy, their usage becomes more and more impractical, and numerical approaches are more than necessary when one wants to go deeper. This issue has been identified since long ago and many numerical techniques have been developed. We shall cover only the most basic schemes, but also providing a basis for more formal approaches.</p>
-<hr>
-<h2 id="*-Class-activities-with-CoCalc">* <a href="https://cocalc.com/projects/a8330cfb-9dfb-442e-9b2b-ba664e31a685/files/numerical-calculus.ipynb?fullscreen=default&amp;session=default">Class activities with CoCalc</a><a class="anchor-link" href="#*-Class-activities-with-CoCalc"> </a></h2><h3 id="Bibliography">Bibliography<a class="anchor-link" href="#Bibliography"> </a></h3><ul>
-<li><a href="https://github.com/restrepo/Calculus">https://github.com/restrepo/Calculus</a></li>
-<li>Thomas J. Sargent John Stachurski, <a href="https://python-programming.quantecon.org/scipy.html">Python Programming for Quantitative Economics GitHub</a></li>
-</ul>
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-<ul>
-<li><a href="#Numerical-Differentiation">Numerical Differentiation</a> <ul>
-<li><a href="#Example-1">Example 1</a></li>
-<li><a href="#n+1-point-formula">(n+1)-point formula</a></li>
-<li><a href="#Endpoint-formulas">Endpoint formulas</a></li>
-<li><a href="#Midpoint-formulas">Midpoint formulas</a></li>
-</ul>
-</li>
-<li><a href="./numerical-calculus-integration.ipynb">Numerical Integration</a>    </li>
-</ul>
-<hr>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%</span><span class="k">pylab</span> inline
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">import</span> <span class="nn">scipy.interpolate</span> <span class="k">as</span> <span class="nn">interp</span>
-</pre></div>
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-<h1 id="Numerical-Differentiation">Numerical Differentiation<a class="anchor-link" href="#Numerical-Differentiation"> </a></h1>
-</div>
-</div>
-</div>
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-<p>According to the formal definition of differentiation, given a function $f(x)$ such that $f(x)\in C^1[a,b]$, the first order derivative is given by</p>
-$$\frac{d}{dx}f(x) = f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}$$<p>However, when $f(x)$ exhibits a complex form or is a numerical function (only a discrete set of points are known), this expression becomes unfeasible. In spite of this, this formula gives us a very first rough way to calculate numerical derivatives by taking a finite interval $h$, i.e.</p>
-$$f'(x) \approx \frac{f(x+h)-f(x)}{h}$$<p>where the function must be known at least in $x_0$ and $x_1 = x_0+h$, and $h$ should be small enough.</p>
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-<h2 id="Example-1">Example 1<a class="anchor-link" href="#Example-1"> </a></h2>
-</div>
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-<p>Evaluate the first derivative of the next function using the previous numerical scheme at the point $x_0=2.0$ and using $h=0.5,\ 0.1,\  0.05$</p>
-<p>$f(x) = \sqrt{1+\cos^2(x)}$</p>
-<p>Compare with the real function and plot the tangent line using the found values of the slope.</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">misc</span>
-</pre></div>
-
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>It is used as:</p>
-<div class="highlight"><pre><span></span><span class="n">misc</span><span class="o">.</span><span class="n">derivative</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">dx</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">args</span><span class="o">=</span><span class="p">(),</span> <span class="n">order</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-</pre></div>
-<p><strong>Parameters</strong><br/>
-<code>func</code> : function →
-    Input function.<br/>
-<code>x0</code> : float →
-    The point at which <code>n</code>-th derivative is found.<br/>
-<code>dx</code> : float, optional →
-    Spacing.<br/>
-<code>n</code> : int, optional →
-    Order of the derivative. Default is 1.<br/>
-<code>args</code> : tuple, optional →
-    Arguments<br/>
-<code>order</code> : int, optional → 
-    Number of points to use, must be odd.<br/></p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">function</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span> <span class="mi">1</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
-
-<span class="c1">#X value</span>
-<span class="n">x0</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="n">xmin</span> <span class="o">=</span> <span class="mf">1.8</span>
-<span class="n">xmax</span> <span class="o">=</span> <span class="mf">2.2</span>
-
-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">,</span> <span class="mi">100</span> <span class="p">)</span>
-
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span> <span class="mi">1</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;function&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">10</span> <span class="p">)</span>
-</pre></div>
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-<pre>[&lt;matplotlib.lines.Line2D at 0x7fdd94000e80&gt;]</pre>
-</div>
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-<img src="../images/features/numerical-calculus_12_1.png"
->
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-<p><strong>Activity</strong>: We now check for the impact of the change of the spacing <code>dx</code>.  Try from <code>dx=0.5</code> and then small values</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
-    <span class="n">dx</span><span class="o">=</span><span class="nb">eval</span><span class="p">(</span><span class="nb">input</span><span class="p">(</span><span class="s1">&#39;dx=&#39;</span><span class="p">))</span>
-    <span class="nb">print</span><span class="p">(</span> <span class="n">misc</span><span class="o">.</span><span class="n">derivative</span><span class="p">(</span><span class="n">function</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="n">dx</span><span class="o">=</span><span class="n">dx</span><span class="p">)</span> <span class="p">)</span>
-</pre></div>
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-</div>
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-<div class="output_subarea output_stream output_stdout output_text">
-<pre>0.27884081567829266
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-</div>
-</div>
-<div class="jb_output_wrapper }}">
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-<div class="output_subarea output_stream output_stdout output_text">
-<pre>0.3462499420237386
-</pre>
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-<div class="output_area">
-
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-</div>
-</div>
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-<div class="output_subarea output_stream output_stdout output_text">
-<pre>0.3493266965195363
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
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-</div>
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-<div class="output_subarea output_stream output_stdout output_text">
-<pre>0.34935758881293744
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
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-</div>
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-<div class="output_subarea output_stream output_stdout output_text">
-<pre>0.3493579009417047
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Compare with:</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">misc</span><span class="o">.</span><span class="n">derivative</span><span class="p">(</span><span class="n">function</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="n">dx</span><span class="o">=</span><span class="mf">1E-12</span><span class="p">)</span>
-</pre></div>
-
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-<pre>0.34927616354707425</pre>
-</div>
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-</div>
-</div>
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-<h2 id="Implementation-of-the-derivate-of-the-function-inside-a-full-range">Implementation of the derivate of the function inside a full range<a class="anchor-link" href="#Implementation-of-the-derivate-of-the-function-inside-a-full-range"> </a></h2><p>We now generalize the <code>derivative</code> function to allow the evaluation of the derivate in a full range of values. It will be designed such that the evaluation in just one point can be still possible. In this way, the new function  can be used as a full replacement of <code>derivative</code> function.</p>
-<p>To implement this function we need three keys ingredients of <code>python</code>:</p>
-<ol>
-<li><code>try</code> and <code>except</code> python progamming sctructure</li>
-<li>Function definition with generic mandatory and optional arguments</li>
-<li>Conversion of a float function into a vectorized fuction</li>
-</ol>
-<h3 id="try-and-except-python-progamming-sctructure"><code>try</code> and <code>except</code> python progamming sctructure<a class="anchor-link" href="#try-and-except-python-progamming-sctructure"> </a></h3><p>First we introduce the <code>try</code> and <code>except</code> python progamming sctructure, wich is used to bypass one python error. For example a zero dimension array has not a shape attribute, so that the following error, of type <code>IndexError</code>, is generated:</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">nn</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</pre></div>
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-<pre>
-<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
-<span class="ansi-red-fg">IndexError</span>                                Traceback (most recent call last)
-<span class="ansi-green-fg">&lt;ipython-input-9-e1814928a8a6&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span>
-<span class="ansi-green-fg">----&gt; 1</span><span class="ansi-red-fg"> </span>nn<span class="ansi-blue-fg">=</span>np<span class="ansi-blue-fg">.</span>array<span class="ansi-blue-fg">(</span><span class="ansi-cyan-fg">3</span><span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">.</span>shape<span class="ansi-blue-fg">[</span><span class="ansi-cyan-fg">0</span><span class="ansi-blue-fg">]</span>
-
-<span class="ansi-red-fg">IndexError</span>: tuple index out of range</pre>
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>To bypass that error we use the following code:</p>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">try</span><span class="p">:</span> 
-    <span class="n">nn</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
-    <span class="n">nn</span><span class="o">=-</span><span class="mi">1</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<p>so that <code>nn</code> takes the values assigned in the <code>except</code> part:</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">nn</span>
-</pre></div>
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-<pre>-1</pre>
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-<p><strong>Activity</strong>: A float has the method <code>is_integer()</code> to check if the decimal part is zero. Creates a function <code>is_integer(x)</code> which works even in the case that the number is already an integer</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span><span class="o">=</span><span class="mi">2</span>
-<span class="n">x</span><span class="o">.</span><span class="n">is_integer</span><span class="p">()</span>
-</pre></div>
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-<pre>
-<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
-<span class="ansi-red-fg">AttributeError</span>                            Traceback (most recent call last)
-<span class="ansi-green-fg">&lt;ipython-input-64-3027cbe30ddf&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span>
-<span class="ansi-green-intense-fg ansi-bold">      1</span> x<span class="ansi-blue-fg">=</span><span class="ansi-cyan-fg">2</span>
-<span class="ansi-green-fg">----&gt; 2</span><span class="ansi-red-fg"> </span>x<span class="ansi-blue-fg">.</span>is_integer<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">)</span>
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-<span class="ansi-red-fg">AttributeError</span>: &#39;int&#39; object has no attribute &#39;is_integer&#39;</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span><span class="o">=</span><span class="mf">2.3</span>
-<span class="n">x</span><span class="o">.</span><span class="n">is_integer</span><span class="p">()</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">type</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">==</span><span class="nb">int</span>
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-<pre><code>str,int,float,list,dict</code></pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">isinstance</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="nb">list</span><span class="p">)</span>
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-    <span class="c1"># Returns true if x is integer</span>
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-<span class="ansi-cyan-fg">  File </span><span class="ansi-green-fg">&#34;&lt;ipython-input-14-dd49574938f5&gt;&#34;</span><span class="ansi-cyan-fg">, line </span><span class="ansi-green-fg">3</span>
-<span class="ansi-red-fg">    ....</span>
-        ^
-<span class="ansi-red-fg">SyntaxError</span><span class="ansi-red-fg">:</span> invalid syntax
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-<h3 id="Function-parameters">Function parameters<a class="anchor-link" href="#Function-parameters"> </a></h3><p>In python the function can be defined with mandatory and optional arguments</p>
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-    <span class="k">return</span> <span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">-</span><span class="n">c</span><span class="o">+</span><span class="n">d</span>
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-<ul>
-<li><code>a</code>,<code>b</code>: is like a <em>tuple</em> of mandatory arguments → <code>(a,b)</code></li>
-<li><code>c=2</code>,<code>d=3</code>: is like a <em>dictionary</em> of optional arguments → <code>{'c':2,'d':3}</code></li>
-</ul>
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-<h3 id="Function-definition-with-generic-mandatory-and-optional-arguments">Function definition with generic mandatory and optional arguments<a class="anchor-link" href="#Function-definition-with-generic-mandatory-and-optional-arguments"> </a></h3><p>We need to introduce another important concept about functions. There is a powerfull way to define a function with generic arguments:</p>
-<ol>
-<li>Instead of the mandatory arguments we can use the generic list pointer: <code>*args</code></li>
-<li>Instead of the optional arguments we can use the generic dictionary pointer: <code>**kwargs</code></li>
-</ol>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span><span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="k">if</span> <span class="n">args</span><span class="p">:</span>
-        <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;*args is the tuple of mandatory arguments: </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">args</span><span class="p">))</span>
-        <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;This allows to have a dynamic return according to input:&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-    <span class="k">if</span> <span class="n">kwargs</span><span class="p">:</span>
-        <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;*kwargs is the dictionary of optional arguments: </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">kwargs</span><span class="p">))</span>
-        <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;This allows to have a dynamic return according to input:&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">kwargs</span><span class="o">.</span><span class="n">keys</span><span class="p">())[</span><span class="mi">0</span><span class="p">]),</span><span class="nb">type</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">kwargs</span><span class="o">.</span><span class="n">values</span><span class="p">())[</span><span class="mi">0</span><span class="p">])</span>
-</pre></div>
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-<hr>
-<p><strong>Activity</strong>; Check the function with any type of mandatory or optional argument</p>
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-<ul>
-<li>Check the function without arguments</li>
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-<li>Check the function with  a mandatory argument</li>
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-<li>Check the function with several mandatory arguments</li>
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-<li>Check the function with several optional arguments</li>
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-<pre>*kwargs is the dictionary of optional arguments: {&#39;c&#39;: 3}
-This allows to have a dynamic return according to input:
-</pre>
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-<pre>
-<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
-<span class="ansi-red-fg">TypeError</span>                                 Traceback (most recent call last)
-<span class="ansi-green-fg">&lt;ipython-input-75-8a4d05152912&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span>
-<span class="ansi-green-fg">----&gt; 1</span><span class="ansi-red-fg"> </span>f<span class="ansi-blue-fg">(</span>c<span class="ansi-blue-fg">=</span><span class="ansi-cyan-fg">3</span><span class="ansi-blue-fg">)</span>
-
-<span class="ansi-green-fg">&lt;ipython-input-71-72e15b8fe5b9&gt;</span> in <span class="ansi-cyan-fg">f</span><span class="ansi-blue-fg">(*args, **kwargs)</span>
-<span class="ansi-green-intense-fg ansi-bold">      7</span>         print<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">&#39;*kwargs is the dictionary of optional arguments: {}&#39;</span><span class="ansi-blue-fg">.</span>format<span class="ansi-blue-fg">(</span>kwargs<span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">      8</span>         print<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">&#39;This allows to have a dynamic return according to input:&#39;</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-fg">----&gt; 9</span><span class="ansi-red-fg">         </span><span class="ansi-green-fg">return</span> type<span class="ansi-blue-fg">(</span>kwargs<span class="ansi-blue-fg">.</span>keys<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">[</span><span class="ansi-cyan-fg">0</span><span class="ansi-blue-fg">]</span><span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">,</span>type<span class="ansi-blue-fg">(</span>kwargs<span class="ansi-blue-fg">.</span>values<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">[</span><span class="ansi-cyan-fg">0</span><span class="ansi-blue-fg">]</span><span class="ansi-blue-fg">)</span>
-
-<span class="ansi-red-fg">TypeError</span>: &#39;dict_keys&#39; object does not support indexing</pre>
-</div>
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-<li>Check the function with one of the optional arguments equal to some list</li>
-</ul>
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-<ul>
-<li>Check the function with one string as mandatory argument, and with one of the optional arguments equal to some list</li>
-</ul>
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-<h3 id="Conversion-of-a-float-function-into-a-vectorized-function">Conversion of a float function into a vectorized function<a class="anchor-link" href="#Conversion-of-a-float-function-into-a-vectorized-function"> </a></h3><p>An important <code>numpy</code> function is:</p>
-
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-<p>Which convert a function which takes a float as an argument into a new function which que takes <code>numpy arrays</code> as an argument. The problem is that the converted function does not longer return a float when the argument is a float:</p>
-
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-<p>It is used as:</p>
-<div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="n">pyfunc</span><span class="p">,</span><span class="o">...</span><span class="p">)</span>
-</pre></div>
-<p><strong>Parameters</strong>:
-<code>pyfunc</code>: A python function or method.</p>
-<p>...</p>
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-<span class="n">sinv</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">)</span>
-<span class="n">sinv</span><span class="p">([</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.7</span><span class="p">])</span>
-</pre></div>
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-<pre>array([0.47942554, 0.64421769])</pre>
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-<h3 id="Vectorization-of-Scipy-method-derivative">Vectorization of Scipy method <code>derivative</code><a class="anchor-link" href="#Vectorization-of-Scipy-method-derivative"> </a></h3><p>The problem is that <code>derivative</code> only works for one a point</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">misc</span><span class="o">.</span><span class="n">derivative</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">dx</span><span class="o">=</span><span class="mf">1E-6</span><span class="p">)</span>
-</pre></div>
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-<pre>0.5403023058958567</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">misc</span><span class="o">.</span><span class="n">derivative</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">,[</span><span class="mi">1</span><span class="p">,</span><span class="mf">1.2</span><span class="p">],</span><span class="n">dx</span><span class="o">=</span><span class="mf">1E-6</span><span class="p">)</span>
-</pre></div>
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-<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
-<span class="ansi-red-fg">TypeError</span>                                 Traceback (most recent call last)
-<span class="ansi-green-fg">&lt;ipython-input-13-287829b0db34&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span>
-<span class="ansi-green-fg">----&gt; 1</span><span class="ansi-red-fg"> </span>misc<span class="ansi-blue-fg">.</span>derivative<span class="ansi-blue-fg">(</span>np<span class="ansi-blue-fg">.</span>sin<span class="ansi-blue-fg">,</span><span class="ansi-blue-fg">[</span><span class="ansi-cyan-fg">1</span><span class="ansi-blue-fg">,</span><span class="ansi-cyan-fg">1.2</span><span class="ansi-blue-fg">]</span><span class="ansi-blue-fg">,</span>dx<span class="ansi-blue-fg">=</span><span class="ansi-cyan-fg">1E-6</span><span class="ansi-blue-fg">)</span>
-
-<span class="ansi-green-fg">/usr/local/lib/python3.5/dist-packages/scipy/misc/common.py</span> in <span class="ansi-cyan-fg">derivative</span><span class="ansi-blue-fg">(func, x0, dx, n, args, order)</span>
-<span class="ansi-green-intense-fg ansi-bold">    116</span>     ho <span class="ansi-blue-fg">=</span> order <span class="ansi-blue-fg">&gt;&gt;</span> <span class="ansi-cyan-fg">1</span>
-<span class="ansi-green-intense-fg ansi-bold">    117</span>     <span class="ansi-green-fg">for</span> k <span class="ansi-green-fg">in</span> range<span class="ansi-blue-fg">(</span>order<span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">:</span>
-<span class="ansi-green-fg">--&gt; 118</span><span class="ansi-red-fg">         </span>val <span class="ansi-blue-fg">+=</span> weights<span class="ansi-blue-fg">[</span>k<span class="ansi-blue-fg">]</span><span class="ansi-blue-fg">*</span>func<span class="ansi-blue-fg">(</span>x0<span class="ansi-blue-fg">+</span><span class="ansi-blue-fg">(</span>k<span class="ansi-blue-fg">-</span>ho<span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">*</span>dx<span class="ansi-blue-fg">,</span><span class="ansi-blue-fg">*</span>args<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    119</span>     <span class="ansi-green-fg">return</span> val <span class="ansi-blue-fg">/</span> product<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">(</span>dx<span class="ansi-blue-fg">,</span><span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">*</span>n<span class="ansi-blue-fg">,</span>axis<span class="ansi-blue-fg">=</span><span class="ansi-cyan-fg">0</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    120</span> 
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-<span class="ansi-red-fg">TypeError</span>: can only concatenate list (not &#34;float&#34;) to list</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">array_derivate</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">,[</span><span class="mi">1</span><span class="p">,</span><span class="mf">1.2</span><span class="p">],</span><span class="n">dx</span><span class="o">=</span><span class="mf">1E-6</span><span class="p">)</span>
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-<pre>array([0.54030231, 0.36235775])</pre>
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-<pre>array(0.54030231)</pre>
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-<h2 id="Code-for-the-derivate-of-the-function-inside-a-full-range">Code for the derivate of the function inside a full range<a class="anchor-link" href="#Code-for-the-derivate-of-the-function-inside-a-full-range"> </a></h2><p>To recover the evaluation of a float into a float we can force the <code>IndexError</code> in order to use of the pure <code>scipy</code> <code>derivative</code> function when a float is given as input. The full implemention combine the three previos ingredients into a very compact pythonic-way function definition as shwon below</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">misc</span>
-<span class="k">def</span> <span class="nf">derivate</span><span class="p">(</span><span class="n">func</span><span class="p">,</span><span class="n">x0</span><span class="p">,</span><span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Vectorized replacement of scipy.misc derivative:</span>
-<span class="sd">        from scipy.misc import derivative</span>
-<span class="sd">    For usage check the derivative help, e.g, in jupyter: </span>
-<span class="sd">        from scipy.misc import derivative</span>
-<span class="sd">        derivative?</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="c1">#x0: can be an array or a list  </span>
-        <span class="n">nn</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="c1"># force error if float is used </span>
-        <span class="n">fp</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="n">misc</span><span class="o">.</span><span class="n">derivative</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
-        <span class="n">fp</span><span class="o">=</span><span class="n">misc</span><span class="o">.</span><span class="n">derivative</span>
-        
-    <span class="k">return</span> <span class="n">fp</span><span class="p">(</span><span class="n">func</span><span class="p">,</span><span class="n">x0</span><span class="p">,</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-</pre></div>
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-<p>which behaves exactly as the <code>scipy</code> <code>derivative</code> function when a float argument is used:</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">derivate</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">dx</span><span class="o">=</span><span class="mf">1E-6</span><span class="p">)</span>
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-</pre></div>
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-<pre>array([ 0.54030231, -0.41614684])</pre>
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-<p>and as a vectorized function when an array argument is used:</p>
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-<pre>(0.5403023058958567, -0.41614683654600526)</pre>
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-<pre>array([-7.77156117e+08, -1.11022302e+09])</pre>
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-<p>Let see now the implementation of the <code>derivate</code> function as full replacement of the <code>derivative</code> function:</p>
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-<h3 id="Test">Test<a class="anchor-link" href="#Test"> </a></h3><p>Let us check the implementation with the previous function</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">func</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span> <span class="mi">1</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
-</pre></div>
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-<p>but now evaluated for a list of values</p>
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-<span class="n">funcp</span><span class="o">=</span><span class="n">derivate</span><span class="p">(</span><span class="n">func</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">dx</span><span class="o">=</span><span class="mf">1E-3</span><span class="p">)</span>
-<span class="n">funcp</span>
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-<pre>array([0.21576117, 0.34935759, 0.41006125])</pre>
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-<p>For the prevous <code>X</code> array:</p>
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-<span class="n">xmin</span> <span class="o">=</span> <span class="mf">1.8</span>
-<span class="n">xmax</span> <span class="o">=</span> <span class="mf">2.2</span>
-
-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">,</span> <span class="mi">100</span> <span class="p">)</span>
-</pre></div>
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-<p>We have:</p>
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-<pre>(100,)</pre>
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-<span class="n">xmax</span> <span class="o">=</span> <span class="mf">2.2</span>
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-<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">,</span> <span class="mi">100</span> <span class="p">)</span>
-
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span> <span class="mi">1</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$f(x)$&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">derivate</span><span class="p">(</span><span class="n">func</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">dx</span><span class="o">=</span><span class="mf">1E-3</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$f&#39;(x)$&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">derivate</span><span class="p">(</span><span class="n">func</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">dx</span><span class="o">=</span><span class="mf">1E-3</span><span class="p">,</span><span class="n">n</span><span class="o">=</span><span class="mi">2</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;blue&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$f&#39;&#39;(x)$&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">&#39;best&#39;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;$x$&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
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-<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">,</span> <span class="mi">100</span> <span class="p">)</span>
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-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">derivate</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">X</span><span class="p">,</span><span class="n">dx</span><span class="o">=</span><span class="mf">1E-3</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$f&#39;(x)$&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="p">,</span> <span class="s1">&#39;b:&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$-\sin x$&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">10</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">derivate</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">X</span><span class="p">,</span><span class="n">dx</span><span class="o">=</span><span class="mf">1E-3</span><span class="p">,</span><span class="n">n</span><span class="o">=</span><span class="mi">4</span><span class="p">,</span><span class="n">order</span><span class="o">=</span><span class="mi">5</span><span class="p">),</span> <span class="s1">&#39;k:&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$f^{(iv)}(x)$&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">&#39;best&#39;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;$x$&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">misc</span><span class="o">.</span><span class="n">derivative</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="n">n</span><span class="o">=</span><span class="mi">4</span><span class="p">,</span><span class="n">dx</span><span class="o">=</span><span class="mf">1E-3</span><span class="p">,</span><span class="n">order</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
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-<pre>-1.3010426069826051e-06</pre>
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-<p><strong>Activity</strong>: Implement the full derivative function by using <code>isinstance()</code> instead of <code>try</code> and <code>except</code></p>
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-<span class="k">def</span> <span class="nf">derivate</span><span class="p">(</span><span class="n">func</span><span class="p">,</span><span class="n">x0</span><span class="p">,</span><span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Vectorized replacement of scipy.misc derivative:</span>
-<span class="sd">        from scipy.misc import derivative</span>
-<span class="sd">    For usage check the derivative help, e.g, in jupyter: </span>
-<span class="sd">        from scipy.misc import derivative</span>
-<span class="sd">        derivative?</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x0</span><span class="p">,</span><span class="nb">list</span><span class="p">):</span>
-        <span class="nb">print</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span>
-        <span class="n">fp</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="n">misc</span><span class="o">.</span><span class="n">derivative</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">fp</span><span class="o">=</span><span class="n">misc</span><span class="o">.</span><span class="n">derivative</span>
-    <span class="k">return</span> <span class="n">fp</span><span class="p">(</span><span class="n">func</span><span class="p">,</span><span class="n">x0</span><span class="p">,</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">type</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span> <span class="p">)</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">isinstance</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">]),</span><span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span> <span class="p">)</span>
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-<pre>True</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">derivate</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">]),</span><span class="n">dx</span><span class="o">=</span><span class="mf">1E-6</span><span class="p">)</span>
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-<pre>array([-0.90929743, -0.14112001])</pre>
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-<h2 id="Example:-Heat-transfer-in-a-1D-bar">Example: Heat transfer in a 1D bar<a class="anchor-link" href="#Example:-Heat-transfer-in-a-1D-bar"> </a></h2>
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-<p>Fourier's Law of thermal conduction describes the diffusion of heat. Situations in which there are gradients of heat, a flux that tends to homogenise the temperature arises as a consequence of collisions of particles within a body. The Fourier's Law is giving by</p>
-$$ q = -k\nabla T = -k\left( \frac{dT}{dx}\hat{i} + \frac{dT}{dy}\hat{j} + \frac{dT}{dz}\hat{k}\right)$$<p>where T is the temperature, $\nabla T$ its gradient and k is the material's conductivity. In the next example it is shown the magnitud of the heat flux in a 1D bar(wire).</p>
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-    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span>
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-<span class="k">def</span> <span class="nf">Temp</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span>
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-<span class="c1"># Points where function is known</span>
-<span class="n">Xn</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
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-<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">7</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Xn</span><span class="p">,</span><span class="n">derivate</span><span class="p">(</span><span class="n">Temp</span><span class="p">,</span><span class="n">Xn</span><span class="p">)</span><span class="o">*</span><span class="mi">10</span><span class="p">)</span>
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-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span> <span class="s2">&quot;$x$&quot;</span><span class="p">,</span><span class="n">fontsize</span> <span class="o">=</span><span class="mi">15</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span> <span class="sa">r</span><span class="s2">&quot;$\frac</span><span class="si">{dT}{dx}</span><span class="s2">$&quot;</span><span class="p">,</span><span class="n">fontsize</span> <span class="o">=</span><span class="mi">20</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span> <span class="s2">&quot; Magnitud heat flux transfer in 1D bar&quot;</span> <span class="p">)</span>
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-<pre>Text(0.5,1,&#39; Magnitud heat flux transfer in 1D bar&#39;)</pre>
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-<p>Construct a density map of the magnitud of the heat flux of a 2D bar. Consider the temperature profile as 
-$$ T(x,y) = x^3 + 3x-1+y^2  $$</p>
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-<h2 id="Activity">Activity<a class="anchor-link" href="#Activity"> </a></h2><p>The Poisson's equation relates the matter content of a body with the gravitational potential through the next equation</p>
-$$\nabla^2 \phi = 4\pi G \rho$$$$\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d\phi}{dr}\right)= 4\pi G \rho$$<p>where $\phi$ is the potential, $\rho$ the density and $G$ the gravitational constant.</p>
-<p>Taking <a href="https://raw.githubusercontent.com/sbustamante/ComputationalMethods/master/data/M1.00-STRUC.dat">these data</a> and using the three-point Midpoint formula, find the density field from the potential (seventh column in the file) and plot it against the radial coordinate. (<strong>Tip:</strong> Use $G=1$)</p>
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-<p>The radar stations A and B, separated by the distance a = 500 m, track the plane
-C by recording the angles $\alpha$ and $\beta$ at 1-second intervals. The successive readings are</p>
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-<p>calculate the speed v using the 3 point approximantion at t = 10 ,12 and 14 s. Calculate the x component of the acceleration of the plane at = 12 s. The coordinates of the plane can be shown to be</p>
-\begin{equation}
-x = a\frac{\tan \beta}{\tan \beta- \tan \alpha}\\
-y = a\frac{\tan \alpha\tan \beta}{\tan \beta- \tan \alpha}
-\end{equation}<p><img src="https://github.com/restrepo/ComputationalMethods/blob/master/material/figures/radar.png?raw=1"></p>
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-<h1 id="Numerical-Integration"><a href="./numerical-calculus-integration.ipynb">Numerical Integration</a><a class="anchor-link" href="#Numerical-Integration"> </a></h1>
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-<h3 id="One-implementation-of-derivative-algorithm">One implementation of <code>derivative</code> algorithm<a class="anchor-link" href="#One-implementation-of-derivative-algorithm"> </a></h3>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Function to evaluate</span>
-<span class="k">def</span> <span class="nf">function</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span> <span class="mi">1</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
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-<span class="c1">#X value</span>
-<span class="n">x0</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="n">xmin</span> <span class="o">=</span> <span class="mf">1.8</span>
-<span class="n">xmax</span> <span class="o">=</span> <span class="mf">2.2</span>
-<span class="c1">#h step</span>
-<span class="n">hs</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.05</span><span class="p">]</span>
-
-<span class="c1">#Calculating derivatives</span>
-<span class="n">dfs</span> <span class="o">=</span> <span class="p">[]</span>
-<span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">hs</span><span class="p">:</span>
-    <span class="n">dfs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="p">(</span><span class="n">function</span><span class="p">(</span><span class="n">x0</span><span class="o">+</span><span class="n">h</span><span class="p">)</span><span class="o">-</span><span class="n">function</span><span class="p">(</span><span class="n">x0</span><span class="p">))</span><span class="o">/</span><span class="n">h</span> <span class="p">)</span>
-    
-<span class="c1">#Plotting</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">8</span><span class="p">)</span> <span class="p">)</span>
-<span class="c1">#X array</span>
-<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">,</span> <span class="mi">100</span> <span class="p">)</span>
-<span class="n">Y</span> <span class="o">=</span> <span class="n">function</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;function&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">10</span> <span class="p">)</span>
-<span class="c1">#Slopes</span>
-<span class="n">Xslp</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">x0</span><span class="p">,</span><span class="mi">3</span><span class="p">]</span>
-<span class="n">Yslp</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
-<span class="k">for</span> <span class="n">df</span><span class="p">,</span> <span class="n">h</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">dfs</span><span class="p">,</span> <span class="n">hs</span><span class="p">):</span>
-    <span class="c1">#First point</span>
-    <span class="n">Yslp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">function</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span><span class="o">+</span><span class="n">df</span><span class="o">*</span><span class="p">(</span><span class="n">Xslp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="n">Xslp</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-    <span class="c1">#Second point</span>
-    <span class="n">Yslp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">function</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span>
-    <span class="c1">#Third point</span>
-    <span class="n">Yslp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">function</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span><span class="o">+</span><span class="n">df</span><span class="o">*</span><span class="p">(</span><span class="n">Xslp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">Xslp</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-    <span class="c1">#Plotting this slope</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">Xslp</span><span class="p">,</span> <span class="n">Yslp</span><span class="p">,</span> <span class="n">linewidth</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;slope$=</span><span class="si">%1.2f</span><span class="s2">$ for $h=</span><span class="si">%1.2f</span><span class="s2">$&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">df</span><span class="p">,</span><span class="n">h</span><span class="p">)</span> <span class="p">)</span>
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-<span class="c1">#Format</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;y&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span> <span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">1.15</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span> <span class="n">loc</span> <span class="o">=</span> <span class="s2">&quot;upper left&quot;</span> <span class="p">)</span>
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-<p>A generalization of the previous formula is given by the (n+1)-point formula, where first-order derivatives are calculated using more than one point, what makes it a much better approximation for many problems. it is controled by the <code>order</code> option of the <code>derivative</code> function of <code>scipy.misc</code></p>
-<p><strong>Theorem</strong></p>
-<p>For a function $f(x)$ such that $f(x)\in C^{n+1}[a,b]$, the next expression is always satisfied</p>
-$$f(x) = P(x) + \frac{f^{(n+1)}(\xi(x))}{(n+1)!}(x-x_0)(x-x_1)\cdots(x-x_n)$$<p>where $\{x_i\}_i$ is a set of point where the function is mapped, $\xi(x)$ is some function of $x$ such that $\xi\in[a,b]$, and $P(x)$ is the associated Lagrange interpolant polynomial.</p>
-<p>As $n$ becomes higher, the approximation should be better as the error term becomes neglectable.</p>
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-<p>Taking the previous expression, and differenciating, we obtain</p>
-$$f(x) = \sum_{k=0}^n f(x_k)L_{n,k}(x) + \frac{(x-x_0)(x-x_1)\cdots(x-x_n)}{(n+1)!}f^{(n+1)}(\xi(x))$$$$f'(x_j) = \sum_{k=0}^n f(x_k)L'_{n,k}(x_j) + \frac{f^{(n+1)}(\xi(x_j))}{(n+1)!} \prod_{k=0,k\neq j}^{n}(x_j-x_k)$$<p>where $L_{n,k}$ is the $k$-th Lagrange basis functions for $n$ points, $L'_{n,k}$ is its first derivative.</p>
-<p>Note that the last expressions is evaluated in $x_j$ rather than a general $x$ value, the cause of this is because this expression is not longer valid for another value not within the set $\{x_i\}_i$, however this is not an inconvenient when handling real applications.</p>
-<p>This formula constitutes the <strong>(n+1)-point approximation</strong> and it comprises a generalization of almost all the existing schemes to differentiate numerically. Next, we shall derive some very used formulas.</p>
-<p>For example, the form that takes this derivative polynomial for 3 points $(x_i,y_i)$ is the following</p>
-$$f'(x_j) = f(x_0)\left[ \frac{2x_j-x_1-x_2}{(x_0-x_1)(x_0-x_2)}\right] + 
-f(x_1)\left[ \frac{2x_j-x_0-x_2}{(x_1-x_0)(x_1-x_2)}\right] +
-f(x_2)\left[ \frac{2x_j-x_0-x_1}{(x_2-x_0)(x_2-x_1)}\right] $$<p>
-$$\hspace{2cm} + \frac{1}{6} f^{(3)}(\epsilon_j) \prod_{k=0,k\neq j}^{n}(x_j-x_k)$$</p>
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-<p>Endpoint formulas are based on evaluating the derivative at the first of a set of points, i.e., if we want to evaluate $f'(x)$ at $x_i$, we then need $(x_i$, $x_{i+1}=x_i+h$, $x_{i+2}=x_i+2h$, $\cdots)$. For the sake of simplicity, it is usually assumed that the set $\{x_i\}_i$ is equally spaced such that $x_k = x_0+k\cdot h$.</p>
-<p><strong>Three-point Endpoint Formula</strong></p>
-$$f'(x_i) = \frac{1}{2h}[-3f(x_i)+4f(x_i+h)-f(x_i+2h)] + \frac{h^2}{3}f^{(3)}(\xi)$$<p>with $\xi\in[x_i,x_i+2h]$</p>
-<p><strong>Five-point Endpoint Formula</strong></p>
-$$f'(x_i) = \frac{1}{12h}[-25f(x_i)+48f(x_i+h)-36f(x_i+2h)+16f(x_i+3h)-3f(x_i+4h)] + \frac{h^4}{5}f^{(5)}(\xi)$$<p>with $\xi\in[x_i,x_i+4h]$</p>
-<p>Endpoint formulas are especially useful near to the end of a set of points, where no further points exist.</p>
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-<p>On the other hand, Midpoint formulas are based on evaluating the derivative at the middle of a set of points, i.e., if we want to evaluate $f'(x)$ at $x_i$, we then need $(\cdots$, $x_{i-2} = x_i - 2h$, $x_{i-1} = x_i - h$, $x_i$, $x_{i+1}=x_i+h$, $x_{i+2}=x_i+2h$, $\cdots)$.</p>
-<p><strong>Three-point Midpoint Formula</strong></p>
-$$f'(x_i) = \frac{1}{2h}[f(x_i+h)-f(x_i-h)] + \frac{h^2}{6}f^{(3)}(\xi)$$<p>with $\xi\in[x_i-h,x_i+h]$</p>
-<p><strong>Five-point Midpoint Formula</strong></p>
-$$f'(x_i) = \frac{1}{12h}[f(x_i-2h)-8f(x_i-h)+8f(x_i+h)-f(x_i+2h)] + \frac{h^4}{30}f^{(5)}(\xi)$$<p>with $\xi\in[x_i-2h,x_i+2h]$</p>
-<p>As Midpoint formulas required one iteration less than Endpoint ones, they are more often used for numerical applications. Furthermore, the round-off error is smaller as well. However, near to the end of a set of points, they are no longer useful as no further points exists, and Endpoint formulas are preferable.</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Temperature profile </span>
-<span class="k">def</span> <span class="nf">Temp</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span>
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-<span class="c1">#Derivative three end point </span>
-
-<span class="k">def</span> <span class="nf">TEP</span><span class="p">(</span> <span class="n">Yn</span><span class="p">,</span><span class="n">i</span><span class="p">,</span> <span class="n">h</span><span class="o">=</span><span class="mf">0.01</span><span class="p">,</span><span class="n">right</span><span class="o">=</span><span class="mi">0</span> <span class="p">):</span>
-    <span class="n">suma</span> <span class="o">=</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">Yn</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">+</span><span class="mi">4</span><span class="o">*</span><span class="n">Yn</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">right</span><span class="o">*</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">Yn</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">right</span><span class="o">*</span><span class="mi">2</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">suma</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">h</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">right</span><span class="p">)</span>
-
-<span class="c1">#Derivative mid point </span>
-<span class="k">def</span> <span class="nf">TMP</span><span class="p">(</span> <span class="n">Ynh</span><span class="p">,</span><span class="n">Ynmh</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="mf">0.01</span> <span class="p">):</span>    
-    <span class="k">return</span> <span class="p">(</span><span class="n">Ynh</span><span class="o">-</span><span class="n">Ynmh</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">h</span><span class="p">)</span>
-
-<span class="c1"># Points where function is known</span>
-<span class="n">Xn</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
-<span class="n">Tn</span> <span class="o">=</span> <span class="n">Temp</span><span class="p">(</span><span class="n">Xn</span><span class="p">)</span>
-<span class="c1">#Magnitude of heat flux array</span>
-<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">Xn</span><span class="p">))</span>
-<span class="c1">#Left end derivative</span>
-<span class="n">Q</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">TEP</span><span class="p">(</span><span class="n">Tn</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-
-<span class="c1">#Mid point derivatives</span>
-<span class="n">index</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">Xn</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">xrange</span><span class="p">(</span> <span class="mi">1</span><span class="p">,</span><span class="n">index</span> <span class="p">):</span>    
-    <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span>  <span class="n">TMP</span><span class="p">(</span> <span class="n">Tn</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">],</span><span class="n">Tn</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="p">)</span>
-
-<span class="c1">#Right end derivative      </span>
-<span class="n">Q</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">TEP</span><span class="p">(</span> <span class="n">Tn</span><span class="p">,</span><span class="n">index</span><span class="p">,</span><span class="n">right</span><span class="o">=</span><span class="mi">1</span> <span class="p">)</span> 
-
-<span class="c1">#Plotting </span>
-<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">7</span><span class="p">)</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Xn</span><span class="p">,</span><span class="n">Q</span><span class="p">)</span>
-
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span> <span class="s2">&quot;x&quot;</span><span class="p">,</span><span class="n">fontsize</span> <span class="o">=</span><span class="mi">15</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span> <span class="s2">&quot;$</span><span class="se">\\</span><span class="s2">frac</span><span class="si">{dT}{dx}</span><span class="s2">$&quot;</span><span class="p">,</span><span class="n">fontsize</span> <span class="o">=</span><span class="mi">20</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span> <span class="s2">&quot; Magnitud heat flux transfer in 1D bar&quot;</span> <span class="p">)</span>
-</pre></div>
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-  Fixed point
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-    <main class="jupyter-page">
-    <div id="page-info"><div id="page-title">Fixed point</div>
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-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/one-variable-equations-fixed-point.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<h1 id="One-Variable-Equations.-Part-II">One Variable Equations. Part II<a class="anchor-link" href="#One-Variable-Equations.-Part-II"> </a></h1><p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/one-variable-equations-fixed-point.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<hr>
-<ul>
-<li><a href="./one-variable-equations.ipynb#Bisection-Method">Bisection Method</a> <ul>
-<li><a href="./one-variable-equations.ipynb#Steps-BM">Steps</a></li>
-<li><a href="./one-variable-equations#Stop-condition-BM">Stop condition</a></li>
-<li><a href="./one-variable-equations#Error-analysis-BM">Error analysis</a></li>
-<li><a href="./one-variable-equations#Example-1">Example 1</a></li>
-<li><a href="./one-variable-equations#Example-2">Example 2</a></li>
-</ul>
-</li>
-<li><a href="#Fixed-point-Iteration">Fixed-point Iteration</a><ul>
-<li><a href="#Steps-FP">Steps</a></li>
-<li><a href="#Example-3">Example 3</a></li>
-<li><a href="#Stop-condition-FP">Stop condition</a></li>
-<li><a href="#Example-4">Example 4</a></li>
-<li><a href="#ACTIVITY-FP">Activity</a></li>
-</ul>
-</li>
-<li><a href="#Newton-Raphson-Method">Newton-Raphson Method</a><ul>
-<li><a href="#Derivation-NM">Derivation</a></li>
-<li><a href="#Steps-NM">Steps</a></li>
-<li><a href="#Example-5">Example 5</a></li>
-<li><a href="#Stop-condition-NM">Stop condition</a></li>
-<li><a href="#Convergence-NM">Convergence</a></li>
-</ul>
-</li>
-<li><a href="#Secant-Method">Secant Method</a><ul>
-<li><a href="#Derivation-SM">Derivation</a></li>
-<li><a href="#Steps-SM">Steps</a></li>
-</ul>
-</li>
-</ul>
-<h2 id="Bibliography">Bibliography<a class="anchor-link" href="#Bibliography"> </a></h2><p>[1] <a href="https://drive.google.com/file/d/0BxoOXsn2EUNIQUdFVkctR2xWRUk/view?usp=sharing">Kiusalaas, Numerical Methods in Engineering with Python</a><br/>
-[2] <a href="https://drive.google.com/file/d/0BxoOXsn2EUNIekRUMFVPYXVsSTg/view?usp=sharing">Jensen, Computational_Physics</a>.  Companion repos: <a href="https://github.com/mhjensen">https://github.com/mhjensen</a> <a href="http://compphysics.github.io/ComputationalPhysics/doc/web/course">Web page</a>  <br/>
-[3] E. Ayres, <a href="http://bit.ly/CPwPython">Computational Physics With Python</a><br/></p>
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-</pre>
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-<pre>/usr/local/lib/python3.7/dist-packages/IPython/core/magics/pylab.py:160: UserWarning: pylab import has clobbered these variables: [&#39;rc&#39;, &#39;f&#39;]
-`%matplotlib` prevents importing * from pylab and numpy
-  &#34;\n`%matplotlib` prevents importing * from pylab and numpy&#34;
-</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">animation</span><span class="p">,</span> <span class="n">rc</span>
-<span class="n">rc</span><span class="p">(</span><span class="s1">&#39;animation&#39;</span><span class="p">,</span> <span class="n">html</span><span class="o">=</span><span class="s1">&#39;jshtml&#39;</span><span class="p">)</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">integrate</span>
-<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">optimize</span>
-</pre></div>
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-<h1 id="Fixed-point-Iteration">Fixed-point Iteration<a class="anchor-link" href="#Fixed-point-Iteration"> </a></h1>
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-<p>Although in many cases the use of Bisection is more than enough, there are some pathological situations where the use of more advanced methods is required.
-One of the advantages featured by this method is one does not have to give an interval where the solution is within, instead, from a seed the algorithm will converge towards the required solution.</p>
-
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-<ol>
-<li>Take your function $f(x)$ and rewrite it like 
-$$g(x) = x - f(x)\,.$$</li>
-</ol>
-<p>In this way, find the root of $f(x_0)$ is equivalent to find the value when 
-$$g(x_0)=x_0\,.$$
-This $x_0$ is called the fixed-point of $g(x)$. Note that $g(x)|_{f=0}=x$ is just and straight line of slope one and zero intercept. Therefore, the fixed points corresponds to the points where $g(x)$ crosses that straight line.</p>
-<ol>
-<li>Give a guest to the solution (root of $f(x)$). This value would be the seed $p_0$, and correspond to the point: $(p_0,g(p_0))$
-<img src="https://raw.githubusercontent.com/restrepo/ComputationalMethods/master/material/fp.svg?sanitize=true" alt="IMG"></li>
-<li>Check the distance to the straight line by goint to the point: $(p_0,g(p_0))$
-<img src="https://raw.githubusercontent.com/restrepo/ComputationalMethods/master/material/fp1.svg?sanitize=true" alt="IMG"></li>
-<li>Go to the point on the straight line: $p_1 = g(p_0)$ →  $(g(p_0),g(p_0))$
-<img src="https://raw.githubusercontent.com/restrepo/ComputationalMethods/master/material/fp2.svg?sanitize=true" alt="IMG">   </li>
-<li>The next guest to the solution will be given by $p_1$ →  $(p_1,g(p_1))$. 
-<img src="https://raw.githubusercontent.com/restrepo/ComputationalMethods/master/material/fp3.svg?sanitize=true" alt="IMG">   </li>
-<li>Compare $p_1$ with $g(p_1)$: If the stop condition is not satisfied, then repeat step 2 but now with $p_1$.</li>
-<li>The End!</li>
-</ol>
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-<p><strong>Example</strong>: Find the roots of the next function:</p>
-$$f(x) = \frac{x^2-1}{3}\,.$$<p>We define</p>
-<p><strong>Step 1:</strong></p>
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-    <span class="k">return</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mf">3.0</span>
-
-<span class="n">g</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">-</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-</pre></div>
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-<p><strong>Step 2:</strong></p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p0</span> <span class="o">=</span> <span class="mf">0.1</span>
-<span class="p">(</span><span class="n">px</span><span class="p">,</span><span class="n">py</span><span class="p">)</span><span class="o">=</span><span class="p">(</span><span class="n">p0</span><span class="p">,</span><span class="n">g</span><span class="p">(</span><span class="n">p0</span><span class="p">))</span>
-<span class="p">(</span><span class="n">px</span><span class="p">,</span><span class="n">py</span><span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="p">(</span><span class="n">px</span><span class="p">,</span><span class="n">py</span><span class="p">)</span><span class="o">=</span><span class="p">(</span><span class="n">g</span><span class="p">(</span><span class="n">p0</span><span class="p">),</span> <span class="n">g</span><span class="p">(</span><span class="n">p0</span><span class="p">)</span> <span class="p">)</span>
-<span class="p">(</span><span class="n">px</span><span class="p">,</span><span class="n">py</span><span class="p">)</span>
-</pre></div>
-
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-<pre>(0.43000000000000005, 0.43000000000000005)</pre>
-</div>
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-</div>
-</div>
-</div>
-</div>
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-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Step 4</strong>: Suggest a solution:</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p1</span><span class="o">=</span><span class="n">g</span><span class="p">(</span><span class="n">p0</span><span class="p">)</span>
-<span class="p">(</span><span class="n">px</span><span class="p">,</span><span class="n">py</span><span class="p">)</span><span class="o">=</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">g</span><span class="p">(</span><span class="n">p1</span><span class="p">)</span> <span class="p">)</span>
-<span class="p">(</span><span class="n">px</span><span class="p">,</span><span class="n">py</span><span class="p">)</span>
-</pre></div>
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-<pre>(0.43000000000000005, 0.7017)</pre>
-</div>
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-</div>
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-</div>
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-</div>
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><img src="https://raw.githubusercontent.com/restrepo/ComputationalMethods/master/material/figures/fp3.png" alt=""></p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Step 5</strong>: Since $p_1$ is still quite differente from $g(p_1)$ we need to continue with another iteration</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Implementation of algorithm</p>
-
-</div>
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-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Defining Fixed-point iteration function</span>
-<span class="k">def</span> <span class="nf">FixedPoint_Animation</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">pini</span><span class="p">,</span> <span class="n">Nmax</span><span class="p">,</span> <span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span> <span class="p">):</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">-</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="c1">#Initial condition</span>
-    <span class="n">pi</span> <span class="o">=</span> <span class="p">[</span><span class="n">pini</span><span class="p">,]</span>
-    <span class="n">px</span> <span class="o">=</span> <span class="p">[</span><span class="n">pini</span><span class="p">,</span><span class="n">pini</span><span class="p">,]</span>
-    <span class="n">py</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,]</span>
-    <span class="c1">#Iterations</span>
-    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Nmax</span><span class="o">+</span><span class="mi">3</span><span class="p">):</span>
-        <span class="n">pi</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">g</span><span class="p">(</span><span class="n">pi</span><span class="p">[</span><span class="n">n</span><span class="p">])</span> <span class="p">)</span>
-        <span class="n">px</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">g</span><span class="p">(</span><span class="n">pi</span><span class="p">[</span><span class="n">n</span><span class="p">])</span> <span class="p">)</span>
-        <span class="n">px</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">g</span><span class="p">(</span><span class="n">pi</span><span class="p">[</span><span class="n">n</span><span class="p">])</span> <span class="p">)</span>
-        <span class="n">py</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">g</span><span class="p">(</span><span class="n">pi</span><span class="p">[</span><span class="n">n</span><span class="p">])</span> <span class="p">)</span>
-        <span class="n">py</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">g</span><span class="p">(</span><span class="n">pi</span><span class="p">[</span><span class="n">n</span><span class="p">])</span> <span class="p">)</span>
-    
-    <span class="n">py</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">g</span><span class="p">(</span><span class="n">pi</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span> <span class="p">)</span>
-    <span class="n">pi</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="n">pi</span> <span class="p">)</span>
-    <span class="n">px</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="n">px</span> <span class="p">)</span>
-    <span class="n">py</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="n">py</span> <span class="p">)</span>
-    
-    <span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Result:&quot;</span><span class="p">,</span> <span class="n">pi</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-    
-    <span class="c1">#Array X-axis</span>
-    <span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">xmin</span><span class="p">,</span><span class="n">xmax</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
-    
-    <span class="c1">#Initializing Figure</span>
-    <span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">7</span><span class="p">)</span> <span class="p">)</span>
-    <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
-    <span class="c1">#Graphic iterations</span>
-    <span class="n">fixedpoint</span><span class="p">,</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="p">[],</span> <span class="p">[],</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="n">linewidth</span> <span class="o">=</span> <span class="mi">3</span> <span class="p">)</span>
-    <span class="c1">#Function g</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">g</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;green&quot;</span><span class="p">,</span> <span class="n">linewidth</span> <span class="o">=</span> <span class="mi">2</span> <span class="p">)</span>
-    <span class="c1">#Identity funcion</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;blue&quot;</span><span class="p">,</span> <span class="n">linewidth</span> <span class="o">=</span> <span class="mi">2</span> <span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span> <span class="p">(</span><span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">)</span> <span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span> <span class="p">(</span><span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">)</span> <span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span> <span class="s2">&quot;X axis&quot;</span> <span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span> <span class="s2">&quot;Y axis&quot;</span> <span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span> <span class="s2">&quot;Fixed-Point iteration&quot;</span> <span class="p">)</span>
-        
-    <span class="k">def</span> <span class="nf">init</span><span class="p">():</span>
-        <span class="n">fixedpoint</span><span class="o">.</span><span class="n">set_data</span><span class="p">([],</span> <span class="p">[])</span>
-        <span class="k">return</span> <span class="n">fixedpoint</span><span class="p">,</span>
-    
-    <span class="k">def</span> <span class="nf">animate</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-        <span class="c1">#Setting new data</span>
-        <span class="n">fixedpoint</span><span class="o">.</span><span class="n">set_data</span><span class="p">(</span> <span class="n">px</span><span class="p">[:</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="p">],</span> <span class="n">py</span><span class="p">[:</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="p">]</span> <span class="p">)</span>
-        <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span> <span class="s2">&quot;Fixed-Point. Iteration </span><span class="si">%d</span><span class="s2">&quot;</span><span class="o">%</span><span class="k">i</span> )
-        <span class="k">return</span> <span class="n">fixedpoint</span><span class="p">,</span>
-    
-    <span class="k">return</span> <span class="n">animation</span><span class="o">.</span><span class="n">FuncAnimation</span><span class="p">(</span><span class="n">fig</span><span class="p">,</span> <span class="n">animate</span><span class="p">,</span> <span class="n">init_func</span><span class="o">=</span><span class="n">init</span><span class="p">,</span><span class="n">frames</span><span class="o">=</span><span class="n">Nmax</span><span class="p">,</span> <span class="n">interval</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span> <span class="n">blit</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Find the roots of the next functions:</p>
-$$f_1(x) = \frac{x^2-1}{3}$$<p>using Fixed-Point iteration.</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">f1</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mf">3.0</span>
-
-<span class="n">FixedPoint_Animation</span><span class="p">(</span> <span class="n">f1</span><span class="p">,</span> <span class="n">pini</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">,</span> <span class="n">Nmax</span> <span class="o">=</span> <span class="mi">15</span><span class="p">,</span> <span class="n">xmin</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">xmax</span> <span class="o">=</span> <span class="mf">1.2</span> <span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
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-<pre>Result: 0.9999999890733635
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-
-<link rel="stylesheet"
-href="https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/css/font-awesome.min.css">
-<script language="javascript">
-  function isInternetExplorer() {
-    ua = navigator.userAgent;
-    /* MSIE used to detect old browsers and Trident used to newer ones*/
-    return ua.indexOf("MSIE ") > -1 || ua.indexOf("Trident/") > -1;
-  }
-
-  /* Define the Animation class */
-  function Animation(frames, img_id, slider_id, interval, loop_select_id){
-    this.img_id = img_id;
-    this.slider_id = slider_id;
-    this.loop_select_id = loop_select_id;
-    this.interval = interval;
-    this.current_frame = 0;
-    this.direction = 0;
-    this.timer = null;
-    this.frames = new Array(frames.length);
-
-    for (var i=0; i<frames.length; i++)
-    {
-     this.frames[i] = new Image();
-     this.frames[i].src = frames[i];
-    }
-    var slider = document.getElementById(this.slider_id);
-    slider.max = this.frames.length - 1;
-    if (isInternetExplorer()) {
-        // switch from oninput to onchange because IE <= 11 does not conform
-        // with W3C specification. It ignores oninput and onchange behaves
-        // like oninput. In contrast, Mircosoft Edge behaves correctly.
-        slider.setAttribute('onchange', slider.getAttribute('oninput'));
-        slider.setAttribute('oninput', null);
-    }
-    this.set_frame(this.current_frame);
-  }
-
-  Animation.prototype.get_loop_state = function(){
-    var button_group = document[this.loop_select_id].state;
-    for (var i = 0; i < button_group.length; i++) {
-        var button = button_group[i];
-        if (button.checked) {
-            return button.value;
-        }
-    }
-    return undefined;
-  }
-
-  Animation.prototype.set_frame = function(frame){
-    this.current_frame = frame;
-    document.getElementById(this.img_id).src =
-            this.frames[this.current_frame].src;
-    document.getElementById(this.slider_id).value = this.current_frame;
-  }
-
-  Animation.prototype.next_frame = function()
-  {
-    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));
-  }
-
-  Animation.prototype.previous_frame = function()
-  {
-    this.set_frame(Math.max(0, this.current_frame - 1));
-  }
-
-  Animation.prototype.first_frame = function()
-  {
-    this.set_frame(0);
-  }
-
-  Animation.prototype.last_frame = function()
-  {
-    this.set_frame(this.frames.length - 1);
-  }
-
-  Animation.prototype.slower = function()
-  {
-    this.interval /= 0.7;
-    if(this.direction > 0){this.play_animation();}
-    else if(this.direction < 0){this.reverse_animation();}
-  }
-
-  Animation.prototype.faster = function()
-  {
-    this.interval *= 0.7;
-    if(this.direction > 0){this.play_animation();}
-    else if(this.direction < 0){this.reverse_animation();}
-  }
-
-  Animation.prototype.anim_step_forward = function()
-  {
-    this.current_frame += 1;
-    if(this.current_frame < this.frames.length){
-      this.set_frame(this.current_frame);
-    }else{
-      var loop_state = this.get_loop_state();
-      if(loop_state == "loop"){
-        this.first_frame();
-      }else if(loop_state == "reflect"){
-        this.last_frame();
-        this.reverse_animation();
-      }else{
-        this.pause_animation();
-        this.last_frame();
-      }
-    }
-  }
-
-  Animation.prototype.anim_step_reverse = function()
-  {
-    this.current_frame -= 1;
-    if(this.current_frame >= 0){
-      this.set_frame(this.current_frame);
-    }else{
-      var loop_state = this.get_loop_state();
-      if(loop_state == "loop"){
-        this.last_frame();
-      }else if(loop_state == "reflect"){
-        this.first_frame();
-        this.play_animation();
-      }else{
-        this.pause_animation();
-        this.first_frame();
-      }
-    }
-  }
-
-  Animation.prototype.pause_animation = function()
-  {
-    this.direction = 0;
-    if (this.timer){
-      clearInterval(this.timer);
-      this.timer = null;
-    }
-  }
-
-  Animation.prototype.play_animation = function()
-  {
-    this.pause_animation();
-    this.direction = 1;
-    var t = this;
-    if (!this.timer) this.timer = setInterval(function() {
-        t.anim_step_forward();
-    }, this.interval);
-  }
-
-  Animation.prototype.reverse_animation = function()
-  {
-    this.pause_animation();
-    this.direction = -1;
-    var t = this;
-    if (!this.timer) this.timer = setInterval(function() {
-        t.anim_step_reverse();
-    }, this.interval);
-  }
-</script>
-
-<style>
-.animation {
-    display: inline-block;
-    text-align: center;
-}
-input[type=range].anim-slider {
-    width: 374px;
-    margin-left: auto;
-    margin-right: auto;
-}
-.anim-buttons {
-    margin: 8px 0px;
-}
-.anim-buttons button {
-    padding: 0;
-    width: 36px;
-}
-.anim-state label {
-    margin-right: 8px;
-}
-.anim-state input {
-    margin: 0;
-    vertical-align: middle;
-}
-</style>
-
-<div class="animation">
-  <img id="_anim_img8fb0cf15a17146bf8e9a1a978c5a9bcb">
-  <div class="anim-controls">
-    <input id="_anim_slider8fb0cf15a17146bf8e9a1a978c5a9bcb" type="range" class="anim-slider"
-           name="points" min="0" max="1" step="1" value="0"
-           oninput="anim8fb0cf15a17146bf8e9a1a978c5a9bcb.set_frame(parseInt(this.value));"></input>
-    <div class="anim-buttons">
-      <button title="Decrease speed" onclick="anim8fb0cf15a17146bf8e9a1a978c5a9bcb.slower()">
-          <i class="fa fa-minus"></i></button>
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-        <i class="fa fa-fast-backward"></i></button>
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-          <i class="fa fa-step-backward"></i></button>
-      <button title="Play backwards" onclick="anim8fb0cf15a17146bf8e9a1a978c5a9bcb.reverse_animation()">
-          <i class="fa fa-play fa-flip-horizontal"></i></button>
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-          <i class="fa fa-play"></i></button>
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-          <i class="fa fa-step-forward"></i></button>
-      <button title="Last frame" onclick="anim8fb0cf15a17146bf8e9a1a978c5a9bcb.last_frame()">
-          <i class="fa fa-fast-forward"></i></button>
-      <button title="Increase speed" onclick="anim8fb0cf15a17146bf8e9a1a978c5a9bcb.faster()">
-          <i class="fa fa-plus"></i></button>
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-      <input type="radio" name="state" value="loop" id="_anim_radio2_8fb0cf15a17146bf8e9a1a978c5a9bcb"
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-      <label for="_anim_radio2_8fb0cf15a17146bf8e9a1a978c5a9bcb">Loop</label>
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-      <label for="_anim_radio3_8fb0cf15a17146bf8e9a1a978c5a9bcb">Reflect</label>
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-</div>
-
-
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-ghcREQlAKngREZEA9P9S85QYf4KPRwAAAABJRU5ErkJggg==\
-"
-
-
-    /* set a timeout to make sure all the above elements are created before
-       the object is initialized. */
-    setTimeout(function() {
-        anim8fb0cf15a17146bf8e9a1a978c5a9bcb = new Animation(frames, img_id, slider_id, 500.0,
-                                 loop_select_id);
-    }, 0);
-  })()
-</script>
-
-</div>
-
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/one-variable-equations-fixed-point_24_2.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Ejemplo 3</strong>:
-$$f_2(x) = x-\cos x$$</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">f2</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">x</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-<span class="n">FixedPoint_Animation</span><span class="p">(</span> <span class="n">f2</span><span class="p">,</span> <span class="n">pini</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">,</span> <span class="n">Nmax</span> <span class="o">=</span> <span class="mi">15</span><span class="p">,</span> <span class="n">xmin</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">xmax</span> <span class="o">=</span> <span class="mf">1.2</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>Result: 0.7387663516682054
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-
-<link rel="stylesheet"
-href="https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/css/font-awesome.min.css">
-<script language="javascript">
-  function isInternetExplorer() {
-    ua = navigator.userAgent;
-    /* MSIE used to detect old browsers and Trident used to newer ones*/
-    return ua.indexOf("MSIE ") > -1 || ua.indexOf("Trident/") > -1;
-  }
-
-  /* Define the Animation class */
-  function Animation(frames, img_id, slider_id, interval, loop_select_id){
-    this.img_id = img_id;
-    this.slider_id = slider_id;
-    this.loop_select_id = loop_select_id;
-    this.interval = interval;
-    this.current_frame = 0;
-    this.direction = 0;
-    this.timer = null;
-    this.frames = new Array(frames.length);
-
-    for (var i=0; i<frames.length; i++)
-    {
-     this.frames[i] = new Image();
-     this.frames[i].src = frames[i];
-    }
-    var slider = document.getElementById(this.slider_id);
-    slider.max = this.frames.length - 1;
-    if (isInternetExplorer()) {
-        // switch from oninput to onchange because IE <= 11 does not conform
-        // with W3C specification. It ignores oninput and onchange behaves
-        // like oninput. In contrast, Mircosoft Edge behaves correctly.
-        slider.setAttribute('onchange', slider.getAttribute('oninput'));
-        slider.setAttribute('oninput', null);
-    }
-    this.set_frame(this.current_frame);
-  }
-
-  Animation.prototype.get_loop_state = function(){
-    var button_group = document[this.loop_select_id].state;
-    for (var i = 0; i < button_group.length; i++) {
-        var button = button_group[i];
-        if (button.checked) {
-            return button.value;
-        }
-    }
-    return undefined;
-  }
-
-  Animation.prototype.set_frame = function(frame){
-    this.current_frame = frame;
-    document.getElementById(this.img_id).src =
-            this.frames[this.current_frame].src;
-    document.getElementById(this.slider_id).value = this.current_frame;
-  }
-
-  Animation.prototype.next_frame = function()
-  {
-    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));
-  }
-
-  Animation.prototype.previous_frame = function()
-  {
-    this.set_frame(Math.max(0, this.current_frame - 1));
-  }
-
-  Animation.prototype.first_frame = function()
-  {
-    this.set_frame(0);
-  }
-
-  Animation.prototype.last_frame = function()
-  {
-    this.set_frame(this.frames.length - 1);
-  }
-
-  Animation.prototype.slower = function()
-  {
-    this.interval /= 0.7;
-    if(this.direction > 0){this.play_animation();}
-    else if(this.direction < 0){this.reverse_animation();}
-  }
-
-  Animation.prototype.faster = function()
-  {
-    this.interval *= 0.7;
-    if(this.direction > 0){this.play_animation();}
-    else if(this.direction < 0){this.reverse_animation();}
-  }
-
-  Animation.prototype.anim_step_forward = function()
-  {
-    this.current_frame += 1;
-    if(this.current_frame < this.frames.length){
-      this.set_frame(this.current_frame);
-    }else{
-      var loop_state = this.get_loop_state();
-      if(loop_state == "loop"){
-        this.first_frame();
-      }else if(loop_state == "reflect"){
-        this.last_frame();
-        this.reverse_animation();
-      }else{
-        this.pause_animation();
-        this.last_frame();
-      }
-    }
-  }
-
-  Animation.prototype.anim_step_reverse = function()
-  {
-    this.current_frame -= 1;
-    if(this.current_frame >= 0){
-      this.set_frame(this.current_frame);
-    }else{
-      var loop_state = this.get_loop_state();
-      if(loop_state == "loop"){
-        this.last_frame();
-      }else if(loop_state == "reflect"){
-        this.first_frame();
-        this.play_animation();
-      }else{
-        this.pause_animation();
-        this.first_frame();
-      }
-    }
-  }
-
-  Animation.prototype.pause_animation = function()
-  {
-    this.direction = 0;
-    if (this.timer){
-      clearInterval(this.timer);
-      this.timer = null;
-    }
-  }
-
-  Animation.prototype.play_animation = function()
-  {
-    this.pause_animation();
-    this.direction = 1;
-    var t = this;
-    if (!this.timer) this.timer = setInterval(function() {
-        t.anim_step_forward();
-    }, this.interval);
-  }
-
-  Animation.prototype.reverse_animation = function()
-  {
-    this.pause_animation();
-    this.direction = -1;
-    var t = this;
-    if (!this.timer) this.timer = setInterval(function() {
-        t.anim_step_reverse();
-    }, this.interval);
-  }
-</script>
-
-<style>
-.animation {
-    display: inline-block;
-    text-align: center;
-}
-input[type=range].anim-slider {
-    width: 374px;
-    margin-left: auto;
-    margin-right: auto;
-}
-.anim-buttons {
-    margin: 8px 0px;
-}
-.anim-buttons button {
-    padding: 0;
-    width: 36px;
-}
-.anim-state label {
-    margin-right: 8px;
-}
-.anim-state input {
-    margin: 0;
-    vertical-align: middle;
-}
-</style>
-
-<div class="animation">
-  <img id="_anim_img444c5976d82f45cba3b1021ae0331688">
-  <div class="anim-controls">
-    <input id="_anim_slider444c5976d82f45cba3b1021ae0331688" type="range" class="anim-slider"
-           name="points" min="0" max="1" step="1" value="0"
-           oninput="anim444c5976d82f45cba3b1021ae0331688.set_frame(parseInt(this.value));"></input>
-    <div class="anim-buttons">
-      <button title="Decrease speed" onclick="anim444c5976d82f45cba3b1021ae0331688.slower()">
-          <i class="fa fa-minus"></i></button>
-      <button title="First frame" onclick="anim444c5976d82f45cba3b1021ae0331688.first_frame()">
-        <i class="fa fa-fast-backward"></i></button>
-      <button title="Previous frame" onclick="anim444c5976d82f45cba3b1021ae0331688.previous_frame()">
-          <i class="fa fa-step-backward"></i></button>
-      <button title="Play backwards" onclick="anim444c5976d82f45cba3b1021ae0331688.reverse_animation()">
-          <i class="fa fa-play fa-flip-horizontal"></i></button>
-      <button title="Pause" onclick="anim444c5976d82f45cba3b1021ae0331688.pause_animation()">
-          <i class="fa fa-pause"></i></button>
-      <button title="Play" onclick="anim444c5976d82f45cba3b1021ae0331688.play_animation()">
-          <i class="fa fa-play"></i></button>
-      <button title="Next frame" onclick="anim444c5976d82f45cba3b1021ae0331688.next_frame()">
-          <i class="fa fa-step-forward"></i></button>
-      <button title="Last frame" onclick="anim444c5976d82f45cba3b1021ae0331688.last_frame()">
-          <i class="fa fa-fast-forward"></i></button>
-      <button title="Increase speed" onclick="anim444c5976d82f45cba3b1021ae0331688.faster()">
-          <i class="fa fa-plus"></i></button>
-    </div>
-    <form title="Repetition mode" action="#n" name="_anim_loop_select444c5976d82f45cba3b1021ae0331688"
-          class="anim-state">
-      <input type="radio" name="state" value="once" id="_anim_radio1_444c5976d82f45cba3b1021ae0331688"
-             >
-      <label for="_anim_radio1_444c5976d82f45cba3b1021ae0331688">Once</label>
-      <input type="radio" name="state" value="loop" id="_anim_radio2_444c5976d82f45cba3b1021ae0331688"
-             checked>
-      <label for="_anim_radio2_444c5976d82f45cba3b1021ae0331688">Loop</label>
-      <input type="radio" name="state" value="reflect" id="_anim_radio3_444c5976d82f45cba3b1021ae0331688"
-             >
-      <label for="_anim_radio3_444c5976d82f45cba3b1021ae0331688">Reflect</label>
-    </form>
-  </div>
-</div>
-
-
-<script language="javascript">
-  /* Instantiate the Animation class. */
-  /* The IDs given should match those used in the template above. */
-  (function() {
-    var img_id = "_anim_img444c5976d82f45cba3b1021ae0331688";
-    var slider_id = "_anim_slider444c5976d82f45cba3b1021ae0331688";
-    var loop_select_id = "_anim_loop_select444c5976d82f45cba3b1021ae0331688";
-    var frames = new Array(15);
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">g</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">x</span><span class="o">-</span><span class="n">f1</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">optimize</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">help</span><span class="p">(</span><span class="n">optimize</span><span class="o">.</span><span class="n">fixed_point</span><span class="p">)</span>
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-<pre>Help on function fixed_point in module scipy.optimize.minpack:
-
-fixed_point(func, x0, args=(), xtol=1e-08, maxiter=500, method=&#39;del2&#39;)
-    Find a fixed point of the function.
-    
-    Given a function of one or more variables and a starting point, find a
-    fixed point of the function: i.e., where ``func(x0) == x0``.
-    
-    Parameters
-    ----------
-    func : function
-        Function to evaluate.
-    x0 : array_like
-        Fixed point of function.
-    args : tuple, optional
-        Extra arguments to `func`.
-    xtol : float, optional
-        Convergence tolerance, defaults to 1e-08.
-    maxiter : int, optional
-        Maximum number of iterations, defaults to 500.
-    method : {&#34;del2&#34;, &#34;iteration&#34;}, optional
-        Method of finding the fixed-point, defaults to &#34;del2&#34;,
-        which uses Steffensen&#39;s Method with Aitken&#39;s ``Del^2``
-        convergence acceleration [1]_. The &#34;iteration&#34; method simply iterates
-        the function until convergence is detected, without attempting to
-        accelerate the convergence.
-    
-    References
-    ----------
-    .. [1] Burden, Faires, &#34;Numerical Analysis&#34;, 5th edition, pg. 80
-    
-    Examples
-    --------
-    &gt;&gt;&gt; from scipy import optimize
-    &gt;&gt;&gt; def func(x, c1, c2):
-    ...    return np.sqrt(c1/(x+c2))
-    &gt;&gt;&gt; c1 = np.array([10,12.])
-    &gt;&gt;&gt; c2 = np.array([3, 5.])
-    &gt;&gt;&gt; optimize.fixed_point(func, [1.2, 1.3], args=(c1,c2))
-    array([ 1.4920333 ,  1.37228132])
-
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-    <span class="k">return</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mf">3.0</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">optimize</span><span class="o">.</span><span class="n">fixed_point</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">-</span><span class="n">f1</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="mf">0.1</span><span class="p">)</span>
-</pre></div>
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-<pre>array(1.)</pre>
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-<p>g(x)=x-f(x)=x_0
-g(x_0)=x_0-f(x_0)=x_0</p>
-
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-</pre></div>
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-<pre>0.0</pre>
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-</pre></div>
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-<pre>(1.0, 1.0)</pre>
-</div>
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-<h2 id="Stop-conditions-FP">Stop conditions FP<a class="anchor-link" href="#Stop-conditions-FP"> </a></h2>
-</div>
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-<p>The stop conditions are the same than Bisection:</p>
-<ul>
-<li><p>A fixed distance between the last two steps (absolute convergence):</p>
-<p>$$|p_i - p_{i-1}|&lt;\epsilon$$</p>
-</li>
-<li><p>A fixed relative distance between the last two steps (relative convergence):</p>
-<p>$$\frac{|p_i - p_{i-1}|}{|p_i|}&lt;\epsilon\ \ \ \ \ p_i \neq 0$$</p>
-</li>
-<li><p>Function tolerance:</p>
-<p>$$f(p_i)&lt; \epsilon$$</p>
-</li>
-<li><p>Computational stop:</p>
-<p>If $N&gt;N_{max}$, stop!</p>
-</li>
-</ul>
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-</pre></div>
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-<p><strong>Activity</strong>: Use the <code>optimize.fixed_point</code> to find the root of <code>f2(x)</code></p>
-
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-<h2 id="Example-4">Example 4<a class="anchor-link" href="#Example-4"> </a></h2>
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-<p>There is a type of functions whose roots coincides with a local or global minima or maxima. This condition implies the function does not cross the x-axis at the root and the problem cannot be solved by using Bisection as the criterion of different signs cannot be satisfied. Fixed-point iteration is a more feasible alternative when solving these problems.</p>
-<p>As an example, here we shall solve the roots of the function</p>
-<p>$f(x) = x^2\cos(2x)$</p>
-<p>For this, we plot the function first:</p>
-
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-<p>It is clear there are three root within the plotted interval, i.e. $x_0 = -\pi/4$, $x_1 = 0$ and $x_2 = \pi/4$. Using bisection for the first and third roots we obtain:</p>
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-    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-
-<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;X axis&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Y axis&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<div class="output_wrapper">
-<div class="output">
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-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/one-variable-equations-fixed-point_43_0.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span> <span class="s2">&quot;Root x_1&quot;</span><span class="p">,</span> <span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">b</span><span class="o">=-</span><span class="mf">0.5</span><span class="p">)</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span> <span class="s2">&quot;Root x_2&quot;</span><span class="p">,</span> <span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<div class="output_wrapper">
-<div class="output">
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-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>Root x_1 -0.7853981633979856
-Root x_2 0.7853981633979856
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>However, when applying bisection to the middle root, we obtain:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span> <span class="s2">&quot;Root x_0&quot;</span><span class="p">,</span> <span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="o">=-</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_text output_error">
-<pre>
-<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
-<span class="ansi-red-fg">ValueError</span>                                Traceback (most recent call last)
-<span class="ansi-green-fg">&lt;ipython-input-33-76a08fa07ea3&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span>
-<span class="ansi-green-fg">----&gt; 1</span><span class="ansi-red-fg"> </span>print<span class="ansi-blue-fg">(</span> <span class="ansi-blue-fg">&#34;Root x_0&#34;</span><span class="ansi-blue-fg">,</span> optimize<span class="ansi-blue-fg">.</span>bisect<span class="ansi-blue-fg">(</span> f<span class="ansi-blue-fg">,</span> a<span class="ansi-blue-fg">=</span><span class="ansi-blue-fg">-</span><span class="ansi-cyan-fg">0.5</span><span class="ansi-blue-fg">,</span> b<span class="ansi-blue-fg">=</span><span class="ansi-cyan-fg">0.5</span><span class="ansi-blue-fg">)</span> <span class="ansi-blue-fg">)</span>
-
-<span class="ansi-green-fg">/usr/local/lib/python3.7/dist-packages/scipy/optimize/zeros.py</span> in <span class="ansi-cyan-fg">bisect</span><span class="ansi-blue-fg">(f, a, b, args, xtol, rtol, maxiter, full_output, disp)</span>
-<span class="ansi-green-intense-fg ansi-bold">    547</span>     <span class="ansi-green-fg">if</span> rtol <span class="ansi-blue-fg">&lt;</span> _rtol<span class="ansi-blue-fg">:</span>
-<span class="ansi-green-intense-fg ansi-bold">    548</span>         <span class="ansi-green-fg">raise</span> ValueError<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">&#34;rtol too small (%g &lt; %g)&#34;</span> <span class="ansi-blue-fg">%</span> <span class="ansi-blue-fg">(</span>rtol<span class="ansi-blue-fg">,</span> _rtol<span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-fg">--&gt; 549</span><span class="ansi-red-fg">     </span>r <span class="ansi-blue-fg">=</span> _zeros<span class="ansi-blue-fg">.</span>_bisect<span class="ansi-blue-fg">(</span>f<span class="ansi-blue-fg">,</span> a<span class="ansi-blue-fg">,</span> b<span class="ansi-blue-fg">,</span> xtol<span class="ansi-blue-fg">,</span> rtol<span class="ansi-blue-fg">,</span> maxiter<span class="ansi-blue-fg">,</span> args<span class="ansi-blue-fg">,</span> full_output<span class="ansi-blue-fg">,</span> disp<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    550</span>     <span class="ansi-green-fg">return</span> results_c<span class="ansi-blue-fg">(</span>full_output<span class="ansi-blue-fg">,</span> r<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    551</span> 
-
-<span class="ansi-red-fg">ValueError</span>: f(a) and f(b) must have different signs</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Now, applying Fixed-point iteration:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">FixedPoint_Animation</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">pini</span> <span class="o">=</span> <span class="mf">0.4</span><span class="p">,</span> <span class="n">Nmax</span> <span class="o">=</span> <span class="mi">50</span><span class="p">,</span> <span class="n">xmin</span> <span class="o">=</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">xmax</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>Result: 0.017317568681637058
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-
-<link rel="stylesheet"
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-    /* MSIE used to detect old browsers and Trident used to newer ones*/
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-
-  /* Define the Animation class */
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-  Animation.prototype.next_frame = function()
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-  Animation.prototype.previous_frame = function()
-  {
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-  Animation.prototype.first_frame = function()
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-  Animation.prototype.last_frame = function()
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-        this.reverse_animation();
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-.anim-state input {
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-<p><strong>Activity</strong>: Comprobar con scipy y fixed_point</p>
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-<h2 id="ACTIVITY-FP">ACTIVITY FP<a class="anchor-link" href="#ACTIVITY-FP"> </a></h2>
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-<p>When a new planet is discovered, there are different methods to estimate its physical properties. Many times is only possible to estimate either the planet mass or the planet radius and the other property has to be predicted through computer modelling.</p>
-<p>If one has the planet mass, a very rough way to estimate its radius is to assume certain composition (mean density) and a homogeneous distribution (a very bad assumption!). For example, for the planet <a href="http://es.wikipedia.org/wiki/Gliese_832_c">Gliese 832c</a> with a mass $M= 5.40 M_{\oplus}$, if we assume an earth-like composition, i.e. $\bar \rho_{\oplus} = 5520\ kg/m^3$, we obtain:</p>
-$$R_{g832c} = \left( \frac{3 M_{g832c}}{ 4 \pi \bar\rho_{\oplus} } \right)^{1/3} \approx 1.75 R_{\oplus}$$<p>That would be the planet radius if the composition where exactly equal to earth's.</p>
-<p>A more realistic approach is assuming an internal one-layer density profile like:</p>
-$$\rho(r) = \rho_0 \exp\left( -\frac{r}{L} \right)$$<p>where $\rho_0$ is the density at planet centre and $L$ is a characteristic lenght depending on the composition. From numerical models of planet interiors, the estimated parameters for a planet of are $M= 5.40 M_{\oplus}$ are approximately $\rho_0 = 18000\ kg/m^3$ and $L = 6500\ km$.</p>
-<p>Integrating over the planet volume, we obtain the total mass as</p>
-$$M = 4\pi \int_0^R \rho(r)r^2dr$$<p>This is a function of the mass in terms of the planet radius.</p>
-<p>Solving the equation $M(R) = M_{g832c}$ it would be possible to find a more realistic planet radius. However when using numerical models, it is not possible to approach the solution from the left side as a negative mass makes no sense.</p>
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-<p><font color='red'>    
-In an IPython notebook, solve the previous problem and find the radius of **Gliese 832c** using your own version of the Fixed-point iteration algorithm.
-</font></p>
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-<h1 id="Newton-Raphson-Method">Newton-Raphson Method<a class="anchor-link" href="#Newton-Raphson-Method"> </a></h1>
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-<p>Although Fixed-point iteration is an efficient algorithm as compared with Bisection, the Newton-Raphson method is an acceletared convergent scheme where the roots of a function are easily found with just a few iterations.</p>
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-<h2 id="Derivation-NM">Derivation NM<a class="anchor-link" href="#Derivation-NM"> </a></h2>
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-<p>Although this method can be presented from an algorithmic point of view, the mathematical deduction is very useful as it allows us to understand the essence of the approximation as well as estimating easily the convergence errors.</p>
-<p>Let be $f(x)$ a continuous and differentiable function defined within an interval $[a,b]$ (i.e. $f\in \mathcal{C}^2[a,b]$), and $p$ is a root of the function such that $f(p) = 0$. If we give an initial an enough close guess $p_0$ to this root, such that $|p-p_0|&lt;\epsilon$, where $\epsilon$ is adequately small, we can expand the function by using a second order taylor serie, yielding:</p>
-$$f(p) = f(p_0) + (p-p_0)f'(p_0) + \frac{(p-p_0)^2}{2}f''(p_0) + \mathcal{O}^3(|p-p_0|)$$<p>but as $f(p) = 0$ and $|p-p_0|^2&lt;\epsilon^2$ is an even smaller quantity, we can readily neglect from second order terms, obtaining</p>
-$$p \approx p_0 - \frac{f(p_0)}{f'(p_0)} \equiv p_1$$<p>If we repeat this process but now using $p_1$ as our guess to the root instead of $p_0$ we shall obtain:</p>
-$$p \approx p_1 - \frac{f(p_1)}{f'(p_1)} \equiv p_2$$<p>and so...</p>
-$$p \approx p_n - \frac{f(p_n)}{f'(p_n)} \equiv p_{n+1}$$<p>where each new iteration is a better approximation to the real root.</p>
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-<h2 id="Steps-NM">Steps NM<a class="anchor-link" href="#Steps-NM"> </a></h2>
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-<ol>
-<li>Take your function $f(x)$ and derive it, $f'(x)$.</li>
-<li>Give a guest to the solution (root of $f(x)$). This value would be the seed $p_0$.</li>
-<li><p>The next guest to the solution will be given by</p>
-<p>$$p_{n+1} = p_n - \frac{f(p_n)}{f'(p_{n})}$$</p>
-</li>
-<li><p>If the stop condition is not satisfied, then repeat step 3.</p>
-</li>
-<li>The End!</li>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Defining Newton Method</span>
-<span class="k">def</span> <span class="nf">NewtonRaphson_Animation</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">fp</span><span class="p">,</span> <span class="n">pini</span><span class="p">,</span> <span class="n">Nmax</span><span class="p">,</span> <span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span> <span class="p">):</span>
-    <span class="c1">#Initial condition</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="p">[</span><span class="n">pini</span><span class="p">,]</span>
-    <span class="n">p_dash</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">p_der</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="c1">#Iterations</span>
-    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Nmax</span><span class="p">):</span>
-        <span class="n">p</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">p</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">-</span> <span class="n">f</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="n">n</span><span class="p">])</span><span class="o">/</span><span class="n">fp</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="n">n</span><span class="p">])</span> <span class="p">)</span>
-        <span class="n">p_dash</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">p</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="p">)</span>
-        <span class="n">p_dash</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">p</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="p">)</span>
-        <span class="n">p_der</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="mi">0</span> <span class="p">)</span>
-        <span class="n">p_der</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">f</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="n">n</span><span class="p">])</span> <span class="p">)</span>
-    
-    <span class="n">p</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="n">p</span> <span class="p">)</span>
-    <span class="n">p_dash</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="n">p_dash</span> <span class="p">)</span>
-    <span class="n">p_der</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="n">p_der</span> <span class="p">)</span>
-    
-    <span class="nb">print</span> <span class="p">(</span>  <span class="s2">&quot;Result:&quot;</span><span class="p">,</span> <span class="n">p</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>    
-    <span class="c1">#Array X-axis</span>
-    <span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">xmin</span><span class="p">,</span><span class="n">xmax</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
-    
-    <span class="c1">#Initializing Figure</span>
-    <span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">7</span><span class="p">)</span> <span class="p">)</span>
-    <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
-    <span class="c1">#Graphic iterations</span>
-    <span class="n">dash</span><span class="p">,</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="p">[],</span> <span class="p">[],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;gray&quot;</span><span class="p">,</span> <span class="n">linewidth</span> <span class="o">=</span> <span class="mi">2</span> <span class="p">)</span>
-    <span class="n">derivative</span><span class="p">,</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="p">[],</span> <span class="p">[],</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="n">linewidth</span> <span class="o">=</span> <span class="mi">3</span> <span class="p">)</span>
-    <span class="c1">#Function f</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">X</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;green&quot;</span><span class="p">,</span> <span class="n">linewidth</span> <span class="o">=</span> <span class="mi">2</span> <span class="p">)</span>
-    <span class="c1">#Horizontal line</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">hlines</span><span class="p">(</span> <span class="mi">0</span><span class="p">,</span> <span class="n">xmin</span><span class="p">,</span><span class="n">xmax</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span> <span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span> <span class="p">(</span><span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">)</span> <span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span> <span class="s2">&quot;X axis&quot;</span> <span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span> <span class="s2">&quot;Y axis&quot;</span> <span class="p">)</span>
-    <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span> <span class="s2">&quot;Fixed-Point iteration&quot;</span> <span class="p">)</span>
-        
-    <span class="k">def</span> <span class="nf">init</span><span class="p">():</span>
-        <span class="n">dash</span><span class="o">.</span><span class="n">set_data</span><span class="p">([],</span> <span class="p">[])</span>
-        <span class="n">derivative</span><span class="o">.</span><span class="n">set_data</span><span class="p">([],</span> <span class="p">[])</span>
-        <span class="k">return</span> <span class="n">dash</span><span class="p">,</span> <span class="n">derivative</span>
-    
-    <span class="k">def</span> <span class="nf">animate</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-        <span class="c1">#Setting new data</span>
-        <span class="n">dash</span><span class="o">.</span><span class="n">set_data</span><span class="p">(</span> <span class="n">p_dash</span><span class="p">[:</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="o">+</span><span class="mi">2</span><span class="p">],</span> <span class="n">p_der</span><span class="p">[:</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="o">+</span><span class="mi">2</span><span class="p">]</span> <span class="p">)</span>
-        <span class="n">derivative</span><span class="o">.</span><span class="n">set_data</span><span class="p">(</span> <span class="n">p_dash</span><span class="p">[</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">:</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="o">+</span><span class="mi">3</span><span class="p">],</span> <span class="n">p_der</span><span class="p">[</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">:</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="o">+</span><span class="mi">3</span><span class="p">]</span> <span class="p">)</span>
-        <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span> <span class="s2">&quot;Newthon-Raphson Method. Iteration </span><span class="si">%d</span><span class="s2">&quot;</span><span class="o">%</span><span class="k">i</span> )
-        <span class="k">return</span> <span class="n">dash</span><span class="p">,</span> <span class="n">derivative</span>
-    
-    <span class="k">return</span> <span class="n">animation</span><span class="o">.</span><span class="n">FuncAnimation</span><span class="p">(</span><span class="n">fig</span><span class="p">,</span> <span class="n">animate</span><span class="p">,</span> <span class="n">init_func</span><span class="o">=</span><span class="n">init</span><span class="p">,</span><span class="n">frames</span><span class="o">=</span><span class="n">Nmax</span><span class="p">,</span> <span class="n">interval</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span> <span class="n">blit</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
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-<h2 id="Example-5">Example 5<a class="anchor-link" href="#Example-5"> </a></h2>
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-<p>Find one root of the function:</p>
-<p>$f(x) = x^2 - x$</p>
-<p>with derivative</p>
-<p>$f'(x) = 2x -1$</p>
-<p>using the Newton-Raphson method.</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Defining the function</span>
-<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span> 
-    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">x</span>
-<span class="c1">#Defining the derivative</span>
-<span class="k">def</span> <span class="nf">df</span><span class="p">(</span><span class="n">x</span><span class="p">):</span> 
-    <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span>
-<span class="c1">#Calculating root</span>
-<span class="n">NewtonRaphson_Animation</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">df</span><span class="p">,</span> <span class="n">pini</span> <span class="o">=</span> <span class="mf">0.55</span><span class="p">,</span> <span class="n">Nmax</span> <span class="o">=</span> <span class="mi">10</span><span class="p">,</span> <span class="n">xmin</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">xmax</span> <span class="o">=</span> <span class="mf">3.5</span> <span class="p">)</span>
-</pre></div>
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-<link rel="stylesheet"
-href="https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/css/font-awesome.min.css">
-<script language="javascript">
-  function isInternetExplorer() {
-    ua = navigator.userAgent;
-    /* MSIE used to detect old browsers and Trident used to newer ones*/
-    return ua.indexOf("MSIE ") > -1 || ua.indexOf("Trident/") > -1;
-  }
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-  /* Define the Animation class */
-  function Animation(frames, img_id, slider_id, interval, loop_select_id){
-    this.img_id = img_id;
-    this.slider_id = slider_id;
-    this.loop_select_id = loop_select_id;
-    this.interval = interval;
-    this.current_frame = 0;
-    this.direction = 0;
-    this.timer = null;
-    this.frames = new Array(frames.length);
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-    for (var i=0; i<frames.length; i++)
-    {
-     this.frames[i] = new Image();
-     this.frames[i].src = frames[i];
-    }
-    var slider = document.getElementById(this.slider_id);
-    slider.max = this.frames.length - 1;
-    if (isInternetExplorer()) {
-        // switch from oninput to onchange because IE <= 11 does not conform
-        // with W3C specification. It ignores oninput and onchange behaves
-        // like oninput. In contrast, Mircosoft Edge behaves correctly.
-        slider.setAttribute('onchange', slider.getAttribute('oninput'));
-        slider.setAttribute('oninput', null);
-    }
-    this.set_frame(this.current_frame);
-  }
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-  Animation.prototype.get_loop_state = function(){
-    var button_group = document[this.loop_select_id].state;
-    for (var i = 0; i < button_group.length; i++) {
-        var button = button_group[i];
-        if (button.checked) {
-            return button.value;
-        }
-    }
-    return undefined;
-  }
-
-  Animation.prototype.set_frame = function(frame){
-    this.current_frame = frame;
-    document.getElementById(this.img_id).src =
-            this.frames[this.current_frame].src;
-    document.getElementById(this.slider_id).value = this.current_frame;
-  }
-
-  Animation.prototype.next_frame = function()
-  {
-    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));
-  }
-
-  Animation.prototype.previous_frame = function()
-  {
-    this.set_frame(Math.max(0, this.current_frame - 1));
-  }
-
-  Animation.prototype.first_frame = function()
-  {
-    this.set_frame(0);
-  }
-
-  Animation.prototype.last_frame = function()
-  {
-    this.set_frame(this.frames.length - 1);
-  }
-
-  Animation.prototype.slower = function()
-  {
-    this.interval /= 0.7;
-    if(this.direction > 0){this.play_animation();}
-    else if(this.direction < 0){this.reverse_animation();}
-  }
-
-  Animation.prototype.faster = function()
-  {
-    this.interval *= 0.7;
-    if(this.direction > 0){this.play_animation();}
-    else if(this.direction < 0){this.reverse_animation();}
-  }
-
-  Animation.prototype.anim_step_forward = function()
-  {
-    this.current_frame += 1;
-    if(this.current_frame < this.frames.length){
-      this.set_frame(this.current_frame);
-    }else{
-      var loop_state = this.get_loop_state();
-      if(loop_state == "loop"){
-        this.first_frame();
-      }else if(loop_state == "reflect"){
-        this.last_frame();
-        this.reverse_animation();
-      }else{
-        this.pause_animation();
-        this.last_frame();
-      }
-    }
-  }
-
-  Animation.prototype.anim_step_reverse = function()
-  {
-    this.current_frame -= 1;
-    if(this.current_frame >= 0){
-      this.set_frame(this.current_frame);
-    }else{
-      var loop_state = this.get_loop_state();
-      if(loop_state == "loop"){
-        this.last_frame();
-      }else if(loop_state == "reflect"){
-        this.first_frame();
-        this.play_animation();
-      }else{
-        this.pause_animation();
-        this.first_frame();
-      }
-    }
-  }
-
-  Animation.prototype.pause_animation = function()
-  {
-    this.direction = 0;
-    if (this.timer){
-      clearInterval(this.timer);
-      this.timer = null;
-    }
-  }
-
-  Animation.prototype.play_animation = function()
-  {
-    this.pause_animation();
-    this.direction = 1;
-    var t = this;
-    if (!this.timer) this.timer = setInterval(function() {
-        t.anim_step_forward();
-    }, this.interval);
-  }
-
-  Animation.prototype.reverse_animation = function()
-  {
-    this.pause_animation();
-    this.direction = -1;
-    var t = this;
-    if (!this.timer) this.timer = setInterval(function() {
-        t.anim_step_reverse();
-    }, this.interval);
-  }
-</script>
-
-<style>
-.animation {
-    display: inline-block;
-    text-align: center;
-}
-input[type=range].anim-slider {
-    width: 374px;
-    margin-left: auto;
-    margin-right: auto;
-}
-.anim-buttons {
-    margin: 8px 0px;
-}
-.anim-buttons button {
-    padding: 0;
-    width: 36px;
-}
-.anim-state label {
-    margin-right: 8px;
-}
-.anim-state input {
-    margin: 0;
-    vertical-align: middle;
-}
-</style>
-
-<div class="animation">
-  <img id="_anim_img1b72341ad9f4449295a399c10d97e5d6">
-  <div class="anim-controls">
-    <input id="_anim_slider1b72341ad9f4449295a399c10d97e5d6" type="range" class="anim-slider"
-           name="points" min="0" max="1" step="1" value="0"
-           oninput="anim1b72341ad9f4449295a399c10d97e5d6.set_frame(parseInt(this.value));"></input>
-    <div class="anim-buttons">
-      <button title="Decrease speed" onclick="anim1b72341ad9f4449295a399c10d97e5d6.slower()">
-          <i class="fa fa-minus"></i></button>
-      <button title="First frame" onclick="anim1b72341ad9f4449295a399c10d97e5d6.first_frame()">
-        <i class="fa fa-fast-backward"></i></button>
-      <button title="Previous frame" onclick="anim1b72341ad9f4449295a399c10d97e5d6.previous_frame()">
-          <i class="fa fa-step-backward"></i></button>
-      <button title="Play backwards" onclick="anim1b72341ad9f4449295a399c10d97e5d6.reverse_animation()">
-          <i class="fa fa-play fa-flip-horizontal"></i></button>
-      <button title="Pause" onclick="anim1b72341ad9f4449295a399c10d97e5d6.pause_animation()">
-          <i class="fa fa-pause"></i></button>
-      <button title="Play" onclick="anim1b72341ad9f4449295a399c10d97e5d6.play_animation()">
-          <i class="fa fa-play"></i></button>
-      <button title="Next frame" onclick="anim1b72341ad9f4449295a399c10d97e5d6.next_frame()">
-          <i class="fa fa-step-forward"></i></button>
-      <button title="Last frame" onclick="anim1b72341ad9f4449295a399c10d97e5d6.last_frame()">
-          <i class="fa fa-fast-forward"></i></button>
-      <button title="Increase speed" onclick="anim1b72341ad9f4449295a399c10d97e5d6.faster()">
-          <i class="fa fa-plus"></i></button>
-    </div>
-    <form title="Repetition mode" action="#n" name="_anim_loop_select1b72341ad9f4449295a399c10d97e5d6"
-          class="anim-state">
-      <input type="radio" name="state" value="once" id="_anim_radio1_1b72341ad9f4449295a399c10d97e5d6"
-             >
-      <label for="_anim_radio1_1b72341ad9f4449295a399c10d97e5d6">Once</label>
-      <input type="radio" name="state" value="loop" id="_anim_radio2_1b72341ad9f4449295a399c10d97e5d6"
-             checked>
-      <label for="_anim_radio2_1b72341ad9f4449295a399c10d97e5d6">Loop</label>
-      <input type="radio" name="state" value="reflect" id="_anim_radio3_1b72341ad9f4449295a399c10d97e5d6"
-             >
-      <label for="_anim_radio3_1b72341ad9f4449295a399c10d97e5d6">Reflect</label>
-    </form>
-  </div>
-</div>
-
-
-<script language="javascript">
-  /* Instantiate the Animation class. */
-  /* The IDs given should match those used in the template above. */
-  (function() {
-    var img_id = "_anim_img1b72341ad9f4449295a399c10d97e5d6";
-    var slider_id = "_anim_slider1b72341ad9f4449295a399c10d97e5d6";
-    var loop_select_id = "_anim_loop_select1b72341ad9f4449295a399c10d97e5d6";
-    var frames = new Array(10);
-
-  frames[0] = "data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAfgAAAH4CAYAAACmKP9/AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90\
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-"
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-
-</div>
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-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
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-<div class="output_png output_subarea ">
-<img src="../images/features/one-variable-equations-fixed-point_63_2.png"
->
-</div>
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-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Stop-conditions-NM">Stop conditions NM<a class="anchor-link" href="#Stop-conditions-NM"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>The stop conditions are the same than Bisection and Fixed-point iteration:</p>
-<ul>
-<li><p>A fixed distance between the last two steps (absolute convergence):</p>
-<p>$$|p_i - p_{i-1}|&lt;\epsilon$$</p>
-</li>
-<li><p>A fixed relative distance between the last two steps (relative convergence):</p>
-<p>$$\frac{|p_i - p_{i-1}|}{|p_i|}&lt;\epsilon\ \ \ \ \ p_i \neq 0$$</p>
-</li>
-<li><p>Function tolerance:</p>
-<p>$$f(p_i)&lt; \epsilon$$</p>
-</li>
-<li><p>Computational stop:</p>
-<p>If $N&gt;N_{max}$, stop!</p>
-</li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Convergence-NM">Convergence NM<a class="anchor-link" href="#Convergence-NM"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>It is possible to demonstrate by means of the previous derivation procedure, that the convergence of the Newton-Raphson method is quadratic, i.e., if $p$ is the exact root and $p_n$ is the $n$-th iteration, then</p>
-$$|p_{n+1}-p|\leq C |p_n-p|^2$$<p>for a fixed ans positive constant $C$.</p>
-<p>This implies, if the initial guess is good enough such that |p_0-p| is small, the convergence is achieved very fast as each iteration improves the precision twice in the order of magnitude, e.g., if $|p_0-p|\sim 10^{-1}$, $|p_1-p|\sim 10^{-2}$, $|p_2-p|\sim 10^{-4}$, $|p_2-p|\sim 10^{-8}$ and so.</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Defining Newton Method</span>
-<span class="k">def</span> <span class="nf">NewtonRaphson</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">fp</span><span class="p">,</span> <span class="n">pini</span><span class="p">,</span> <span class="n">Nmax</span> <span class="p">):</span>
-    <span class="c1">#Initial condition</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">pini</span>
-    <span class="c1">#Iterations</span>
-    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span>  <span class="nb">range</span><span class="p">(</span><span class="n">Nmax</span><span class="p">):</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">p</span> <span class="o">-</span> <span class="n">f</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">/</span><span class="n">fp</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-    <span class="c1">#Final result</span>
-    <span class="k">return</span> <span class="n">p</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>ACTIVITY</strong></p>
-<p>In an IPython notebook, copy the latter routine NewtonRaphson and the Bisection routine provided in this notebook (<em>and if you have it already, the Fixed-point iteration routine as well</em>) and find the root of the next function using all the methods.</p>
-<p>$f(x) = x - \cos(x)$</p>
-<p>Plot in the same figure the convergence of each method as a function of the number of iterations.</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>See the <strong>Summary</strong> section at the end for the implementation in Scipy</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<hr>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h1 id="Secant-Method">Secant Method<a class="anchor-link" href="#Secant-Method"> </a></h1>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>The Newton-Raphson method is highly efficient as the convergence is accelerated, however there is a weakness with it: one needs to know the derivative of the function beforehand. This aspect may be complicated when dealing with numerical functions or even very complicated analytical functions. Numerical methods to derive the input function can be applied, but this extra procedure may involve an extra computing time that compensates the time spent by using other methods like Bisection.</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Derivation-SM">Derivation SM<a class="anchor-link" href="#Derivation-SM"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Retaking the iterative expression obtained from the Newton-Raphson method:</p>
-$$p_{n+1} = p_n - \frac{f(p_n)}{f'(p_{n})}\,,$$<p>or, changing $n \to n-1$</p>
-$$p_{n} = p_{n-1} - \frac{f(p_{n-1})}{f'(p_{n-1})}\,.$$<p>The derivative can be approximated as</p>
-$$f'(p_n) = \lim_{p_{n}\rightarrow p_{n-1}} \frac{f(p_{n})-f(p_{n-1})}{p_{n}-p_{n-1}} $$<p>As we know, the convergence of the NR method is quadratic, so $p_{n-1}$ should be close enough to $p_n$ such that one can assume $p_{n-1}\rightarrow p_n$ and the previous term is:</p>
-$$f'(p_n) \approx \frac{f(p_{n})-f(p_{n-1})}{p_{n}-p_{n-1}} $$<p>or, changing $n \to n-1$</p>
-$$f'(p_{n-1}) \approx \frac{f(p_{n-1})-f(p_{n-2})}{p_{n-1}-p_{n-2}} $$<p>The final expression for the $n$-th iteration of the root is then:</p>
-$$p_{n} = p_{n-1} - \frac{f(p_{n-1})}{f'(p_{n-1})}\,.$$$$p_n = p_{n-1} - \frac{ f(p_{n-1})(p_{n-1}-p_{n-2}) }{f(p_{n-1})-f(p_{n-2})}$$<p>In this consists the Secant method, what is just an approximation to the Newton-Raphson method, but without the derivative term.</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Steps-SM">Steps SM<a class="anchor-link" href="#Steps-SM"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<ol>
-<li>Give the input function $f(x)$.</li>
-<li>Give two guests to the solution (root of $f(x)$). These values would be the seeds $p_0$, $p_1$.</li>
-<li><p>The next guest to the solution will be given by</p>
-<p>$$p_n = p_{n-1} - \frac{ f(p_{n-1})(p_{n-1}-p_{n-2}) }{f(p_{n-1})-f(p_{n-2})}$$</p>
-</li>
-<li><p>If the stop condition is not satisfied, then repeat step 3.</p>
-</li>
-<li>The End!</li>
-</ol>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>See the <strong>Summary</strong> section at the end for the implementation in Scipy</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>ACTIVITY</strong></p>
-<p>In an IPython notebook and based on the routine NewtonRaphson, write your own routine SecantMethod that performs the previous steps for the Secant Method. Test your code with the function $f(x)$:</p>
-<p>$f(x) = x - \cos(x)$
-&lt;/font&gt;</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>ACTIVITY</strong></p>
-<p>It is known that light rays are deflected when they pass near by a gravitational field and that this deviation is proportional to the body mass which the light is interacting with and inversely proportional to the passing distance. 
-Since it is common finding very massive structures in the universe and the measures that are done to study it involve photons, it makes sense to study what happens to a light source image when the rays get close to a grumpy object like a dark matter halo.</p>
-<p>In order to study the light deflection in these cases, it will be used the simplest model, gravitational lens theory, where the len is a very massive object. A sketch of a typical system is shown in the figure below. The source plane is the light source or image that is going to be affected,  $\eta$ is the distance from a image point to the line of sight and $\beta$ the subtended angle by the point. 
-The lens plane corresponds to the mass that affects the light coming from the source, $\xi$ is the new image point distance to the line of sight, $\theta$ is the subtended angle by the new point position. Then, $\alpha$ is the deflection angle.</p>
-<p>Since from observations $\theta$ is known, the problem to be solved per pixel usually is</p>
-\begin{equation}
-\beta = \theta - \hat{\alpha}(\theta) 
-\end{equation}<p>but $\alpha$ also depends on $\theta$ besides the len halo properties. This would allow construct the real image
-from the distorted and magnified one.</p>
-<p><img src="https://github.com/restrepo/ComputationalMethods/blob/master/material/figures/lente1.png?raw=1"></p>
-<p>This equation can also be written in terms of distances</p>
-\begin{equation}
-\vec{\eta}  = \frac{D_s}{D_d} \vec{\xi} - D_{ds}\alpha ( \vec{\xi }) 
-\end{equation}<p>The solution to the lens equation is easier to get if it is assumed that the len is axially symmetric. In this case, the deflection angle  takes the next form</p>
-$$ \hat{\alpha}(\vec{\xi}) = \frac{\vec{\xi}}{|\vec{\xi}|^2} \frac{8G\pi}{c^2} \int_0^\xi d\xi'\xi'\Sigma(\xi')$$<p>The quantity $\Sigma$ is the surface mass density, i.e., the len's mass enclosed inside $\xi$ circle per area unit.<br>
-It is important to notice that the direction of $\alpha$ is the same as $\xi$ and consequently $\eta$.</p>
-<p>The problem to be solved is the next: Given the positions of a square find the image distorsion due to gravitational lensing, i.e., find the root of \xi in the trascendal equation it satisfies. Use the routines given below and all of 
-the data for the len and image that is going to be distorted.</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Superficial density of the lens</span>
-<span class="k">def</span> <span class="nf">Sup_density</span><span class="p">(</span><span class="n">radio</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">radio</span><span class="o">*</span><span class="n">M</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="o">&lt;</span><span class="n">radio</span><span class="p">])</span><span class="o">/</span><span class="n">radio</span><span class="o">**</span><span class="mf">2.</span>
-
-<span class="c1">#Deviation angle due to the gravitational len</span>
-<span class="k">def</span> <span class="nf">des_angle</span><span class="p">(</span> <span class="n">radio</span> <span class="p">):</span>
-    <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="mi">4</span><span class="o">*</span><span class="n">G</span><span class="o">*</span><span class="n">integrate</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span> <span class="n">Sup_density</span> <span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="n">radio</span> <span class="p">)[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="p">(</span><span class="n">radio</span><span class="o">*</span><span class="n">c</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-<span class="c1">#Len equation</span>
-<span class="k">def</span> <span class="nf">Len_equation</span><span class="p">(</span><span class="n">radio</span><span class="p">,</span> <span class="n">eta</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">eta</span> <span class="o">-</span> <span class="n">Ds</span><span class="o">*</span><span class="n">radio</span><span class="o">/</span><span class="n">Dd</span> <span class="o">-</span> <span class="n">Dds</span><span class="o">*</span><span class="n">des_angle</span><span class="p">(</span> <span class="n">radio</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Len distribution generated </span>
-<span class="n">M</span> <span class="o">=</span> <span class="mf">3e7</span>
-<span class="n">L</span> <span class="o">=</span> <span class="mf">1e5</span>
-<span class="n">puntos</span> <span class="o">=</span> <span class="mi">6</span>
-<span class="n">Ds</span> <span class="o">=</span> <span class="mi">1000</span>
-<span class="n">Dd</span> <span class="o">=</span> <span class="mi">900</span>
-<span class="n">Dds</span> <span class="o">=</span> <span class="n">Ds</span> <span class="o">-</span> <span class="n">Dd</span>
-<span class="n">G</span> <span class="o">=</span> <span class="mf">4.302e-3</span><span class="c1"># pc M_sun**-1(km/s)**2</span>
-<span class="n">c</span> <span class="o">=</span> <span class="mf">3e6</span> <span class="c1"># km/s </span>
-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">L</span><span class="p">,</span><span class="n">puntos</span><span class="p">)</span>
-<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">L</span><span class="p">,</span><span class="n">puntos</span><span class="p">)</span>
-<span class="n">X</span><span class="p">,</span><span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
-<span class="c1">#Generating meshgrid of points </span>
-<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">puntos</span><span class="o">*</span><span class="n">puntos</span><span class="p">)</span>
-<span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span><span class="n">puntos</span><span class="o">*</span><span class="n">puntos</span><span class="p">)</span>
-<span class="n">r</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">X</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">Y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-<span class="c1">#Image to be distorted</span>
-<span class="n">Li</span> <span class="o">=</span> <span class="mi">5</span>
-<span class="n">ni</span> <span class="o">=</span> <span class="mi">8</span>
-<span class="n">X0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="n">Li</span><span class="p">,</span><span class="n">Li</span><span class="p">,</span><span class="n">ni</span><span class="p">)</span>
-<span class="n">Y0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="n">Li</span><span class="p">,</span><span class="n">Li</span><span class="p">,</span><span class="n">ni</span><span class="p">)</span>
-<span class="c1">#Generating meshgrid of points </span>
-<span class="n">X0</span><span class="p">,</span><span class="n">Y0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">X0</span><span class="p">,</span><span class="n">Y0</span><span class="p">)</span>
-<span class="n">r0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span> <span class="n">X0</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">Y0</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
-<span class="n">theta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">ni</span><span class="p">,</span><span class="n">ni</span><span class="p">))</span>
-<span class="n">epsilon</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">ni</span><span class="p">,</span><span class="n">ni</span><span class="p">))</span>
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-<span class="n">yc</span> <span class="o">=</span> <span class="n">epsilon</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">6</span><span class="p">))</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xc</span><span class="p">,</span><span class="n">yc</span><span class="p">,</span><span class="s1">&#39;b.&#39;</span><span class="p">);</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X0</span><span class="p">,</span><span class="n">Y0</span><span class="p">,</span><span class="s2">&quot;b.&quot;</span><span class="p">);</span>
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-<h2 id="Summary">Summary<a class="anchor-link" href="#Summary"> </a></h2>
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-<ul>
-<li>optimize.bisect</li>
-<li>optimize.fixed_point</li>
-<li>optimize.newton</li>
-</ul>
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-<p>In <code>scipy.optimize</code> the <code>newton</code> method contains both the <code>secant</code> and <code>Newton-Raphson</code> algorithms. The las one is used when the derivative of the function is used explicitly</p>
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-<div class=" highlight hl-ipython3"><pre><span></span>optimize.newton<span class="o">?</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">optimize</span> 
-<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">g</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">x</span><span class="o">-</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">fprime</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="s2">&quot;Derivate of f=x**2*np.cos(2x)&quot;</span>
-    <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">root1</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">fixed_point</span><span class="p">(</span><span class="n">g</span><span class="p">,</span><span class="o">-</span><span class="mf">1.0</span><span class="p">)</span>
-<span class="n">root2</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">fixed_point</span><span class="p">(</span><span class="n">g</span><span class="p">,</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="n">root3</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">fixed_point</span><span class="p">(</span><span class="n">g</span><span class="p">,</span><span class="mf">0.75</span><span class="p">)</span>
-
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;las raices son x=</span><span class="si">{}</span><span class="s2"> x=</span><span class="si">{}</span><span class="s2"> x=</span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">root1</span><span class="p">,</span><span class="n">root2</span><span class="p">,</span><span class="n">root3</span><span class="p">))</span>
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-<pre>las raices son x=-0.7853981633974483 x=-4.363697583627993e-09 x=0.7853981633974483
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f</span><span class="p">(</span><span class="n">root1</span><span class="p">),</span><span class="n">f</span><span class="p">(</span><span class="n">root2</span><span class="p">),</span><span class="n">f</span><span class="p">(</span><span class="n">root3</span><span class="p">)</span>
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-<pre>(3.777118574576472e-17, 1.9041856601360782e-17, 3.777118574576472e-17)</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Secant</span>
-<span class="n">root1</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">newton</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="n">root2</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">newton</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="mf">0.4</span><span class="p">,</span><span class="n">tol</span><span class="o">=</span><span class="mf">1E-16</span><span class="p">,</span><span class="n">maxiter</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
-<span class="n">root3</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">newton</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="mf">0.5</span><span class="p">)</span>
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-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;las raices son x=</span><span class="si">{}</span><span class="s2"> x=</span><span class="si">{}</span><span class="s2"> x=</span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">root1</span><span class="p">,</span><span class="n">root2</span><span class="p">,</span><span class="n">root3</span><span class="p">))</span>
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-<pre>las raices son x=-0.7853981633974487 x=1.0433087260535526e-16 x=-0.7853981633974482
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Newton Raphson</span>
-<span class="n">root1</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">newton</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">fprime</span><span class="p">)</span>
-<span class="n">root2</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">newton</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="mf">0.1</span><span class="p">,</span><span class="n">fprime</span><span class="p">,</span><span class="n">tol</span><span class="o">=</span><span class="mf">1E-16</span><span class="p">)</span>
-<span class="n">root3</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">newton</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="n">fprime</span><span class="p">)</span>
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-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;las raices son x=</span><span class="si">{}</span><span class="s2"> x=</span><span class="si">{}</span><span class="s2"> x=</span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">root1</span><span class="p">,</span><span class="n">root2</span><span class="p">,</span><span class="n">root3</span><span class="p">))</span>
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-<pre>las raices son x=-0.7853981633974483 x=8.642260125895126e-17 x=-0.7853981633974484
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">root2</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">newton</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">fprime</span><span class="p">)</span>
-<span class="n">root2</span>
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-<h2 id="Brent's-method.">Brent's method.<a class="anchor-link" href="#Brent's-method."> </a></h2><p>Find a root of a function in a bracketing interval using Brent's method.</p>
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-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span> <span class="p">)</span>
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-<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
-<span class="ansi-red-fg">ValueError</span>                                Traceback (most recent call last)
-<span class="ansi-green-fg">&lt;ipython-input-62-2741c04b08c2&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span>
-<span class="ansi-green-fg">----&gt; 1</span><span class="ansi-red-fg"> </span>brentq<span class="ansi-blue-fg">(</span>f<span class="ansi-blue-fg">,</span><span class="ansi-blue-fg">-</span><span class="ansi-cyan-fg">0.5</span><span class="ansi-blue-fg">,</span><span class="ansi-cyan-fg">0.5</span><span class="ansi-blue-fg">)</span>
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-<span class="ansi-green-fg">/usr/local/lib/python3.7/dist-packages/scipy/optimize/zeros.py</span> in <span class="ansi-cyan-fg">brentq</span><span class="ansi-blue-fg">(f, a, b, args, xtol, rtol, maxiter, full_output, disp)</span>
-<span class="ansi-green-intense-fg ansi-bold">    774</span>     <span class="ansi-green-fg">if</span> rtol <span class="ansi-blue-fg">&lt;</span> _rtol<span class="ansi-blue-fg">:</span>
-<span class="ansi-green-intense-fg ansi-bold">    775</span>         <span class="ansi-green-fg">raise</span> ValueError<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">&#34;rtol too small (%g &lt; %g)&#34;</span> <span class="ansi-blue-fg">%</span> <span class="ansi-blue-fg">(</span>rtol<span class="ansi-blue-fg">,</span> _rtol<span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">)</span>
-<span class="ansi-green-fg">--&gt; 776</span><span class="ansi-red-fg">     </span>r <span class="ansi-blue-fg">=</span> _zeros<span class="ansi-blue-fg">.</span>_brentq<span class="ansi-blue-fg">(</span>f<span class="ansi-blue-fg">,</span> a<span class="ansi-blue-fg">,</span> b<span class="ansi-blue-fg">,</span> xtol<span class="ansi-blue-fg">,</span> rtol<span class="ansi-blue-fg">,</span> maxiter<span class="ansi-blue-fg">,</span> args<span class="ansi-blue-fg">,</span> full_output<span class="ansi-blue-fg">,</span> disp<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    777</span>     <span class="ansi-green-fg">return</span> results_c<span class="ansi-blue-fg">(</span>full_output<span class="ansi-blue-fg">,</span> r<span class="ansi-blue-fg">)</span>
-<span class="ansi-green-intense-fg ansi-bold">    778</span> 
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-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/one-variable-equations.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
-<h1 id="One-Variable-Equations">One Variable Equations<a class="anchor-link" href="#One-Variable-Equations"> </a></h1><p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/one-variable-equations.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
-<p>Throughout this section and the next ones we shall cover the topic of solutions to one variable equations. Many different problems in physics and astronomy require the use of complex expressions, even with implicit dependence of variables. When it is necessary to solve for one of those variable, an analytical approach is not usually the best solution, because of its complexity or even because it does not exist at all. Different approaches for dealing with this comprehend series expansions and numerical solutions. Among the most widely used numerical approaches are the Bisection or Binary-search method, fixed-point iteration, Newton's methods.</p>
-<p>For further details see for example Chap. 5 of Ref. (<a href="../index.ipynb#Bibliography">1</a>) or Chap. 4 of Ref. (<a href="../index.ipynb#Bibliography">3</a>)</p>
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-<li><a href="#Steps-BM">Steps</a></li>
-<li><a href="#Stop-condition-BM">Stop condition</a></li>
-<li><a href="#Error-analysis-BM">Error analysis</a></li>
-<li><a href="#Example-1">Example 1</a></li>
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-<li><a href="./one-variable-equations-fixed-point.ipynb#Fixed-point-Iteration">Fixed-point Iteration</a><ul>
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-<li><a href="./one-variable-equations-fixed-point.ipynb#Stop-condition-FP">Stop condition</a></li>
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-<li><a href="./one-variable-equations-fixed-point.ipynb#Steps-NM">Steps</a></li>
-<li><a href="./one-variable-equations-fixed-point.ipynb#Example-5">Example 5</a></li>
-<li><a href="./one-variable-equations-fixed-point.ipynb#Stop-condition-NM">Stop condition</a></li>
-<li><a href="./one-variable-equations-fixed-point.ipynb#Convergence-NM">Convergence</a></li>
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-<li><a href="./one-variable-equations-fixed-point.ipynb#Secant-Method">Secant Method</a><ul>
-<li><a href="./one-variable-equations-fixed-point.ipynb#Derivation-SM">Derivation</a></li>
-<li><a href="./one-variable-equations-fixed-point.ipynb#Steps-SM">Steps</a></li>
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-<h2 id="Bibliography">Bibliography<a class="anchor-link" href="#Bibliography"> </a></h2><p>[1] <a href="https://drive.google.com/file/d/0BxoOXsn2EUNIQUdFVkctR2xWRUk/view?usp=sharing">Kiusalaas, Numerical Methods in Engineering with Python</a><br/>
-[2] <a href="https://drive.google.com/file/d/0BxoOXsn2EUNIekRUMFVPYXVsSTg/view?usp=sharing">Jensen, Computational_Physics</a>.  Companion repos: <a href="https://github.com/mhjensen">https://github.com/mhjensen</a> <a href="http://compphysics.github.io/ComputationalPhysics/doc/web/course">Web page</a>  <br/>
-[3] E. Ayres, <a href="http://bit.ly/CPwPython">Computational Physics With Python</a><br/></p>
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-<h1 id="Bisection-Method">Bisection Method<a class="anchor-link" href="#Bisection-Method"> </a></h1>
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-<p>The Bisection method exploits the <a href="http://en.wikipedia.org/wiki/Intermediate_value_theorem">intermediate value theorem</a>, where a continuous and differentiable function $f$ must have a zero between an interval $[a,b]$ such that $f(a)f(b)&lt;0$, or equivalently, there must be a value $p\in[a,b]$ such that $f(p)=0$. Below the algorithmm is stated explicitly.</p>
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-</div><ol>
-<li><p>There must be selected two values $a$ and $b$ such that $f(a)f(b)&lt;0$ and $p\in[a,b]$ where $f(p)=0$. In other words, though we do not know the value of the root, we must know that there is at least one within the selected interval.</p>
-</li>
-<li><p>To begin, it must be set $a_1=a$ and $b_1=b$.</p>
-</li>
-<li><p>Calculate the mid-point $p_1$ as</p>
-<p>$$p_1 = a_1 + \frac{b_1-a_1}{2} = \frac{a_1+b_1}{2}$$</p>
-</li>
-<li><p>Evaluate the function in $p_1$, if the <a href="#Stop-Condition">stop condition</a> is true, go to step 6.</p>
-</li>
-<li><p>If the <a href="#Stop-Condition">stop condition</a> is not satisfied, then:</p>
-<ol>
-<li><p>If $f(p_1)f(a_1) &gt; 0$, $p\in(p_1,b_1)$. Then set $a_2=p_1$ and $b_2=b_1$</p>
-</li>
-<li><p>If $f(p_1)f(a_1) &lt; 0$, $p\in(a_1,p_1)$. Then set $a_2=a_1$ and $b_2=p_1$</p>
-</li>
-<li><p>Go to step 3 using $p_2$, $a_2$ and $b_2$ instead of $p_1$, $a_1$ and $b_1$. For next iterations the index increases until the <a href="#Stop-Condition">stop condition</a> is reached.</p>
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-<p>There are several different stop conditions for this algorithm. The most used are stated below:</p>
-<ul>
-<li><p>A fixed distance between the last two steps (absolute convergence): → <code>xtol</code></p>
-<p>$$|p_i - p_{i-1}|&lt;\epsilon$$</p>
-</li>
-<li><p>A fixed relative distance between the last two steps (relative convergence): → <code>rtol</code></p>
-<p>$$\frac{|p_i - p_{i-1}|}{|p_i|}&lt;\epsilon\ \ \ \ \ p_i \neq 0$$</p>
-</li>
-<li><p>Function tolerance:</p>
-<p>$$f(p_i)&lt; \epsilon$$</p>
-</li>
-</ul>
-<p>All these conditions should lead to a desired convergence expressed by the $\epsilon$ value. However, the first and the third conditions present some problems when the function has a derivative very large or close to $0$ as evaluated in the root value. When the function is very inclined, the first condition fails as a convergence in the $x$ axis does not guarantee a convergence in the $y$ axis, so the found root $p$ may be far from the real value. When the function is very flat ($dF/dx\rightarrow 0$), the third condition fails due to an analogous reason.</p>
-<p>A final stop condition which does not have mathematical motivation but a computational one, is a maximum number of allowed iterations. This condition should be used not only for this algorithm but for all iteration-based numerical methods. This condition guarantees a finite computing time and prevents undesired infinite bucles.</p>
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-<li>If $N&gt;N_{max}$, stop!</li>
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-<p>If we suppose $f\in C[a,b]$ and $f(a)f(b)&lt;0$, the Bisection method generates a sequence of numbers $\left\{p_i\right\}_{i=1}^\infty$ approximating a root $p$ of $f$ as:</p>
-$$|p_i-p|\leq \frac{b-a}{2^n},\ \ \ \ \ i\geq 1$$<p>From this, we can conclude the convergence rate of the method is</p>
-$$p_i = p + \mathcal{O}\left( \frac{1}{2^i} \right)$$<p>This expression allows us to estimate the maximum number of required iterations for achieving a desired precision. The next figure sketches the number of iterations required for some precision.</p>
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-<span class="n">i</span><span class="p">[:</span><span class="mi">3</span><span class="p">]</span>
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-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">exp2</span><span class="p">(</span><span class="o">-</span><span class="n">i</span><span class="p">))</span>
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-<span class="c1">#Array of iterations</span>
-<span class="n">Niter</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">100</span> <span class="p">)</span>
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-<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span> <span class="p">)</span>
-<span class="c1">#plt.semilogy( Niter, 2.0**-Niter, color=&quot;green&quot;, lw = 2 )</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">semilogy</span><span class="p">(</span> <span class="n">Niter</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">exp2</span><span class="p">(</span><span class="o">-</span><span class="n">Niter</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;green&quot;</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Number of Iterations&quot;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Absolute Error $|p_n-p|$&quot;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="n">f</span><span class="o">=</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">s</span>
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-<p>Find the root of the function $f$</p>
-<p>$f(x) = x^3 - 2$</p>
-<p>for $20$ iterations, show the result and the relative error in each iteration.</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Defining Bisection function</span>
-<span class="k">def</span> <span class="nf">Bisection</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">Nmax</span><span class="p">,</span> <span class="n">printer</span><span class="o">=</span><span class="kc">False</span> <span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Find the root of function f between a and b</span>
-<span class="sd">    </span>
-<span class="sd">    f: function</span>
-<span class="sd">    a,b: the initial interval</span>
-<span class="sd">    Nmax: Nmax de interations</span>
-<span class="sd">    printer: bool, to print internal steps</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c1">#verifying the STEP1, a and b with different signs</span>
-    <span class="kn">import</span> <span class="nn">sys</span>
-    <span class="k">if</span> <span class="n">f</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">&gt;</span><span class="mi">0</span><span class="p">:</span>
-        <span class="n">sys</span><span class="o">.</span><span class="n">exit</span><span class="p">(</span><span class="s2">&quot;Error, f(a) and f(b) should have opposite signs&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="kc">False</span>
-    <span class="c1">#Assigning the current extreme values, STEP2</span>
-    <span class="n">ai</span> <span class="o">=</span> <span class="n">a</span>
-    <span class="n">bi</span> <span class="o">=</span> <span class="n">b</span>
-    <span class="c1">#Iterations</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">while</span> <span class="n">n</span><span class="o">&lt;=</span><span class="n">Nmax</span><span class="p">:</span>
-        <span class="c1">#Bisection, STEP3</span>
-        <span class="n">pi</span> <span class="o">=</span> <span class="p">(</span><span class="n">ai</span><span class="o">+</span><span class="n">bi</span><span class="p">)</span><span class="o">/</span><span class="mf">2.0</span>
-        <span class="c1">#Evaluating function in pi, STEP4 and STEP5</span>
-        <span class="k">if</span> <span class="n">printer</span><span class="p">:</span>
-            <span class="nb">print</span><span class="p">(</span> <span class="s2">&quot;Value for </span><span class="si">{}</span><span class="s2"> iterations: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">pi</span><span class="p">)</span> <span class="p">)</span>
-        <span class="c1">#Condition A</span>
-        <span class="k">if</span> <span class="n">f</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">ai</span><span class="p">)</span><span class="o">&gt;</span><span class="mi">0</span><span class="p">:</span>
-            <span class="n">ai</span> <span class="o">=</span> <span class="n">pi</span>
-        <span class="c1">#Condition B</span>
-        <span class="k">elif</span> <span class="n">f</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">ai</span><span class="p">)</span><span class="o">&lt;</span><span class="mi">0</span><span class="p">:</span>
-            <span class="n">bi</span> <span class="o">=</span> <span class="n">pi</span>
-        <span class="c1">#Condition C: repeat the cycle</span>
-        <span class="n">n</span><span class="o">+=</span><span class="mi">1</span>
-    <span class="c1">#Final result</span>
-    <span class="k">return</span> <span class="n">pi</span>
-</pre></div>
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-    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mf">3.0</span> <span class="o">-</span> <span class="mf">2.0</span>
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-<p>Finding the root of the function. The real root is $\sqrt[3]{2}$, so <code>a</code> and <code>b</code> should enclose this value</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Defining a and b</span>
-<span class="n">a</span> <span class="o">=</span> <span class="mf">0.0</span>
-<span class="n">b</span> <span class="o">=</span> <span class="mf">2.0</span>
-<span class="n">Nmax</span> <span class="o">=</span> <span class="mi">20</span>
-<span class="n">result</span> <span class="o">=</span> <span class="n">Bisection</span><span class="p">(</span><span class="n">function</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">Nmax</span><span class="p">,</span> <span class="n">printer</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-<span class="nb">print</span> <span class="p">(</span> <span class="s2">&quot;Real value:&quot;</span><span class="p">,</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">3</span><span class="p">))</span>
-<span class="nb">print</span> <span class="p">(</span> <span class="s2">&quot;Absolute error&quot;</span><span class="p">,</span> <span class="nb">abs</span><span class="p">((</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span><span class="o">-</span><span class="n">result</span><span class="p">))</span> <span class="p">)</span>
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-<pre>Value for 1 iterations: 1.0
-Value for 2 iterations: 1.5
-Value for 3 iterations: 1.25
-Value for 4 iterations: 1.375
-Value for 5 iterations: 1.3125
-Value for 6 iterations: 1.28125
-Value for 7 iterations: 1.265625
-Value for 8 iterations: 1.2578125
-Value for 9 iterations: 1.26171875
-Value for 10 iterations: 1.259765625
-Value for 11 iterations: 1.2607421875
-Value for 12 iterations: 1.26025390625
-Value for 13 iterations: 1.260009765625
-Value for 14 iterations: 1.2598876953125
-Value for 15 iterations: 1.25994873046875
-Value for 16 iterations: 1.259918212890625
-Value for 17 iterations: 1.2599334716796875
-Value for 18 iterations: 1.2599258422851562
-Value for 19 iterations: 1.2599220275878906
-Value for 20 iterations: 1.2599201202392578
-Real value: 1.2599210498948732
-Absolute error 9.296556153781665e-07
-</pre>
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-<p>Using the error analysis, we can predict the produced error at $20$ iterations by computing:</p>
-$$ \left( \frac{1}{2^{20}}\right) \approx 9.53674316\times 10^{-7} $$<p>This value is very close to the obtained relative error.</p>
-<p>If we were interested in a double precision, i.e. $\epsilon \sim 10^{-17}$, the number of required iterations would be:</p>
-$$ 10^{-17} = \left( \frac{1}{2^{N}}\right) \longrightarrow N = \frac{17}{\log_{10}(2)} \approx 56 $$
-</div>
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-<p>Now the same but using scipy implementation</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">optimize</span> 
-</pre></div>
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-<p>See help with <code>optimize?:</code></p>
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-<div class=" highlight hl-ipython3"><pre><span></span>optimize.bisect<span class="o">?</span>
-</pre></div>
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-<p>Check the help: See <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.bisect.html">here</a></p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">n</span><span class="p">,</span><span class="n">c</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">-</span><span class="n">c</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="mi">3</span><span class="p">,))</span> 
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-<pre>1.2599210498938191</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
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-<pre>1.2599210498938191</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">help</span><span class="p">(</span><span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">)</span>
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-<pre>Help on function bisect in module scipy.optimize.zeros:
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-bisect(f, a, b, args=(), xtol=2e-12, rtol=8.881784197001252e-16, maxiter=100, full_output=False, disp=True)
-    Find root of a function within an interval using bisection.
-    
-    Basic bisection routine to find a zero of the function `f` between the
-    arguments `a` and `b`. `f(a)` and `f(b)` cannot have the same signs.
-    Slow but sure.
-    
-    Parameters
-    ----------
-    f : function
-        Python function returning a number.  `f` must be continuous, and
-        f(a) and f(b) must have opposite signs.
-    a : scalar
-        One end of the bracketing interval [a,b].
-    b : scalar
-        The other end of the bracketing interval [a,b].
-    xtol : number, optional
-        The computed root ``x0`` will satisfy ``np.allclose(x, x0,
-        atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The
-        parameter must be nonnegative.
-    rtol : number, optional
-        The computed root ``x0`` will satisfy ``np.allclose(x, x0,
-        atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The
-        parameter cannot be smaller than its default value of
-        ``4*np.finfo(float).eps``.
-    maxiter : int, optional
-        If convergence is not achieved in `maxiter` iterations, an error is
-        raised. Must be &gt;= 0.
-    args : tuple, optional
-        Containing extra arguments for the function `f`.
-        `f` is called by ``apply(f, (x)+args)``.
-    full_output : bool, optional
-        If `full_output` is False, the root is returned. If `full_output` is
-        True, the return value is ``(x, r)``, where x is the root, and r is
-        a `RootResults` object.
-    disp : bool, optional
-        If True, raise RuntimeError if the algorithm didn&#39;t converge.
-        Otherwise, the convergence status is recorded in a `RootResults`
-        return object.
-    
-    Returns
-    -------
-    x0 : float
-        Zero of `f` between `a` and `b`.
-    r : `RootResults` (present if ``full_output = True``)
-        Object containing information about the convergence. In particular,
-        ``r.converged`` is True if the routine converged.
-    
-    Examples
-    --------
-    
-    &gt;&gt;&gt; def f(x):
-    ...     return (x**2 - 1)
-    
-    &gt;&gt;&gt; from scipy import optimize
-    
-    &gt;&gt;&gt; root = optimize.bisect(f, 0, 2)
-    &gt;&gt;&gt; root
-    1.0
-    
-    &gt;&gt;&gt; root = optimize.bisect(f, -2, 0)
-    &gt;&gt;&gt; root
-    -1.0
-    
-    See Also
-    --------
-    brentq, brenth, bisect, newton
-    fixed_point : scalar fixed-point finder
-    fsolve : n-dimensional root-finding
-
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="General-arguments-and-options-of-a-SciPy-function">General arguments and options of a SciPy function<a class="anchor-link" href="#General-arguments-and-options-of-a-SciPy-function"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Let define a general multiparameter</p>
-<ul>
-<li><code>f</code>
-with a mandatory parameter and two optional parameters, corresponding to a straight line with the slope and the intercept as parameters
-$$
-f(x)=mx+b\,,
-$$</li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">m</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span><span class="n">b</span><span class="o">=-</span><span class="mi">2</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">m</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="n">b</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
-
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f</span><span class="p">(</span><span class="mf">0.66666666666666666</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>0.0</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>We have at least three ways to use a multiparameter function, the first two with the implicit <code>lambda</code> function</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>0.6666666666671972</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="input">
-
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">,)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>0.6666666666671972</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>and the recommended way to used the designed implementation in Scipy through the <code>args</code> parameter which is common to many methods</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
-
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>0.6666666666671972</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<ul>
-<li><code>xtol</code></li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="n">xtol</span><span class="o">=</span><span class="mf">1E-17</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>0.6666666666666671</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<ul>
-<li><code>maxiter</code></li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="n">xtol</span><span class="o">=</span><span class="mf">1E-17</span><span class="p">,</span><span class="n">maxiter</span><span class="o">=</span><span class="mi">10000</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>0.6666666666666671</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="n">xtol</span><span class="o">=</span><span class="mf">1E-17</span><span class="p">,</span><span class="n">rtol</span><span class="o">=</span><span class="mf">1E-15</span><span class="p">,</span><span class="n">maxiter</span><span class="o">=</span><span class="mi">10000</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
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-<div class="output">
-
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-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>0.6666666666666671</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="text_cell_render border-box-sizing rendered_html">
-<ul>
-<li><code>full_output</code></li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x0</span><span class="p">,</span><span class="n">r</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="n">xtol</span><span class="o">=</span><span class="mf">1E-17</span><span class="p">,</span><span class="n">full_output</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x0</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
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-<div class="output">
-
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-<div class="output_area">
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>0.6666666666666671</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">type</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
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-<div class="output_area">
-
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>scipy.optimize.zeros.RootResults</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Check the code</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">r</span><span class="o">.</span><span class="n">root</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
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-
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>0.6666666666666671</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>The help of the function is write directly in the code of the Scipy method <code>bisect</code></p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span>optimize.bisect<span class="o">??</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Check the internal referenced code</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span>optimize._zeros<span class="o">??</span>
-</pre></div>
-
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-
-<div class="output_text output_subarea ">
-<pre><span class="ansi-red-fg">Type:</span>        module
-<span class="ansi-red-fg">String form:</span> &lt;module &#39;scipy.optimize._zeros&#39; from &#39;/usr/local/lib/python3.7/dist-packages/scipy/optimize/_zeros.cpython-37m-x86_64-linux-gnu.so&#39;&gt;
-<span class="ansi-red-fg">File:</span>        /usr/local/lib/python3.7/dist-packages/scipy/optimize/_zeros.cpython-37m-x86_64-linux-gnu.so
-</pre>
-</div>
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-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Example">Example<a class="anchor-link" href="#Example"> </a></h3><p>Find the roots of
-$$x^3=2$$</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">function</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mf">3.0</span> <span class="o">-</span> <span class="mf">2.0</span>
-</pre></div>
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">todo</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span><span class="n">function</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">full_output</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">type</span><span class="p">(</span><span class="n">todo</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>tuple</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">len</span><span class="p">(</span><span class="n">todo</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>2</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">result</span><span class="o">=</span><span class="n">todo</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="n">result</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>1.2599210498938191</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">r</span><span class="o">=</span><span class="n">todo</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-<span class="n">r</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
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-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>      converged: True
-           flag: &#39;converged&#39;
- function_calls: 42
-     iterations: 40
-           root: 1.2599210498938191</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">type</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>scipy.optimize.zeros.RootResults</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">r</span><span class="o">.</span><span class="n">iterations</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>40</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">todo</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span><span class="n">function</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">full_output</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">type</span><span class="p">(</span><span class="n">todo</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>float</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="mf">0.0</span>
-<span class="n">b</span> <span class="o">=</span> <span class="mf">2.0</span>
-
-<span class="n">result</span><span class="p">,</span><span class="n">r</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span><span class="n">function</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">full_output</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">result</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>1.2599210498938191</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mf">3.0</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>1.2599210498948732</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Check <code>r</code> object by using the <code>&lt;TAB&gt;</code> key next</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">r</span><span class="o">.</span><span class="n">iterations</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>40</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">r</span><span class="o">.</span><span class="n">function_calls</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>42</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">r</span><span class="o">.</span><span class="n">iterations</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>9.094947017729282e-13</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">abs</span><span class="p">((</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mf">3.0</span><span class="p">)</span><span class="o">-</span><span class="n">result</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>1.0540457395791236e-12</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">r</span><span class="o">.</span><span class="n">converged</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>True</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">result</span><span class="p">,</span><span class="n">r</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span><span class="n">function</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">xtol</span><span class="o">=</span><span class="mf">1E-17</span><span class="p">,</span><span class="n">full_output</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">r</span><span class="o">.</span><span class="n">iterations</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>51</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">abs</span><span class="p">((</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mf">3.0</span><span class="p">)</span><span class="o">-</span><span class="n">result</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>2.220446049250313e-16</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">r</span><span class="o">.</span><span class="n">iterations</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>4.440892098500626e-16</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="-ACTIVITY-"><font color="red"> ACTIVITY </font><a class="anchor-link" href="#-ACTIVITY-"> </a></h3>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><font color='red'>
-In an IPython notebook, use the scipy implementation and find the first solution to the equation</p>
-<p>$ 7 = \sqrt{x^2+1}+e^x\sin x $</p>
-<p>CLUE: Check graphically (with matplotlib) that the solution is within the interval $[0,2]$.
-&lt;/font&gt;</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><a href="./solutions/sltn_bisect.ipynb">solution</a></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">c</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>4</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Solo usar para constante</span>
-<span class="n">np</span><span class="o">.</span><span class="n">e</span>
-</pre></div>
-
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-</div>
-</div>
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-<div class="output_wrapper">
-<div class="output">
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-<div class="jb_output_wrapper }}">
-<div class="output_area">
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-<pre>2.718281828459045</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-</pre></div>
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-<pre>2.718281828459045</pre>
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-<h2 id="Example-2">Example 2<a class="anchor-link" href="#Example-2"> </a></h2>
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-<p>In orbital mechanics, when solving the central-force problem it becomes necessary to solve the Kepler's equation. This is a transcendental equation that relates the orbital parameters of the trajectory.</p>
-<p><em>Kepler equation:</em> $M = E - \epsilon \sin E$</p>
-<p>where $M$ is the mean anomaly, $E$ the eccentric anomaly and $\epsilon$ the eccentricity. The mean anomaly can be computed with the expression</p>
-$$M = n\ t = \sqrt{ \frac{GM}{a^3} } t$$<p>where $n$ is the mean motion, $G$ the gravitational constant, $M$ the mass of the central body and $a$ the semi-major axis. $t$ is the time where the position in the trajectory will be computed.</p>
-<p>The coordinates $x$ and $y$ as time functions can be recovered by means of the next expressions</p>
-$$x(t) = a(\cos E - \epsilon)$$$$y(t) = b\sin E$$<p>where $b = a \sqrt{1-\epsilon^2}$ is the semi-minor axis of the orbit and the implicit time-dependence of the eccentric anomaly $E$ is computed through the Kepler's equation.</p>
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-<p><strong>Problem:</strong></p>
-<p>For a stallite orbiting the earth in a equatorial trajectory with eccentricity $\epsilon = 0.5$ at a geostationary distance for the semi-major axis, tabulate the positions $x$ and $y$ within the orbital plane in intervals of $15$ min during $5$ hours.</p>
-<p><strong>Parameters:</strong></p>
-<ul>
-<li><p>$\epsilon = 0.5$</p>
-</li>
-<li><p>$a = 35900$ km</p>
-</li>
-<li><p>$G = 6.67384 \times 10^{-11}$ m$^3$ kg$^{-1}$ s$^{-2}$</p>
-</li>
-<li><p>$M_{\oplus} = 5.972\times 10^{24}$ kg</p>
-</li>
-</ul>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#====================================================================</span>
-<span class="c1">#Parameters</span>
-<span class="c1">#====================================================================</span>
-<span class="c1">#Eccentricity</span>
-<span class="n">eps</span> <span class="o">=</span> <span class="mf">0.5</span>
-<span class="c1">#Semi-major axis    [m]</span>
-<span class="n">a</span> <span class="o">=</span> <span class="mf">35900e3</span>
-<span class="c1">#Gravitational constant    [m3kg-1s-2]</span>
-<span class="n">GC</span> <span class="o">=</span> <span class="mf">6.67384e-11</span>
-<span class="c1">#Earth mass    [kg]</span>
-<span class="n">Me</span> <span class="o">=</span> <span class="mf">5.972e24</span>
-
-<span class="c1">#Semi-minor axis    [m]</span>
-<span class="n">b</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">eps</span><span class="o">**</span><span class="mf">2.0</span><span class="p">)</span><span class="o">**</span><span class="mf">0.5</span>
-<span class="c1">#Mean motion</span>
-<span class="n">n</span> <span class="o">=</span> <span class="p">(</span> <span class="n">GC</span><span class="o">*</span><span class="n">Me</span><span class="o">/</span><span class="n">a</span><span class="o">**</span><span class="mf">3.0</span> <span class="p">)</span><span class="o">**</span><span class="mf">0.5</span>
-
-<span class="c1">#Hour to Second</span>
-<span class="n">HR2SC</span> <span class="o">=</span> <span class="mf">3600.</span>
-<span class="c1">#Initial time    [hr]</span>
-<span class="n">t0</span> <span class="o">=</span> <span class="mi">0</span><span class="o">*</span><span class="n">HR2SC</span>
-<span class="c1">#Final time    [hr]</span>
-<span class="n">tf</span> <span class="o">=</span> <span class="mi">5</span><span class="o">*</span><span class="n">HR2SC</span>
-<span class="c1">#Time step    [hr]</span>
-<span class="n">tstep</span> <span class="o">=</span> <span class="mf">0.25</span><span class="o">*</span><span class="n">HR2SC</span>
-<span class="c1">#Number of maxim iterations</span>
-<span class="n">Niter</span> <span class="o">=</span> <span class="mi">56</span>
-<span class="c1">#Root interval</span>
-<span class="n">a0</span> <span class="o">=</span> <span class="o">-</span><span class="mi">10</span>
-<span class="n">b0</span> <span class="o">=</span> <span class="mi">10</span>
-
-<span class="c1">#====================================================================</span>
-<span class="c1">#Kepler Function</span>
-<span class="c1">#====================================================================</span>
-<span class="k">def</span> <span class="nf">kepler</span><span class="p">(</span> <span class="n">E</span> <span class="p">):</span>
-    <span class="n">func</span> <span class="o">=</span> <span class="n">E</span> <span class="o">-</span> <span class="n">eps</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">E</span><span class="p">)</span> <span class="o">-</span> <span class="n">n</span><span class="o">*</span><span class="n">t</span>
-    <span class="k">return</span> <span class="n">func</span>
-
-<span class="c1">#====================================================================</span>
-<span class="c1">#Position function</span>
-<span class="c1">#====================================================================</span>
-<span class="k">def</span> <span class="nf">r</span><span class="p">(</span><span class="n">E</span><span class="p">):</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">E</span><span class="p">)</span><span class="o">-</span><span class="n">eps</span><span class="p">)</span>
-    <span class="n">y</span> <span class="o">=</span> <span class="n">b</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">E</span><span class="p">)</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">x</span><span class="o">/</span><span class="mf">1.e3</span><span class="p">,</span> <span class="n">y</span><span class="o">/</span><span class="mf">1.e3</span><span class="p">]</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#====================================================================</span>
-<span class="c1">#Solving for different times</span>
-<span class="c1">#====================================================================</span>
-<span class="c1">#Time array</span>
-<span class="n">times</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span> <span class="n">t0</span><span class="p">,</span> <span class="n">tf</span><span class="p">,</span> <span class="n">tstep</span> <span class="p">)</span>
-
-<span class="n">rpos</span> <span class="o">=</span> <span class="p">[]</span>
-<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">times</span><span class="p">:</span>
-    <span class="c1">#Finding the new eccentric anomaly</span>
-    <span class="n">E</span> <span class="o">=</span> <span class="n">optimize</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span> <span class="n">kepler</span><span class="p">,</span> <span class="n">a0</span><span class="p">,</span> <span class="n">b0</span> <span class="p">)</span>
-    <span class="c1">#Computing coordinates at this time</span>
-    <span class="n">ri</span> <span class="o">=</span> <span class="n">r</span><span class="p">(</span><span class="n">E</span><span class="p">)</span>
-    <span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;In </span><span class="si">%f</span><span class="s2"> hours, the satellite is located at (</span><span class="si">%f</span><span class="s2">,</span><span class="si">%f</span><span class="s2">) km&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">t</span><span class="o">/</span><span class="n">HR2SC</span><span class="p">,</span> <span class="n">ri</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">ri</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="p">)</span>
-    <span class="n">rpos</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">ri</span> <span class="p">)</span>
-<span class="n">rpos</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">rpos</span><span class="p">)</span>    
-
-<span class="c1">#Plotting</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">rpos</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">rpos</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s2">&quot;o-&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$x$ coordinate [km]&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$y$ coordinate [km]&quot;</span><span class="p">)</span>
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-<pre>In 0.000000 hours, the satellite is located at (17950.000000,0.000000) km
-In 0.250000 hours, the satellite is located at (17454.741542,5146.426647) km
-In 0.500000 hours, the satellite is located at (16033.097675,10023.437750) km
-In 0.750000 hours, the satellite is located at (13848.847528,14430.262364) km
-In 1.000000 hours, the satellite is located at (11104.379909,18261.701894) km
-In 1.250000 hours, the satellite is located at (7989.437466,21493.410338) km
-In 1.500000 hours, the satellite is located at (4657.168469,24151.489610) km
-In 1.750000 hours, the satellite is located at (1221.066166,26286.121177) km
-In 2.000000 hours, the satellite is located at (-2238.747501,27954.872718) km
-In 2.250000 hours, the satellite is located at (-5667.264143,29214.008624) km
-In 2.500000 hours, the satellite is located at (-9027.373695,30114.739827) km
-In 2.750000 hours, the satellite is located at (-12294.392344,30702.085866) km
-In 3.000000 hours, the satellite is located at (-15452.164136,31014.965936) km
-In 3.250000 hours, the satellite is located at (-18490.361033,31086.789919) km
-In 3.500000 hours, the satellite is located at (-21402.633716,30946.195086) km
-In 3.750000 hours, the satellite is located at (-24185.347784,30617.769816) km
-In 4.000000 hours, the satellite is located at (-26836.717825,30122.702148) km
-In 4.250000 hours, the satellite is located at (-29356.211228,29479.336137) km
-In 4.500000 hours, the satellite is located at (-31744.135350,28703.638662) km
-In 4.750000 hours, the satellite is located at (-34001.350021,27809.586870) km
-</pre>
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-    <img src="https://www.python.org/static/community_logos/python-logo-master-v3-TM.png">
-</div><h1 id="PYTHON">PYTHON<a class="anchor-link" href="#PYTHON"> </a></h1><p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/overview-python.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
-<p>Python is an interpreted programming language oriented to easy-readable coding, unlike compiled languages like C/C++ and Fortran, where the syntax usually does not favor the readability. This feature makes Python very interesting when we want to focus on something different than the program structure itself, e.g. on Computational Methods, thereby allowing to optimize our time, to debug syntax errors easily, etc.</p>
-<p><a href="https://www.python.org/">Official page</a></p>
-<p><a href="http://en.wikipedia.org/wiki/Python">Wikipedia</a></p>
-<h2 id="Python-Philosophy">Python Philosophy<a class="anchor-link" href="#Python-Philosophy"> </a></h2><ol>
-<li>Beautiful is better than ugly.</li>
-<li>Explicit is better than implicit.</li>
-<li>Simple is better than complex.</li>
-<li>Complex is better than complicated.</li>
-<li>Flat is better than nested.</li>
-<li>Sparse is better than dense.</li>
-<li>Readability counts.</li>
-<li>Special cases aren't special enough to break the rules. (Although practicality beats purity)</li>
-<li>Errors should never pass silently. (Unless explicitly silenced)</li>
-<li>In the face of ambiguity, refuse the temptation to guess.</li>
-<li>There should be one-- and preferably only one --obvious way to do it. (Although that way may not be obvious at first unless you're Dutch)</li>
-<li>Now is better than never. (Although never is often better than right now)</li>
-<li>If the implementation is hard to explain, it's a bad idea.</li>
-<li>If the implementation is easy to explain, it may be a good idea.</li>
-<li>NameSpaces are one honking great idea -- let's do more of those!</li>
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-<li><a href="#String,-Integer,-Float">String, Integer, Float</a></li>
-<li><a href="#Functions-I">Functions I</a> </li>
-<li><a href="#Hello-World!">Hello World!</a> </li>
-<li><a href="#Arithmetics">Arithmetics</a></li>
-<li><a href="#Lists,-Tuples-and-Dictionaries">Lists, Tuples and Dictionaries</a></li>
-<li><a href="#Bucles-and-Conditionals">Bucles and Conditionals</a></li>
-<li><a href="#Functions-II">Functions II</a></li>
-</ul>
-<h2 id="Biblography">Biblography<a class="anchor-link" href="#Biblography"> </a></h2><p>[1f] Ani Adhikari and John DeNero, <a href="https://www.inferentialthinking.com/chapters/intro.html">Computational and Inferential Thinking</a><br/></p>
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-<h1 id="String,-Integer,-Float">String, Integer, Float<a class="anchor-link" href="#String,-Integer,-Float"> </a></h1><p>The basic types of variables in Python are:</p>
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-<span class="n">hello</span><span class="o">=</span><span class="s1">&#39;hola&#39;</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Integer</span>
-<span class="n">n</span><span class="o">=</span><span class="mi">3</span>
-</pre></div>
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-<p><code>float</code></p>
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-</pre></div>
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-<h1 id="Functions-I">Functions I<a class="anchor-link" href="#Functions-I"> </a></h1><p>Python includes a battery of predefined functions which takes an input and generates an output. For example, to check the type of variable we can used the
-predefined function</p>
-<h2 id="isinistance:"><code>isinistance</code>:<a class="anchor-link" href="#isinistance:"> </a></h2>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">isinstance</span><span class="p">(</span><span class="n">hello</span><span class="p">,</span><span class="nb">str</span><span class="p">)</span>
-</pre></div>
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-<pre>True</pre>
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-<p><strong>Activity</strong>: In the next cell check if <code>n</code>  is a <code>float</code> type of variable</p>
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-<h2 id="print"><code>print</code><a class="anchor-link" href="#print"> </a></h2>
-</div>
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-<p>See: <a href="https://pyformat.info/">https://pyformat.info/</a></p>
-<p>To write the <em>Hello world</em> program in python we must first introduce the concept of function. It is the same in mathematics, were something called function receives a number and return back another number. For example, the function to square a number is
-\begin{equation}
-f(x)=x^2\,,
-\end{equation}
-$x$ is called the argument of the function $f$, and the <em>returned</em> value is the evaluation of $f(x)$.</p>
-<p>In <code>Python</code> there are a lot of such a functions.
-In particular there is a function called <code>print</code> which takes strings (see below) as input and return the same string as output. In this way, the <strong>hello world</strong> program in Python is one of the most simple between all the programming languages:</p>
-
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-<h1 id="Hello-World!">Hello World!<a class="anchor-link" href="#Hello-World!"> </a></h1>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Hello World!&#39;</span><span class="p">)</span>
-</pre></div>
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-<pre>Hello World!
-</pre>
-</div>
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-<p>And also allows scripting: <em>(This code should be copied on a file 'hello.py')</em></p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="ch">#! /usr/bin/python</span>
-
-<span class="c1">#This is a comment</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Hello World!&#39;</span><span class="p">)</span>
-</pre></div>
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-<pre>Hello World!
-</pre>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>The recommended way to print a variable in Python is to use the <code>.format</code> <em>method</em> of the function <code>print</code>:</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">hello</span><span class="o">=</span><span class="s1">&#39;Hello World!&#39;</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">hello</span><span class="p">))</span>
-</pre></div>
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-<pre>Hello World!
-</pre>
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-<p><strong>Activity</strong>: Change the values of the previous string variables to print <code>Hello World!</code> in Spanish</p>
-
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-<h1 id="Functions">Functions<a class="anchor-link" href="#Functions"> </a></h1>
-</div>
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-<p>In <code>Python</code> it is possible also to create new <a href="https://en.wikibooks.org/wiki/Python_Programming/Functions">functions</a>. We illustrate the format to define a function in <code>Python</code> with the implementation of the function $f(x)=x^2$, where to write an exponent: ${}^2$, in Python we must use the format: <code>**2</code>.</p>
-
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-<h2 id="Implicit-functions">Implicit functions<a class="anchor-link" href="#Implicit-functions"> </a></h2>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span>
-</pre></div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="nb">str</span><span class="p">)</span>
-</pre></div>
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-<pre>False</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">type</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-</pre></div>
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-<pre>function</pre>
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-</pre></div>
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-<pre>9</pre>
-</div>
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-<h2 id="Explicit-functions">Explicit functions<a class="anchor-link" href="#Explicit-functions"> </a></h2>
-</div>
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-<p>The standard function build may include the help for the function</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Calculates the square of `x`</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span>
-</pre></div>
-
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-</div>
-</div>
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-</div>
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-<p>f(5)</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">help</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-</pre></div>
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-<pre>Help on function f in module __main__:
-
-f(x)
-    Calculates the square of `x`
-
-</pre>
-</div>
-</div>
-</div>
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-<p>The full list of built-in functions is in <a href="https://docs.python.org/3/library/functions.html">https://docs.python.org/3/library/functions.html</a> and the specific help for a function, for example <code>print</code> can be checked with <a href="https://docs.python.org/3/library/functions.html#print">https://docs.python.org/3/library/functions.html#print</a></p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Multiply two numbers</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="k">return</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span>
-<span class="n">f</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
-
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-</div>
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-<pre>6</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#It is possible to assign default arguments</span>
-<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    Multiply two numbers. By default the second is 2</span>
-<span class="sd">    &#39;&#39;&#39;</span>    
-    <span class="k">return</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span>
-<span class="c1">#When evaluating, we can omit the default argument</span>
-<span class="n">f</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
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-<pre>6</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
-
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-<pre>6</pre>
-</div>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">y</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
-</pre></div>
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-<div class="output_text output_subarea output_execute_result">
-<pre>10</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
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-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#It is possible to specify explicitly the order of the arguments</span>
-<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">    evaluates x to the power y</span>
-<span class="sd">    &#39;&#39;&#39;</span>    
-    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="n">y</span>
-<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;f(1,2)=&#39;</span><span class="p">,</span><span class="n">f</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;f(2,1)=&#39;</span><span class="p">,</span><span class="n">f</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span> <span class="p">)</span>
-</pre></div>
-
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-<pre>f(1,2)= 1
-f(2,1)= 2
-</pre>
-</div>
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-</div>
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-</div>
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-<h2 id="Implicit-functions-of-several-arguments">Implicit functions of several arguments<a class="anchor-link" href="#Implicit-functions-of-several-arguments"> </a></h2>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Implicit functions are usdeful when we want to use a function once.</p>
-
-</div>
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-<div class="inner_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="n">y</span>
-<span class="n">f</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>9</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#It is possible to pass functions as arguments of other function</span>
-<span class="k">def</span> <span class="nf">f2</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">x</span> <span class="p">):</span>
-    <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-
-<span class="c1">#We can define a new function explicitly</span>
-<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">x</span><span class="o">+</span><span class="mi">2</span>
-
-<span class="nb">print</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">f2</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
-
-
-<span class="c1">#Or define the function implicitly</span>
-<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Implicit: f(2)^2 =&quot;</span><span class="p">,</span> <span class="n">f2</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span><span class="n">x</span><span class="o">+</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>4
-16
-Implicit: f(2)^2 = 16
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h1 id="Arithmetics">Arithmetics<a class="anchor-link" href="#Arithmetics"> </a></h1>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Sum">Sum<a class="anchor-link" href="#Sum"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="mf">5.89</span><span class="o">+</span><span class="mf">4.89</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>10.78</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Activity</strong>: Sum strings:
-Hint: use <code>+' '+</code></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Activity</strong>: Sum integers</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>5</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Example</strong></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">hello</span><span class="o">+</span><span class="s1">&#39; &#39;</span><span class="o">+</span><span class="n">world</span><span class="o">+</span><span class="s1">&#39;!&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>Hola Mundo!
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Multiplication">Multiplication<a class="anchor-link" href="#Multiplication"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="mi">120</span><span class="o">*</span><span class="mf">4.5</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>540.0</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><strong>Example</strong> String multiplied by integer:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;=&#39;</span><span class="o">*</span><span class="mi">80</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>================================================================================
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Division"><strong>Division</strong><a class="anchor-link" href="#Division"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Python 3 does support complete division</span>
-<span class="mi">100</span><span class="o">/</span><span class="mi">3</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>33.333333333333336</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="mi">100</span><span class="o">/</span><span class="mf">3.</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>33.333333333333336</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Force integer division</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="mi">100</span><span class="o">//</span><span class="mi">3</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>33</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Module"><strong>Module</strong><a class="anchor-link" href="#Module"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="mi">10</span><span class="o">%</span><span class="k">2</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>0</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="mi">20</span><span class="o">%</span><span class="k">3</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>2</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Power"><strong>Power</strong><a class="anchor-link" href="#Power"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="mi">2</span><span class="o">**</span><span class="mi">6</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>64</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Scientific-notation"><strong>Scientific notation</strong><a class="anchor-link" href="#Scientific-notation"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>$1\times 10^3=10^3=1000$</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="mf">1E3</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>1000.0</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>$2\times 10^3=2000$</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="mf">2E3</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>2000.0</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-$$\frac{ \dfrac{10^{24}}{3}+2.9\times 10^{23}}{10^2}$$
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="p">(</span><span class="mf">1.0e24</span><span class="o">/</span><span class="mf">3.</span> <span class="o">+</span> <span class="mf">2.9e23</span><span class="p">)</span><span class="o">/</span><span class="mf">1e-2</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>6.233333333333333e+25</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">sin</span><span class="o">=</span><span class="mf">0.3</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="o">*</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">isinstance</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span><span class="nb">float</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>False</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Keep the name space</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Use <code>name.last_name</code></p>
-
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-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">sin</span><span class="o">=</span><span class="mf">0.3</span>
-</pre></div>
-
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-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">math</span> <span class="k">as</span> <span class="nn">m</span>
-<span class="kn">import</span> <span class="nn">cmath</span> <span class="k">as</span> <span class="nn">cm</span>
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">import</span> <span class="nn">scipy</span> <span class="k">as</span> <span class="nn">sp</span>
-<span class="c1">#Recommended option:</span>
-<span class="kn">import</span> <span class="nn">numpy.lib.scimath</span> <span class="k">as</span> <span class="nn">sc</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">isinstance</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span><span class="nb">float</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
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-<div class="output">
-
-<div class="jb_output_wrapper }}">
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-
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-<div class="output_text output_subarea output_execute_result">
-<pre>True</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">m</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>0.479425538604203</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">cm</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
-
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-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>(0.479425538604203+0j)</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
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-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>0.479425538604203</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
-
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">sp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stderr output_text">
-<pre>/home/restrepo/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:1: DeprecationWarning: scipy.sin is deprecated and will be removed in SciPy 2.0.0, use numpy.sin instead
-  &#34;&#34;&#34;Entry point for launching an IPython kernel.
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>0.479425538604203</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Complex-numbers">Complex numbers<a class="anchor-link" href="#Complex-numbers"> </a></h2>
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="mi">1</span><span class="n">j</span><span class="o">**</span><span class="mi">2</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
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-<pre>(-1+0j)</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">z</span><span class="o">=</span><span class="mi">2</span><span class="o">+</span><span class="mf">3.2</span><span class="n">j</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">isinstance</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="nb">complex</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>True</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Attributes and methods:</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">z</span><span class="o">.</span><span class="n">real</span><span class="p">,</span><span class="n">z</span><span class="o">.</span><span class="n">imag</span><span class="p">,</span><span class="n">z</span><span class="o">.</span><span class="n">conjugate</span><span class="p">()</span>
-</pre></div>
-
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-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>(2.0, 3.2, (2-3.2j))</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">z</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">z</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>(8+12.8j)</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">z</span><span class="o">*</span><span class="n">z</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
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-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>(-6.240000000000002+12.8j)</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
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-<div class="input">
-
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">z</span><span class="o">*</span><span class="n">z</span><span class="o">.</span><span class="n">conjugate</span><span class="p">()</span>
-</pre></div>
-
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-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>(14.240000000000002+0j)</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><code>math</code> does not work with complex numbers</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">m</span><span class="o">.</span><span class="n">asin</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">0</span><span class="n">j</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_text output_error">
-<pre>
-<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
-<span class="ansi-red-fg">TypeError</span>                                 Traceback (most recent call last)
-<span class="ansi-green-fg">&lt;ipython-input-162-1385535d58fd&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span>
-<span class="ansi-green-fg">----&gt; 1</span><span class="ansi-red-fg"> </span>m<span class="ansi-blue-fg">.</span>asin<span class="ansi-blue-fg">(</span><span class="ansi-cyan-fg">2</span><span class="ansi-blue-fg">+</span><span class="ansi-cyan-fg">0j</span><span class="ansi-blue-fg">)</span>
-
-<span class="ansi-red-fg">TypeError</span>: can&#39;t convert complex to float</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">cm</span><span class="o">.</span><span class="n">asin</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>(1.5707963267948966+1.3169578969248166j)</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><code>numpy</code> requires proper input</p>
-
-</div>
-</div>
-</div>
-</div>
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-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">arcsin</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stderr output_text">
-<pre>/usr/local/lib/python3.7/dist-packages/ipykernel_launcher.py:1: RuntimeWarning: invalid value encountered in arcsin
-  &#34;&#34;&#34;Entry point for launching an IPython kernel.
-</pre>
-</div>
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>nan</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">arcsin</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">0</span><span class="n">j</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>(1.5707963267948966+1.3169578969248166j)</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><code>numpy.lib.scimath</code> imported as <code>sc</code> here, is from <code>sc?</code>:</p>
-<blockquote><p>Wrapper functions to more user-friendly calling of certain math functions
-whose output data-type is different than the input data-type in certain
-domains of the input.
-Function with some parts of its domain in the complex plane like, <code>sqrt</code>, <code>log</code>, <code>log2</code>, <code>logn</code>, <code>log10</code>, <code>power</code>, <code>arccos</code>, <code>arcsin</code>, and <code>arctanh</code>.</p>
-</blockquote>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
-
-<div class="inner_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">lib</span><span class="o">.</span><span class="n">scimath</span><span class="o">.</span><span class="n">arcsin</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>(1.5707963267948966+1.3169578969248166j)</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="input">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">sc</span><span class="o">.</span><span class="n">arcsin</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>(1.5707963267948966+1.3169578969248166j)</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">ipywidgets</span> <span class="k">as</span> <span class="nn">widgets</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell tag_interactive">
-
-<div class="cell border-box-sizing code_cell rendered tag_interactive">
-<div class="input">
-
-<div class="inner_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nd">@widgets</span><span class="o">.</span><span class="n">interact</span>
-<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">)):</span>
-    <span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">sc</span><span class="o">.</span><span class="n">arcsin</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
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-<span class="n">lista</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;abc&quot;</span><span class="p">,</span> <span class="mi">42</span><span class="p">,</span> <span class="mf">3.1415</span><span class="p">]</span>
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-<span class="n">lista</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
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-<span class="n">lista</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
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-<span class="n">lista</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
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-<span class="n">lista</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="s2">&quot;I am second&quot;</span><span class="p">)</span>
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-<span class="k">del</span> <span class="n">lista</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
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-<span class="n">lista</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="s2">&quot;xyz&quot;</span>
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-<div class="highlight"><pre><span></span><span class="n">l</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">end</span><span class="p">]</span>
-</pre></div>
-<p>We can also define the step, like this:</p>
-<div class="highlight"><pre><span></span><span class="n">l</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">end</span><span class="p">:</span><span class="n">step</span><span class="p">]</span>
-</pre></div>
-<p>If <code>start</code> is not passed it is considered 0. If <code>end</code> is not passed it is considered length of array in that dimension. The <code>end</code> can given in reverse order by assigning a minus signus to the index. For example <code>-1</code> means the last element, whicle <code>-2</code> means the penultimate, and so on and so forth.</p>
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-<span class="n">lista</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">3</span><span class="p">]</span>
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-<span class="n">lista</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span>
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-<span class="n">lista</span><span class="p">[::</span><span class="mi">2</span><span class="p">]</span>
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-<span class="n">lista</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
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-<span class="n">lista</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-<span class="n">lista</span>
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-<h3 id="Embedded-lists">Embedded lists<a class="anchor-link" href="#Embedded-lists"> </a></h3>
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-<span class="n">embedded_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">lista</span><span class="p">,</span> <span class="p">[</span><span class="kc">True</span><span class="p">,</span> <span class="mi">42</span><span class="p">]]</span>
-<span class="n">embedded_list</span>
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-<span class="n">embedded_list</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
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-<span class="n">A</span> <span class="o">=</span> <span class="p">[</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">]</span> <span class="p">]</span>
-<span class="n">A</span>
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-<p><strong>Activity</strong>: Obtain entry $A_{01}$ of the previous matrix, where
-\begin{align}
-A=\begin{pmatrix}
-A_{00} &amp; A_{01}\\
-A_{10} &amp; A_{11}\\
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-<span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="s2">&quot;ab&quot;</span><span class="p">,</span><span class="kc">True</span><span class="p">,[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">]]</span> <span class="o">+</span> <span class="p">[</span><span class="mf">3.1415</span><span class="p">,</span><span class="s2">&quot;Pi&quot;</span><span class="p">,</span><span class="s2">&quot;circle&quot;</span><span class="p">]</span>
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-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;after append 2 to [] : </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="n">y</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;after append 5 to [2]: </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
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-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;original: </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">lista</span><span class="p">))</span>
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-<span class="n">tupla</span> <span class="o">=</span> <span class="p">(</span><span class="s2">&quot;abc&quot;</span><span class="p">,</span> <span class="mi">42</span><span class="p">,</span> <span class="mf">3.1415</span><span class="p">)</span>
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-<p><code>any</code> can extract a true value from a comprension tuple (see also <code>all</code>: <a href="https://docs.python.org/3/library/functions.html#all">https://docs.python.org/3/library/functions.html#all</a>). 
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-<p>we can extract the list that contains <code>'H'</code> more easily</p>
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-<pre>[[&#39;A&#39;, &#39;B&#39;, &#39;C&#39;], [&#39;D&#39;, &#39;E&#39;, &#39;F&#39;], [&#39;G&#39;, &#39;H&#39;, &#39;I&#39;]]</pre>
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-<p>With the function <code>zip</code> we can generate dictionary-like tuple from two lists:</p>
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-<span class="n">tupla</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s2">&quot;xy&quot;</span><span class="p">)</span>
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-<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
-<span class="ansi-red-fg">AttributeError</span>                            Traceback (most recent call last)
-<span class="ansi-green-fg">&lt;ipython-input-87-4cc56b436273&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span>
-<span class="ansi-green-intense-fg ansi-bold">      1</span> <span class="ansi-red-fg">#It is not possible to add more elements</span>
-<span class="ansi-green-fg">----&gt; 2</span><span class="ansi-red-fg"> </span>tupla<span class="ansi-blue-fg">.</span>append<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">&#34;xy&#34;</span><span class="ansi-blue-fg">)</span>
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-<span class="ansi-red-fg">AttributeError</span>: &#39;tuple&#39; object has no attribute &#39;append&#39;</pre>
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-<span class="ansi-green-fg">&lt;ipython-input-88-c05ec550c1d0&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span>
-<span class="ansi-green-intense-fg ansi-bold">      1</span> <span class="ansi-red-fg">#It is not possible to delete an element</span>
-<span class="ansi-green-fg">----&gt; 2</span><span class="ansi-red-fg"> </span><span class="ansi-green-fg">del</span> tupla<span class="ansi-blue-fg">[</span><span class="ansi-cyan-fg">0</span><span class="ansi-blue-fg">]</span>
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-<span class="ansi-red-fg">TypeError</span>: &#39;tuple&#39; object doesn&#39;t support item deletion</pre>
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-<span class="n">tupla</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="s2">&quot;xy&quot;</span>
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-<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
-<span class="ansi-red-fg">TypeError</span>                                 Traceback (most recent call last)
-<span class="ansi-green-fg">&lt;ipython-input-89-4e97d1641827&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span>
-<span class="ansi-green-intense-fg ansi-bold">      1</span> <span class="ansi-red-fg">#It is not possible to modify an existing element</span>
-<span class="ansi-green-fg">----&gt; 2</span><span class="ansi-red-fg"> </span>tupla<span class="ansi-blue-fg">[</span><span class="ansi-cyan-fg">0</span><span class="ansi-blue-fg">]</span> <span class="ansi-blue-fg">=</span> <span class="ansi-blue-fg">&#34;xy&#34;</span>
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-<span class="ansi-red-fg">TypeError</span>: &#39;tuple&#39; object does not support item assignment</pre>
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-<h2 id="Dictionaries">Dictionaries<a class="anchor-link" href="#Dictionaries"> </a></h2>
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-<p>Dictionaries are labeled lists: with keys and values. They are extremely useful when manipulating complex data.</p>
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-<span class="n">dictionary</span><span class="o">=</span><span class="p">{</span> <span class="s1">&#39;Kenia&#39;</span>         <span class="p">:</span><span class="s1">&#39;Nairobi&#39;</span><span class="p">,</span>
-             <span class="s1">&#39;Noruega&#39;</span>       <span class="p">:</span><span class="s1">&#39;Oslo&#39;</span><span class="p">,</span>
-             <span class="s1">&#39;Finlandia&#39;</span>     <span class="p">:</span><span class="s1">&#39;Helsinski&#39;</span><span class="p">,</span>
-             <span class="s1">&#39;Rusia&#39;</span>         <span class="p">:</span><span class="s1">&#39;Moscú&#39;</span><span class="p">,</span>
-             <span class="s1">&#39;Rio de Janeiro&#39;</span><span class="p">:</span><span class="s1">&#39;Rio&#39;</span><span class="p">,</span>
-             <span class="s1">&#39;Japón&#39;</span>         <span class="p">:</span><span class="s1">&#39;Tokio&#39;</span><span class="p">,</span>
-             <span class="s1">&#39;Colorado&#39;</span>      <span class="p">:</span><span class="s1">&#39;Denver&#39;</span><span class="p">,</span>
-             <span class="s1">&#39;Alemania&#39;</span>      <span class="p">:</span><span class="s1">&#39;Berlin&#39;</span><span class="p">,</span>
-             <span class="s1">&#39;Colombia&#39;</span>      <span class="p">:</span><span class="s1">&#39;Bogotá&#39;</span><span class="p">}</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">dictionary</span><span class="p">)</span>
-<span class="c1">#Note the order in a dictionary does not matter as one identifies a single element through a string, not a number</span>
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-<pre>{&#39;Kenia&#39;: &#39;Nairobi&#39;, &#39;Noruega&#39;: &#39;Oslo&#39;, &#39;Finlandia&#39;: &#39;Helsinski&#39;, &#39;Rusia&#39;: &#39;Moscú&#39;, &#39;Rio de Janeiro&#39;: &#39;Rio&#39;, &#39;Japón&#39;: &#39;Tokio&#39;, &#39;Colorado&#39;: &#39;Denver&#39;, &#39;Alemania&#39;: &#39;Berlin&#39;, &#39;Colombia&#39;: &#39;Bogotá&#39;}
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-<p>Instead of a number, an element of a dictionary is accessed with the key of the component</p>
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-<pre>{&#39;Kenia&#39;: &#39;Nairobi&#39;,
- &#39;Noruega&#39;: &#39;Oslo&#39;,
- &#39;Finlandia&#39;: &#39;Helsinski&#39;,
- &#39;Rusia&#39;: &#39;Moscú&#39;,
- &#39;Rio de Janeiro&#39;: &#39;Rio&#39;,
- &#39;Japón&#39;: &#39;Tokio&#39;,
- &#39;Colorado&#39;: &#39;Denver&#39;,
- &#39;Alemania&#39;: &#39;Berlin&#39;,
- &#39;Colombia&#39;: &#39;Bogotá&#39;}</pre>
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-<pre>Tokio ♥ Rio
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-<span class="n">dictionary2</span> <span class="o">=</span> <span class="p">{</span> <span class="s2">&quot;Enteros&quot;</span><span class="p">:[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span> 
-                <span class="s2">&quot;Ciudad&quot;</span> <span class="p">:</span><span class="s2">&quot;Medellin&quot;</span><span class="p">,</span> 
-                <span class="s2">&quot;Cédula&quot;</span> <span class="p">:</span><span class="mi">1128400433</span><span class="p">,</span> 
-                <span class="s2">&quot;Colores&quot;</span><span class="p">:[</span><span class="s2">&quot;Amarillo&quot;</span><span class="p">,</span> <span class="s2">&quot;Azul&quot;</span><span class="p">,</span> <span class="s2">&quot;Rojo&quot;</span><span class="p">]</span> <span class="p">}</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">dictionary2</span><span class="p">[</span><span class="s2">&quot;Colores&quot;</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span>
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-<span class="n">dictionary2</span><span class="p">[</span><span class="s2">&quot;Ciudad&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="s2">&quot;Bogota&quot;</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">dictionary2</span><span class="p">)</span>
-</pre></div>
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-<pre>{&#39;Enteros&#39;: [1, 2, 3, 4, 5], &#39;Ciudad&#39;: &#39;Bogota&#39;, &#39;Cédula&#39;: 1128400433, &#39;Colores&#39;: [&#39;Amarillo&#39;, &#39;Azul&#39;, &#39;Rojo&#39;]}
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-<span class="n">dictionary2</span><span class="p">[</span><span class="s2">&quot;Pais&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="s2">&quot;Colombia&quot;</span>
-<span class="n">dictionary2</span><span class="p">[</span><span class="s2">&quot;País&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="s2">&quot;Colombia&quot;</span>
-<span class="nb">print</span><span class="p">(</span> <span class="n">dictionary2</span> <span class="p">)</span>
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-<pre>{&#39;Enteros&#39;: [1, 2, 3, 4, 5], &#39;Ciudad&#39;: &#39;Bogota&#39;, &#39;Cédula&#39;: 1128400433, &#39;Colores&#39;: [&#39;Amarillo&#39;, &#39;Azul&#39;, &#39;Rojo&#39;], &#39;Pais&#39;: &#39;Colombia&#39;, &#39;País&#39;: &#39;Colombia&#39;}
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#dictionary2.update({&#39;Pais&#39;:&#39;Colombia&#39;})</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#The command del can be also used for deleting an element, as a list</span>
-<span class="k">del</span> <span class="n">dictionary2</span><span class="p">[</span><span class="s2">&quot;Pais&quot;</span><span class="p">]</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">dictionary2</span><span class="p">)</span>
-</pre></div>
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-<pre>{&#39;Enteros&#39;: [1, 2, 3, 4, 5], &#39;Ciudad&#39;: &#39;Bogota&#39;, &#39;Cédula&#39;: 1128400433, &#39;Colores&#39;: [&#39;Amarillo&#39;, &#39;Azul&#39;, &#39;Rojo&#39;], &#39;País&#39;: &#39;Colombia&#39;}
-</pre>
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-<p>With the previous <code>zip</code> function to create tuples from lists, we can create a dictionary from the two lists:</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">dict</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span> <span class="p">[</span><span class="s1">&#39;A&#39;</span><span class="p">,</span><span class="s1">&#39;B&#39;</span><span class="p">,</span><span class="s1">&#39;C&#39;</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">]</span>  <span class="p">))</span>
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-<pre>{&#39;A&#39;: 1, &#39;B&#39;: 2, &#39;C&#39;: 3}</pre>
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-<p><strong>Activity</strong>: Creates a diccionary for the values: <code>['xyz',3,4.5]</code> with integer keys starting with zero. In this way, the dictionary could behave as list</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">l</span><span class="o">=</span><span class="p">{}</span>
-<span class="n">l</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">=</span><span class="s1">&#39;xyz&#39;</span>
-<span class="n">l</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">=</span><span class="mi">3</span>
-<span class="n">l</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">=</span><span class="mf">4.5</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">l</span>
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-<pre>{0: &#39;xyz&#39;, 1: 3, 2: 4.5}</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">l</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</pre></div>
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-<pre>&#39;xyz&#39;</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">d</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span><span class="o">=</span><span class="mf">5.5</span>
-</pre></div>
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-<h1 id="Bucles-and-Conditionals">Bucles and Conditionals<a class="anchor-link" href="#Bucles-and-Conditionals"> </a></h1>
-</div>
-</div>
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-<h2 id="if">if<a class="anchor-link" href="#if"> </a></h2>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Conditionals are useful when we want to check some condition.
-The statements <code>elif</code> and <code>else</code> can be used when more than one condition need to be used, or when there is something to do when some condition is not fulfilled.</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="mi">10</span>
-<span class="n">y</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="k">if</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="mi">5</span> <span class="ow">and</span> <span class="n">y</span><span class="o">==</span><span class="mi">2</span><span class="p">:</span>
-    <span class="nb">print</span><span class="p">(</span> <span class="s2">&quot;True&quot;</span> <span class="p">)</span>
-</pre></div>
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-<pre>True
-</pre>
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="mi">4</span>
-<span class="n">y</span> <span class="o">=</span> <span class="mi">3</span>
-<span class="k">if</span> <span class="n">x</span><span class="o">&gt;</span><span class="mi">5</span> <span class="ow">or</span> <span class="n">y</span><span class="o">&lt;</span><span class="mi">2</span><span class="p">:</span>
-    <span class="nb">print</span><span class="p">(</span> <span class="s2">&quot;True 1&quot;</span> <span class="p">)</span>
-<span class="k">elif</span> <span class="n">x</span><span class="o">==</span><span class="mi">4</span><span class="p">:</span>
-    <span class="nb">print</span><span class="p">(</span> <span class="s2">&quot;True 2&quot;</span> <span class="p">)</span>
-<span class="k">else</span><span class="p">:</span>
-    <span class="nb">print</span><span class="p">(</span> <span class="s2">&quot;False&quot;</span> <span class="p">)</span>
-</pre></div>
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-<pre>True 2
-</pre>
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-<h2 id="for">for<a class="anchor-link" href="#for"> </a></h2>
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-<p><code>For</code> cycles are specially useful when we want to sweep a set of elements with a known size.</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">1</span><span class="p">):</span>
-    <span class="nb">print</span><span class="p">(</span> <span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
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-<pre>0 0
-1 1
-2 4
-3 9
-4 16
-</pre>
-</div>
-</div>
-</div>
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-</div>
-
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-</div>
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-<p><strong>Activity</strong>: change print with format</p>
-
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-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">suma</span> <span class="o">=</span> <span class="mi">0</span>
-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">1</span><span class="p">):</span>
-    <span class="n">suma</span> <span class="o">+=</span> <span class="n">i</span><span class="o">**</span><span class="mi">2</span> <span class="c1"># suma = suma + i**2</span>
-<span class="nb">print</span> <span class="p">(</span> <span class="s2">&quot;The result is </span><span class="si">%d</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">suma</span><span class="p">)</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>The result is 285
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">for</span> <span class="n">language</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;Python&#39;</span><span class="p">,</span> <span class="s1">&#39;C&#39;</span><span class="p">,</span> <span class="s1">&#39;C++&#39;</span><span class="p">,</span> <span class="s1">&#39;Ruby&#39;</span><span class="p">,</span> <span class="s1">&#39;Java&#39;</span><span class="p">]:</span>
-    <span class="nb">print</span> <span class="p">(</span> <span class="n">language</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
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-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>Python
-C
-C++
-Ruby
-Java
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>As we see before, <code>for</code> can be used to build comprenhension lists</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">serie</span> <span class="o">=</span> <span class="p">[</span> <span class="n">i</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span> <span class="p">]</span>
-<span class="nb">print</span><span class="p">(</span> <span class="n">serie</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>[1, 4, 9, 16, 25, 36, 49, 64, 81]
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="while">while<a class="anchor-link" href="#while"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p><code>While</code> cycles are specially useful when we want to sweep a set of elements with an  unknown size.</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Before we check the functions <code>input</code> and <code>int</code> of Python</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">edad</span><span class="o">=</span><span class="nb">input</span><span class="p">(</span><span class="s1">&#39;¿What is your age, Jesus?:</span><span class="se">\n</span><span class="s1">&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">int</span><span class="p">(</span><span class="n">edad</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>33</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Bonus: some time the input must be hidden</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">getpass</span>
-</pre></div>
-
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-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">c</span><span class="o">=</span><span class="n">getpass</span><span class="o">.</span><span class="n">getpass</span><span class="p">(</span><span class="s1">&#39;Contraseña&#39;</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
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-<div class="output">
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-<div class="output_area">
-
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<p>Now the <code>while</code> examples</p>
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-<span class="n">number</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">input</span><span class="p">(</span><span class="s2">&quot;Write a negative number: &quot;</span><span class="p">))</span>
-<span class="k">while</span> <span class="n">number</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-    <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;You wrote a positive number. Do it again&quot;</span><span class="p">)</span>
-    <span class="n">number</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">input</span><span class="p">(</span><span class="s2">&quot;Write a negative number: &quot;</span><span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Thank you!&quot;</span><span class="p">)</span>
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-<span class="n">x</span> <span class="o">=</span> <span class="mi">0</span>
-<span class="k">while</span> <span class="n">x</span><span class="o">&lt;</span><span class="mf">0.9</span><span class="p">:</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">()</span>
-    <span class="nb">print</span><span class="p">(</span> <span class="n">x</span> <span class="p">)</span>
-<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;The selected number was&quot;</span><span class="p">,</span> <span class="n">x</span> <span class="p">)</span>
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-<pre>0.8014478544604644
-0.16730340110095598
-0.22609305175080763
-0.3910061658713566
-0.16471583849093963
-0.03893794385343341
-0.9631958511875766
-The selected number was 0.9631958511875766
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-<h1 id="Methods-and-attributes-of-objects-in-Python">Methods and attributes of objects in Python<a class="anchor-link" href="#Methods-and-attributes-of-objects-in-Python"> </a></h1><p>All the objects in <code>Python</code>, including the function and variable types discussed here, are enhanced with special functions called <em>methods</em>, which are implemented after a point of the name of the variable, in the format:</p>
-<div class="highlight"><pre><span></span><span class="n">variable</span><span class="o">.</span><span class="n">method</span><span class="p">()</span>
-</pre></div>
-<p>Some times, the method can even accept some arguments.</p>
-<p>The objects have also attributes which describe some property of the object. They do not end up with parenthesis:</p>
-<div class="highlight"><pre><span></span><span class="n">variable</span><span class="o">.</span><span class="n">attribute</span>
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-<h3 id="Methods">Methods<a class="anchor-link" href="#Methods"> </a></h3><p>For example. The method <code>.keys()</code> of a variable dictionary allows to obtain the list of keys of the dictionary. For example</p>
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-<pre>dict_keys([&#39;Kenia&#39;, &#39;Noruega&#39;, &#39;Finlandia&#39;, &#39;Rusia&#39;, &#39;Rio de Janeiro&#39;, &#39;Japón&#39;, &#39;Colorado&#39;, &#39;Alemania&#39;, &#39;Colombia&#39;])</pre>
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-<p>And the list of values:</p>
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-<pre>dict_values([&#39;Nairobi&#39;, &#39;Oslo&#39;, &#39;Helsinski&#39;, &#39;Moscú&#39;, &#39;Rio&#39;, &#39;Tokio&#39;, &#39;Denver&#39;, &#39;Berlin&#39;, &#39;Bogotá&#39;])</pre>
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-<p>For strings we have for example the conversion to lower case</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="s2">&quot;Juan Valdez&quot;</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span>
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-<pre>&#39;juan valdez&#39;</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="s2">&quot;juan valdez&quot;</span><span class="o">.</span><span class="n">title</span><span class="p">()</span>
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-<pre>&#39;Juan Valdez&#39;</pre>
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-</pre></div>
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-<pre>&#39;JUAN VALDEZ&#39;</pre>
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-<h3 id="Attributes">Attributes<a class="anchor-link" href="#Attributes"> </a></h3>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span><span class="o">=</span><span class="mi">1</span><span class="n">j</span>
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-</pre></div>
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-<pre>1.0</pre>
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-</pre></div>
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-<pre>0.0</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">z</span><span class="o">=</span><span class="mi">3</span><span class="o">+</span><span class="mi">5</span><span class="n">j</span>
-</pre></div>
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-<p>with attributes</p>
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-</pre></div>
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-<pre>3.0</pre>
-</div>
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-</pre></div>
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-<pre>5.0</pre>
-</div>
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-<p>and the method:</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">z</span><span class="o">.</span><span class="n">conjugate</span><span class="p">()</span>
-</pre></div>
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-<pre>(3-5j)</pre>
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-<p>In the notebook, All the methods and attributes of an object can be accessed by using the <code>&lt;TAB&gt;</code> key after write down the point:</p>
-<div class="highlight"><pre><span></span><span class="n">variable</span><span class="o">.&lt;</span><span class="n">TAB</span><span class="o">&gt;</span>
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-<p><strong>Activity</strong>: Check the methods and attributes of the dictionary <code>dictonary</code>. HINT: Check the help for some of them by using a question mark, "?", at the end:</p>
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-<pre><code>variable.method?</code></pre>
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-<h2 id="Unicode">Unicode<a class="anchor-link" href="#Unicode"> </a></h2><p>Is an standard to encode characters. In Python 3, around  <a href="https://stackoverflow.com/a/17043983">120.000 characters</a> can be used to define variables. For example, one right to left arabic variable can be defined as</p>
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-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Arabic character values is: </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">ࢶ</span><span class="p">))</span>
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-<pre>Arabic character values is: 2
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-<p>Spanish example</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">mamá</span><span class="o">=</span><span class="s1">&#39;Lola&#39;</span>
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-<p>In Jupyter lab greek symbols can be accessed by using its LaTeX command follow by the <code>&lt;TAB&gt;</code> key.</p>
-<p><code>\alpha+&lt;TAB&gt;=0.5</code> could convert on the fly to</p>
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-<pre>0.5
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-<p>Alternatively yuo can copy the character from some list of unicode symbols, like <a href="https://en.wikipedia.org/wiki/List_of_Unicode_characters#Greek_and_Coptic">this one</a>.</p>
-<p><strong>Activity</strong>: Define a Greek variable by using a symbol from the previous list</p>
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-<h2 id="Final-remarks">Final remarks<a class="anchor-link" href="#Final-remarks"> </a></h2><p>Make your google Python questions in English and check specially the <a href="https://stackoverflow.com/">https://stackoverflow.com/</a> results.</p>
-<p><strong>Activity</strong>: Make some Python query in Google and paste the example code below</p>
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-<span class="o">&gt;&gt;&gt;</span> <span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="o">&gt;&gt;&gt;</span> <span class="n">Parallel</span><span class="p">(</span><span class="n">n_jobs</span><span class="o">=</span><span class="mi">8</span><span class="p">)(</span><span class="n">delayed</span><span class="p">(</span><span class="n">sqrt</span><span class="p">)(</span><span class="n">i</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">))</span>
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-<h1 id="Scientific-Libraries-with-Python">Scientific Libraries with Python<a class="anchor-link" href="#Scientific-Libraries-with-Python"> </a></h1><p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/scientific-libraries.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
-<p>Although coding with Python is very versatile and allows many advanced features that are useful when manipulating massive data (a common task in science), Python is still a multipurpose language, what implies that scientific routines and functions cannot (should not) be supported within its basic core. Nevertheless, there are many different scientific libraries that can extend the capabilities of Python to scientific implementations in a natural way. One of the most used libraries is NumPy. This introduce the array object as a generalization of Python nested lists. This new object has many linear algebra operations implemented as methods or attributes. This operations are implemented at the low level through fast highly optimized algorithms.</p>
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-<h2 id="Python+NumPy">Python+NumPy<a class="anchor-link" href="#Python+NumPy"> </a></h2><p>In this way, Python+NumPy can be seen as a framework to implement numerical code as linear algebra abstractions which replaces the slow Python loops. The contrary is also true. In this framework each algorithm must be designed to try to avoid the use of Python loops like <code>for</code> or <code>while</code>.</p>
-<p>An ideal program implemented in Python+NumPy does not have explicit Python loops, but only linear algebra abstractions.</p>
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-<h2 id="Extended-Python+Numpy-framework">Extended Python+Numpy framework<a class="anchor-link" href="#Extended-Python+Numpy-framework"> </a></h2><p>All the other high level packages for scientific computation are designed to work around the linear algebra abstractions of the Python+NumPy framework</p>
-<ul>
-<li>Pandas add labels to the Numpy arrays.</li>
-<li>Mathplotlib plotting library. Ti visualize arrays in 1, 2 and 3 dimensions</li>
-<li>SciPy, intended for manipulating NumPy arrays more efficiently and for extending and including numerical methods, respectively. </li>
-</ul>
-<p>Another less used libraries like SymPy are intended for manipulating analytical expressions, i.e. a CAS (Computer Algebraic System).</p>
-<p>Installation of these libraries is often an easy task. In most of the Linux distros you should find them in the official repositories.</p>
-<p>Avoid loops. Use abstractions</p>
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-<h2 id="Official-Pages">Official Pages<a class="anchor-link" href="#Official-Pages"> </a></h2><p>See the official pages of the libraries for new versions, news and manuals.</p>
-<p><strong>NumPy:</strong></p>
-<p><a href="http://www.numpy.org/">http://www.numpy.org/</a></p>
-<p><strong>SciPy</strong></p>
-<p><a href="http://www.scipy.org/">http://www.scipy.org/</a></p>
-<p><strong>SymPy</strong></p>
-<p><a href="http://www.sympy.org/">http://www.sympy.org/</a></p>
-<p><strong>Anaconda</strong></p>
-<p><a href="https://www.anaconda.com/">https://www.anaconda.com/</a></p>
-<p>Anaconda is a self-cointained Python distribution that integrates many standard scientific libraries with Python, along with some generic libraries like MongoDB</p>
-<p>There are many different scientific libraries for Python with many different uses, even for very specific tasks. However, as we are interested in general numerical methods, we will focus only on NumPy and Scipy.</p>
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-<li><a href="#Official-Pages">Official Pages</a></li>
-<li><a href="#NumPy"><strong>NumPy</strong></a><ul>
-<li><a href="#Basic-Use">Basic Use</a><ul>
-<li><a href="#Importing-and-basic-math">Importing and basic math</a></li>
-<li><a href="#Lists-vs-NumPy-arrays">Lists vs NumPy arrays</a></li>
-<li><a href="#Advanced-features-of-arrays">Advanced features of arrays</a></li>
-</ul>
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-<li><a href="#Miscellaneous-Functions">Miscellaneous Functions</a></li>
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-</div><h1 id="NumPy">NumPy<a class="anchor-link" href="#NumPy"> </a></h1><p>NumPy is the fundamental package for scientific computing with Python. It contains among other things:</p>
-<ul>
-<li>a powerful N-dimensional array object</li>
-<li>sophisticated (broadcasting) functions</li>
-<li>tools for integrating C/C++ and Fortran code</li>
-<li>useful linear algebra, Fourier transform, and random number capabilities</li>
-</ul>
-<p>Besides its obvious scientific uses, NumPy can also be used as an efficient multi-dimensional container of generic data. Arbitrary data-types can be defined. This allows NumPy to seamlessly and speedily integrate with a wide variety of databases.</p>
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-<h2 id="Basic-Use">Basic Use<a class="anchor-link" href="#Basic-Use"> </a></h2><h3 id="Importing-and-basic-math">Importing and basic math<a class="anchor-link" href="#Importing-and-basic-math"> </a></h3><p>NumPy can be imported in several different ways. Keeping the standard name space is the recommended way</p>
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-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
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-<div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-<span class="kn">import</span> <span class="nn">scipy</span> <span class="k">as</span> <span class="nn">sp</span>
-<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
-<span class="kn">import</span> <span class="nn">numpy.linalg</span> <span class="k">as</span> <span class="nn">la</span>
-<span class="kn">import</span> <span class="nn">math</span> <span class="k">as</span> <span class="nn">m</span> <span class="c1"># real </span>
-<span class="kn">import</span> <span class="nn">cmath</span> <span class="k">as</span> <span class="nn">cm</span> <span class="c1"># complex</span>
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-<p>If imported through the name space <code>np</code>, NumPy methods and attributes are accessed by using the assigned name space.</p>
-<p>The Basic methods of NumPy include the usual mathematical functions</p>
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-<pre>36315.502674246636 3.9569963710708773 1.8055008581584002 3.1622776601683795
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-<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mf">5.0</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="mf">9.6</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">arcsin</span><span class="p">(</span><span class="mf">0.5</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">arctan</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
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-<pre>-0.9589242746631385 -0.984687855794127 0.5235987755982989 1.373400766945016
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-<p>The basic attributes include some important constants</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span>  <span class="p">(</span> <span class="s2">&quot;The value of PI is </span><span class="si">{:.6f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="p">)</span> <span class="p">)</span>
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-<pre>The value of PI is 3.141593
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span>  <span class="p">(</span> <span class="s2">&quot;The value of e is </span><span class="si">{:.6f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">e</span> <span class="p">)</span> <span class="p">)</span>
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-<pre>The value of e is 2.718282
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-<pre>∞=inf
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="o">*</span>
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-<pre>array([0.90929743, 0.14112001])</pre>
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-<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
-<span class="ansi-red-fg">TypeError</span>                                 Traceback (most recent call last)
-<span class="ansi-green-fg">&lt;ipython-input-17-0612cee6990d&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span>
-<span class="ansi-green-fg">----&gt; 1</span><span class="ansi-red-fg"> </span>sin<span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">[</span><span class="ansi-cyan-fg">2</span><span class="ansi-blue-fg">,</span><span class="ansi-cyan-fg">3</span><span class="ansi-blue-fg">]</span><span class="ansi-blue-fg">)</span>
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-<h3 id="Lists-vs-NumPy-arrays">Lists vs NumPy arrays<a class="anchor-link" href="#Lists-vs-NumPy-arrays"> </a></h3>
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-<p><strong>Supported methods for lists</strong></p>
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-<p><code>x.append   x.count    x.extend   x.index    x.insert   x.pop      x.remove   x.reverse  x.sort</code></p>
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-<p><strong>Supported methods for NumPy arrays</strong></p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span><span class="o">.</span><span class="n">T</span>             <span class="n">x</span><span class="o">.</span><span class="n">clip</span>          <span class="n">x</span><span class="o">.</span><span class="n">dot</span>           <span class="n">x</span><span class="o">.</span><span class="n">item</span>          <span class="n">x</span><span class="o">.</span><span class="n">prod</span>          <span class="n">x</span><span class="o">.</span><span class="n">setfield</span>      <span class="n">x</span><span class="o">.</span><span class="n">take</span>
-<span class="n">x</span><span class="o">.</span><span class="n">all</span>           <span class="n">x</span><span class="o">.</span><span class="n">compress</span>      <span class="n">x</span><span class="o">.</span><span class="n">dtype</span>         <span class="n">x</span><span class="o">.</span><span class="n">itemset</span>       <span class="n">x</span><span class="o">.</span><span class="n">ptp</span>           <span class="n">x</span><span class="o">.</span><span class="n">setflags</span>      <span class="n">x</span><span class="o">.</span><span class="n">tofile</span>
-<span class="n">x</span><span class="o">.</span><span class="n">any</span>           <span class="n">x</span><span class="o">.</span><span class="n">conj</span>          <span class="n">x</span><span class="o">.</span><span class="n">dump</span>          <span class="n">x</span><span class="o">.</span><span class="n">itemsize</span>      <span class="n">x</span><span class="o">.</span><span class="n">put</span>           <span class="n">x</span><span class="o">.</span><span class="n">shape</span>         <span class="n">x</span><span class="o">.</span><span class="n">tolist</span>
-<span class="n">x</span><span class="o">.</span><span class="n">argmax</span>        <span class="n">x</span><span class="o">.</span><span class="n">conjugate</span>     <span class="n">x</span><span class="o">.</span><span class="n">dumps</span>         <span class="n">x</span><span class="o">.</span><span class="n">max</span>           <span class="n">x</span><span class="o">.</span><span class="n">ravel</span>         <span class="n">x</span><span class="o">.</span><span class="n">size</span>          <span class="n">x</span><span class="o">.</span><span class="n">tostring</span>
-<span class="n">x</span><span class="o">.</span><span class="n">argmin</span>        <span class="n">x</span><span class="o">.</span><span class="n">copy</span>          <span class="n">x</span><span class="o">.</span><span class="n">fill</span>          <span class="n">x</span><span class="o">.</span><span class="n">mean</span>          <span class="n">x</span><span class="o">.</span><span class="n">real</span>          <span class="n">x</span><span class="o">.</span><span class="n">sort</span>          <span class="n">x</span><span class="o">.</span><span class="n">trace</span>
-<span class="n">x</span><span class="o">.</span><span class="n">argsort</span>       <span class="n">x</span><span class="o">.</span><span class="n">ctypes</span>        <span class="n">x</span><span class="o">.</span><span class="n">flags</span>         <span class="n">x</span><span class="o">.</span><span class="n">min</span>           <span class="n">x</span><span class="o">.</span><span class="n">repeat</span>        <span class="n">x</span><span class="o">.</span><span class="n">squeeze</span>       <span class="n">x</span><span class="o">.</span><span class="n">transpose</span>
-<span class="n">x</span><span class="o">.</span><span class="n">astype</span>        <span class="n">x</span><span class="o">.</span><span class="n">cumprod</span>       <span class="n">x</span><span class="o">.</span><span class="n">flat</span>          <span class="n">x</span><span class="o">.</span><span class="n">nbytes</span>        <span class="n">x</span><span class="o">.</span><span class="n">reshape</span>       <span class="n">x</span><span class="o">.</span><span class="n">std</span>           <span class="n">x</span><span class="o">.</span><span class="n">var</span>
-<span class="n">x</span><span class="o">.</span><span class="n">base</span>          <span class="n">x</span><span class="o">.</span><span class="n">cumsum</span>        <span class="n">x</span><span class="o">.</span><span class="n">flatten</span>       <span class="n">x</span><span class="o">.</span><span class="n">ndim</span>          <span class="n">x</span><span class="o">.</span><span class="n">resize</span>        <span class="n">x</span><span class="o">.</span><span class="n">strides</span>       <span class="n">x</span><span class="o">.</span><span class="n">view</span>
-<span class="n">x</span><span class="o">.</span><span class="n">byteswap</span>      <span class="n">x</span><span class="o">.</span><span class="n">data</span>          <span class="n">x</span><span class="o">.</span><span class="n">getfield</span>      <span class="n">x</span><span class="o">.</span><span class="n">newbyteorder</span>  <span class="n">x</span><span class="o">.</span><span class="n">round</span>         <span class="n">x</span><span class="o">.</span><span class="n">sum</span>           
-<span class="n">x</span><span class="o">.</span><span class="n">choose</span>        <span class="n">x</span><span class="o">.</span><span class="n">diagonal</span>      <span class="n">x</span><span class="o">.</span><span class="n">imag</span>          <span class="n">x</span><span class="o">.</span><span class="n">nonzero</span>       <span class="n">x</span><span class="o">.</span><span class="n">searchsorted</span>  <span class="n">x</span><span class="o">.</span><span class="n">swapaxes</span>
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-<p><strong>In common</strong></p>
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-<span class="n">x1</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">,</span> <span class="mf">1.9</span><span class="p">]</span>
-<span class="n">x2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">1.6</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.6</span><span class="p">,</span> <span class="mf">6.9</span><span class="p">])</span>
-<span class="nb">print</span><span class="p">(</span> <span class="n">x1</span><span class="p">,</span> <span class="n">x2</span> <span class="p">)</span>
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-<pre>[1.2, 3.5, 1.9] [ 1.6 -2.6  6.9]
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-<span class="n">x1</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">,</span> <span class="mf">1.9</span><span class="p">]</span>
-<span class="n">x1</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mf">5.9</span><span class="p">)</span>
-
-<span class="c1">#For arrays, new elements can be added using append function of NumPy</span>
-<span class="n">x2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">1.6</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.6</span><span class="p">,</span> <span class="mf">6.9</span><span class="p">])</span>
-<span class="n">x2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
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-<span class="nb">print</span> <span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">)</span>
-</pre></div>
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-<pre>[1.2, 3.5, 1.9, 5.9] [ 1.6 -2.6  6.9  3. ]
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-<span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.1</span><span class="p">,</span><span class="mf">3.4</span><span class="p">,</span><span class="mf">1.0</span><span class="p">]</span>
-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="c1">#And a numpy array into a list as well</span>
-<span class="n">x</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
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-<p><strong>Differences</strong></p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Operator + for lists is overloaded for concatenating </span>
-<span class="n">x1</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">]</span>
-<span class="n">x2</span> <span class="o">=</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>
-<span class="nb">print</span> <span class="p">(</span><span class="n">x1</span><span class="o">+</span><span class="n">x2</span><span class="p">)</span>
-</pre></div>
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-<pre>[1, 2, 3, 3, 2, 1]
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Operator + for numpy arrays is overloaded for adding</span>
-<span class="n">x1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="n">x2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="nb">print</span> <span class="p">(</span><span class="n">x1</span><span class="o">+</span><span class="n">x2</span><span class="p">)</span>
-</pre></div>
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-<span class="n">x1</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.2</span><span class="p">,</span><span class="mf">4.8</span><span class="p">,</span><span class="mf">6.9</span><span class="p">]</span>
-<span class="n">x2</span> <span class="o">=</span> <span class="p">[</span><span class="mf">2.6</span><span class="p">,</span><span class="mf">2.8</span><span class="p">,</span><span class="mf">1.1</span><span class="p">]</span>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Multiplication</span>
-<span class="nb">print</span> <span class="n">x1</span><span class="o">*</span><span class="n">x2</span>
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-<span class="nb">print</span> <span class="n">x1</span><span class="o">/</span><span class="n">x2</span>
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-<span class="n">x1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">1.2</span><span class="p">,</span><span class="mf">4.8</span><span class="p">,</span><span class="mf">6.9</span><span class="p">])</span>
-<span class="n">x2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">2.6</span><span class="p">,</span><span class="mf">2.8</span><span class="p">,</span><span class="mf">1.1</span><span class="p">])</span>
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-<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Adding&quot;</span><span class="p">,</span> <span class="n">x1</span><span class="o">+</span><span class="n">x2</span><span class="p">)</span>
-<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Multiplication&quot;</span><span class="p">,</span> <span class="n">x1</span><span class="o">*</span><span class="n">x2</span><span class="p">)</span>
-<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Division&quot;</span><span class="p">,</span> <span class="n">x1</span><span class="o">/</span><span class="n">x2</span><span class="p">)</span>
-<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Subtraction&quot;</span><span class="p">,</span> <span class="n">x1</span><span class="o">-</span><span class="n">x2</span><span class="p">)</span>
-<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Power&quot;</span><span class="p">,</span> <span class="n">x1</span><span class="o">**</span><span class="n">x2</span><span class="p">)</span>
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-<pre>Adding [3.8 7.6 8. ]
-Multiplication [ 3.12 13.44  7.59]
-Division [0.46153846 1.71428571 6.27272727]
-Subtraction [-1.4  2.   5.8]
-Power [ 1.6064649  80.81192733  8.37016462]
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-<p><strong>Note:</strong> Matrices can be represented as NumPy arrays where each element is a row vector. Nevertheless, be careful when multiply arrays, the operator * is overloaded in such a way that single elements are multiplied one by one, quite different from multiplication of matrices.</p>
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-<span class="n">B</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">4</span><span class="p">,</span><span class="mi">3</span><span class="p">],[</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">]])</span>
-<span class="nb">print</span> <span class="p">(</span><span class="s1">&#39;A=</span><span class="se">\n</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">A</span><span class="p">))</span> 
-<span class="nb">print</span> <span class="p">(</span><span class="s1">&#39;B=</span><span class="se">\n</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">B</span><span class="p">))</span>
-<span class="nb">print</span> <span class="p">(</span><span class="s1">&#39;A*B=</span><span class="se">\n</span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">B</span><span class="p">))</span> <span class="c1"># No matrix multiplication</span>
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-<pre>A=
-[[1 2]
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-[[4 3]
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-A*B=
-[[4 6]
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-<h3 id="Advanced-features-of-arrays">Advanced features of arrays<a class="anchor-link" href="#Advanced-features-of-arrays"> </a></h3>
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-<p>NumPy arrays have a number of interesting features that facilitate complex tasks. Next it is shown some of the more useful ones.</p>
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-<h3 id="Masks">Masks<a class="anchor-link" href="#Masks"> </a></h3><p>Superspostion of a logical array unpon the original one. The output filters only the <code>True</code> values</p>
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-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
-<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="kc">False</span><span class="p">,</span> <span class="kc">False</span><span class="p">,</span> <span class="kc">True</span><span class="p">,</span> <span class="kc">True</span><span class="p">])</span>
-<span class="n">x</span><span class="p">[</span><span class="n">y</span><span class="p">]</span>
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-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">x</span><span class="p">:</span>
-    <span class="k">if</span> <span class="n">i</span><span class="o">&gt;</span><span class="mi">2</span><span class="p">:</span>
-        <span class="n">xfin</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        
-<span class="n">xfin</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">xfin</span><span class="p">)</span>
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-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="o">&gt;</span><span class="n">y</span><span class="p">)</span> 
-<span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="o">&lt;</span><span class="n">y</span><span class="p">)</span> 
-<span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="o">==</span><span class="n">y</span><span class="p">)</span> 
-<span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="o">&gt;</span><span class="mi">4</span><span class="p">)</span> 
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-[False  True False  True]
-[ True False False False]
-[False  True  True False]
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-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">])</span>
-<span class="c1">#A new list with numbers greater than 4</span>
-<span class="n">x</span><span class="p">[</span><span class="n">x</span><span class="o">&gt;=</span><span class="mi">4</span><span class="p">]</span>
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-<p>Numpy has logical operators: <code>np.logical_...</code></p>
-<p>For <code>array([1,4,2,6,8,4,3,0,9,1,3,6,7])</code>:</p>
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-<p>For <code>array([1,4,2,6,8,4,3,0,9,1,3,6,7])</code>:</p>
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-<h3 id="Numpy-methods-to-basic-calculations">Numpy methods to basic calculations<a class="anchor-link" href="#Numpy-methods-to-basic-calculations"> </a></h3><p>spreadsheeet like operations</p>
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-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">])</span>
-<span class="c1">#Maximum element</span>
-<span class="nb">print</span><span class="p">(</span> <span class="s2">&quot;Maximum element&quot;</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">max</span><span class="p">()</span> <span class="p">)</span>
-<span class="c1">#Minimum element</span>
-<span class="nb">print</span><span class="p">(</span> <span class="s2">&quot;Minimum element&quot;</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">min</span><span class="p">())</span>
-<span class="c1">#Mean value</span>
-<span class="s2">&quot;Mean value&quot;</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
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-<h3 id="upon-the-indices">upon the indices<a class="anchor-link" href="#upon-the-indices"> </a></h3>
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-<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Sorted arguments&quot;</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">argsort</span><span class="p">()</span> <span class="p">)</span>
-<span class="c1">#Sorted array</span>
-<span class="nb">print</span> <span class="p">(</span> <span class="s2">&quot;Sorted array&quot;</span><span class="p">,</span> <span class="n">x</span><span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">argsort</span><span class="p">()])</span>
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-<pre>Sorted arguments [ 7  0  9  2  6 10  1  5  3 11 12  4  8]
-Sorted array [0 1 1 2 3 3 4 4 6 6 7 8 9]
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-<span class="n">x</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-<span class="n">x</span>
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-<span class="n">np</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span> <span class="n">x</span> <span class="p">)</span>
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-<h2 id="Miscellaneous-methods">Miscellaneous methods<a class="anchor-link" href="#Miscellaneous-methods"> </a></h2>
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-<p>NumPy includes many general purpose functions that complement the capabilities of python. We are interested here specially in functions for creating ordered arrays, storing and loading data as well as histograms, tasks that will be continuously required for the activities of the course.</p>
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-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
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-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span> <span class="p">)</span>
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-<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">9</span><span class="p">])</span>
-<span class="n">x</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
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-<h2 id="Miscellaneous-functions">Miscellaneous functions<a class="anchor-link" href="#Miscellaneous-functions"> </a></h2>
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-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="mi">10</span> <span class="p">)</span> 
-<span class="nb">print</span> <span class="p">(</span><span class="n">x</span><span class="p">)</span>
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-<pre>[-3.14159265 -2.44346095 -1.74532925 -1.04719755 -0.34906585  0.34906585
-  1.04719755  1.74532925  2.44346095  3.14159265]
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-<pre>array([2.00000000e+00, 6.16155028e+00, 1.89823509e+01, 5.84803548e+01,
-       1.80164823e+02, 5.55047308e+02, 1.70997595e+03, 5.26805138e+03,
-       1.62296817e+04, 5.00000000e+04])</pre>
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-<pre>array([2.00000000e+00, 5.55733333e+03, 1.11126667e+04, 1.66680000e+04,
-       2.22233333e+04, 2.77786667e+04, 3.33340000e+04, 3.88893333e+04,
-       4.44446667e+04, 5.00000000e+04])</pre>
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-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mf">0.2</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span> <span class="n">x</span> <span class="p">)</span>
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-<pre>[1.  1.2 1.4 1.6 1.8 2.  2.2 2.4 2.6 2.8 3.  3.2 3.4 3.6 3.8 4.  4.2 4.4
- 4.6 4.8]
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-<span class="n">data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">3.2</span><span class="p">,</span> <span class="mf">2.1</span><span class="p">],[</span><span class="mf">3.1</span><span class="p">,</span> <span class="mf">4.1</span><span class="p">]])</span>
-<span class="n">np</span><span class="o">.</span><span class="n">savetxt</span><span class="p">(</span> <span class="s2">&quot;file.dat&quot;</span><span class="p">,</span> <span class="n">data</span><span class="p">,</span> <span class="n">fmt</span><span class="o">=</span><span class="s2">&quot;</span><span class="si">%1.5e</span><span class="s2">  </span><span class="si">%1.5e</span><span class="s2">&quot;</span> <span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">cat</span> <span class="n">file</span><span class="o">.</span><span class="n">dat</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#In the same way, using the function loadtxt it is possible to load external data files</span>
-<span class="n">data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="s2">&quot;file.dat&quot;</span><span class="p">)</span>
-<span class="c1">#Data is then a multidimensional array with the loaded data</span>
-<span class="nb">print</span><span class="p">(</span> <span class="n">data</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
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-<p>Other useful functions will be covered when needed during the course.</p>
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-</div>
-</div>
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-<h2 id="List-Slices">List Slices<a class="anchor-link" href="#List-Slices"> </a></h2><p>Slices work on list-like objects like numpy arrays, and can also be used to change sub-parts of the list.</p>
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-<span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="mi">3</span><span class="p">]</span><span class="o">=</span><span class="mi">2</span>
-<span class="n">x</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">=</span><span class="mi">8</span>
-<span class="n">x</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">=</span><span class="mi">6</span>
-<span class="n">x</span>
-</pre></div>
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-<pre>array([1., 2., 2., 1., 1., 1., 1., 6., 6., 8.])</pre>
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-<p>Reverse order (arrays do not have the <code>reverse()</code> method)</p>
-
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-</pre></div>
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-<pre>array([8., 6., 6., 1., 1., 1., 1., 2., 2., 1.])</pre>
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-    <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/2/22/Pandas_mark.svg/368px-Pandas_mark.svg.png" width="60px">
-</div><h1 id="Pandas">Pandas<a class="anchor-link" href="#Pandas"> </a></h1>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">numbers</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;even&quot;</span><span class="p">:</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span>   <span class="c1">#  First  key-list</span>
-         <span class="s2">&quot;odd&quot;</span> <span class="p">:</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">9</span><span class="p">]</span> <span class="p">}</span>  <span class="c1">#  Second key-list</span>
-
-<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-<span class="n">pd</span><span class="o">.</span><span class="n">set_option</span><span class="p">(</span><span class="s1">&#39;display.max_colwidth&#39;</span><span class="p">,</span><span class="mi">200</span><span class="p">)</span>
-<span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">numbers</span><span class="p">)</span>
-<span class="n">df</span>
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-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
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-</style>
-<table border="1" class="dataframe">
-  <thead>
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-      <th></th>
-      <th>even</th>
-      <th>odd</th>
-    </tr>
-  </thead>
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-      <th>0</th>
-      <td>0</td>
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-      <td>8</td>
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-<p>In the previous DataFrame, all the column values are converted to Numpy arrays, which is the basic object in Numpy corresponding to generalized nested lists;</p>
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-</pre></div>
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-1    2
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-<div style="float: right;" markdown="1">
-    <img src="https://www.scipy.org/_static/images/scipy_med.png">
-</div><h1 id="SciPy">SciPy<a class="anchor-link" href="#SciPy"> </a></h1>
-</div>
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-<p>SciPy is a collection of mathematical algorithms and convenience functions built on the Numpy extension of Python. It adds significant power to the interactive Python session by providing the user with high-level commands and classes for manipulating and visualizing data. With SciPy an interactive Python session becomes a data-processing and system-prototyping environment rivaling sytems such as MATLAB, IDL, Octave, R-Lab, and SciLab.</p>
-<p>Some of the packages included with SciPy are:</p>
-<ul>
-<li>Special functions (<strong>scipy.special</strong>)</li>
-<li>Integration (<strong>scipy.integrate</strong>)</li>
-<li>Optimization (<strong>scipy.optimize</strong>)</li>
-<li>Interpolation (<strong>scipy.interpolate</strong>)</li>
-<li>Fourier Transforms (<strong>scipy.fftpack</strong>)</li>
-<li>Signal Processing (<strong>scipy.signal</strong>)</li>
-<li>Linear Algebra (<strong>scipy.linalg</strong>)</li>
-</ul>
-
-</div>
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-<p>Each of these packages must be imported separately.</p>
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-<span class="kn">import</span> <span class="nn">scipy.integrate</span> <span class="k">as</span> <span class="nn">integ</span>
-</pre></div>
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-<p>The integrate package then includes the next functions:</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">integ</span><span class="o">.</span><span class="n">Tester</span>        <span class="n">integ</span><span class="o">.</span><span class="n">fixed_quad</span>    <span class="n">integ</span><span class="o">.</span><span class="n">odepack</span>       <span class="n">integ</span><span class="o">.</span><span class="n">quadrature</span>    <span class="n">integ</span><span class="o">.</span><span class="n">test</span>          
-<span class="n">integ</span><span class="o">.</span><span class="n">complex_ode</span>   <span class="n">integ</span><span class="o">.</span><span class="n">newton_cotes</span>  <span class="n">integ</span><span class="o">.</span><span class="n">quad</span>          <span class="n">integ</span><span class="o">.</span><span class="n">romb</span>          <span class="n">integ</span><span class="o">.</span><span class="n">tplquad</span>       
-<span class="n">integ</span><span class="o">.</span><span class="n">cumtrapz</span>      <span class="n">integ</span><span class="o">.</span><span class="n">ode</span>           <span class="n">integ</span><span class="o">.</span><span class="n">quad_explain</span>  <span class="n">integ</span><span class="o">.</span><span class="n">romberg</span>       <span class="n">integ</span><span class="o">.</span><span class="n">trapz</span>         
-<span class="n">integ</span><span class="o">.</span><span class="n">dblquad</span>       <span class="n">integ</span><span class="o">.</span><span class="n">odeint</span>        <span class="n">integ</span><span class="o">.</span><span class="n">quadpack</span>      <span class="n">integ</span><span class="o">.</span><span class="n">simps</span>         <span class="n">integ</span><span class="o">.</span><span class="n">vode</span>  
-</pre></div>
-
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-<p>Almost each of the numerical methods that will be covered during the course can be found in SciPy. In next classes we will explore the offered options by SciPy according to the specific methods covered.</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">name</span><span class="o">=</span><span class="s1">&#39;Diego&#39;</span>
-<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;</span><span class="si">{</span><span class="n">name</span><span class="si">}</span><span class="s2"> </span><span class="si">{</span><span class="n">name</span><span class="si">}</span><span class="s2"> </span><span class="si">{</span><span class="n">name</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
-</pre></div>
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-</pre></div>
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-<pre>&#39;Diego Diego Diego&#39;</pre>
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-  Statistics
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-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/statistics.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
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-<p>Considering the amount of data produced nowadays in many fields, the computer manipulation of this information is not only important but also has become a very active field of study. Despite of the fact of increasing computational facilities, the effort necessary to make statistical analysis has become more complicated, at least in the computational aspect. This is why 
-becomes fundamental to some aspects, leastwise in a raw way.</p>
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-<ul>
-<li><a href="#Data-Adjust">Data Adjust</a> <ul>
-<li><a href="##Linear-least-squares">Linear least square</a></li>
-<li><a href="#Example-1">Example 1</a></li>
-<li><a href="##Non-linear-least-square">Non-linear least square</a></li>
-</ul>
-</li>
-<li><a href="#Random-Numbers">Random Numbers</a>  <ul>
-<li><a href="#Example-2">Example 2</a></li>
-</ul>
-</li>
-</ul>
-<hr>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%</span><span class="k">pylab</span> inline
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>Populating the interactive namespace from numpy and matplotlib
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">animation</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Data-Adjust">Data Adjust<a class="anchor-link" href="#Data-Adjust"> </a></h2><p>See also:</p>
-<ul>
-<li><a href="https://www.mathsisfun.com/data/least-squares-regression.html">https://www.mathsisfun.com/data/least-squares-regression.html</a></li>
-<li><a href="http://blog.mmast.net/least-squares-fitting-numpy-scipy">http://blog.mmast.net/least-squares-fitting-numpy-scipy</a></li>
-<li><a href="https://stackoverflow.com/a/43623588/2268280">https://stackoverflow.com/a/43623588/2268280</a></li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Given a data set, the first approach to find a function that passes through the points would be using a
-interpolation polynomial. But we should take special attention to the way data set is gathered, i.e., usually
-is a sample obtained experimentally or in a way that has associated an intrinsic error. 
-Then, forcing that the approximate function passes through all the points would actually incur in incrementing the error. This is why it is necessary to build a different procedure to build the function that fits the data.</p>
-<p>The fitting functions, though, are build using a Lagrange polynomial and the order of this polynomial constitutes
-the approximation that is going to be used. But the fitting function is not going to take the exact value in
-the known points, they are going to desagree in certain tolerance value.</p>
-<!-- This type of procedure is also used to approximate functions to a simper type of function. Although the procedure
-is very similar is not going to be discussed here because of the lack of use of this kind of procedures.  -->
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Linear-least-squares">Linear least squares<a class="anchor-link" href="#Linear-least-squares"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Approximating a data set to a linear Langrange polynomial would be just to use
-$$
-y_i= f(x_i) = a_1x_i + a_0\,.
-$$
-However, with experimental data, the problem is that the values $y_i$ are not precise, then it is proposed to find the best approximation.
-For the best <em>linear approximation</em>, we need to find  for <em>all</em> points the value of $a_0$ and $a_1$ that <em>minimize</em> the error
-$$
-E = E(a_0,a_1) = \sum_{i=1}^{m}[y_i - (a_1 x_i + a_0)]^2 
-$$
-where the square is more well suited for the minimization procedure since avoids the fluctions for changes of sign.</p>
-<p>To minimize the function of 2 variables, $E(a_0,a_1)$ with respect to $a_0$ and $a_1$,  it is necessary to set its partial derivatives to zero and simultaneously 
-solve to the resulting equations.</p>
-<p>To establish the minimzation equations, it is necesary to take the partial derivatives with respect to $a_0$  and $a_1$ and and equal them to zero
-$$
-\frac{\partial E}{\partial a_0} = 0\,, \hspace{1cm} 
-\frac{\partial E}{\partial a_1} = 0\,.
-$$</p>
-<p>Afterwards,</p>
-\begin{align}
-&amp;0= 2\sum_{i=1}^{m}(y_i -a_1x_i-a_0)(-1)\,, &amp;&amp; 
-&amp;0 =&amp; 2\sum_{i=1}^{m}(y_i -a_1x_i-a_0)(-x_i) \\
-&amp;a_0 m + a_1\sum_{i=1}^{m}x_i = \sum_{i=1}^{m} y_i\,,&amp;&amp; &amp;a_0\sum_{i=1}^{m}x_i  + a_1\sum_{i=1}^{m}x_i^2 = \sum_{i=1}^{m} x_iy_i 
-\end{align}\begin{align}
-a_0= &amp;\frac{ \sum_{i=1}^{m} y_i-a_1\sum_{i=1}^{m}x_i }{m} 
-\end{align}
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Replacing back in the second equation
-\begin{align}
- \left( \frac{ \sum_{i=1}^{m} y_i-a_1\sum_{i=1}^{m}x_i }{m} \right)\sum_{i=1}^{m}x_i  + a_1\sum_{i=1}^{m}x_i^2 = \sum_{i=1}^{m} x_iy_i 
-\end{align}
-\begin{align}
-   \frac{1}{m}\sum_{i=1}^{m} y_i \sum_{i=1}^{m}x_i -a_1\frac{1}{m} \sum_{i=1}^{m}x_i \sum_{i=1}^{m}x_i   + a_1\sum_{i=1}^{m}x_i^2 = \sum_{i=1}^{m} x_iy_i 
-\end{align}
-\begin{align}
-   a_1\left[ \sum_{i=1}^{m}x_i^2 -\frac{1}{m}\left(   \sum_{i=1}^{m}x_i \right)^2  \right]   = \sum_{i=1}^{m} x_iy_i - \frac{1}{m}\sum_{i=1}^{m} y_i \sum_{i=1}^{m}x_i
-\end{align}
-\begin{align}
-   a_1\left[ m\sum_{i=1}^{m}x_i^2 -\left(   \sum_{i=1}^{m}x_i \right)^2  \right]   = m\sum_{i=1}^{m} x_iy_i -\sum_{i=1}^{m} y_i \sum_{i=1}^{m}x_i
-\end{align}</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Therefore
-\begin{align}
-a_1 = \frac{m\sum_{i=1}^{m} x_iy_i - \sum_{i=1}^{m} x_i \sum_{i=1}^{m} y_i }
-{m\sum_{i=1}^{m} x_i^2 - \left(\sum_{i=1}^{m} x_i\right)^2} 
-\end{align}</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Replacing back in $a_0$
-\begin{align}
-ma_0= &amp;\sum_{i=1}^{m} y_i-a_1\sum_{i=1}^{m}x_i \\
- = &amp;\sum_{i=1}^{m} y_i-\left[\frac{m\sum_{i=1}^{m} x_iy_i - \sum_{i=1}^{m} x_i \sum_{i=1}^{m} y_i }
-{m\sum_{i=1}^{m} x_i^2 - \left(\sum_{i=1}^{m} x_i\right)^2}  \right]\sum_{i=1}^{m}x_i \\
-= &amp;\sum_{i=1}^{m} y_i-\frac{m\sum_{i=1}^{m}x_i\sum_{i=1}^{m} x_iy_i - \left( \sum_{i=1}^{m} x_i\right)^2 \sum_{i=1}^{m} y_i }
-{m\sum_{i=1}^{m} x_i^2 - \left(\sum_{i=1}^{m} x_i\right)^2}   \\
-= &amp;\frac{m\sum_{i=1}^{m} x_i^2\sum_{i=1}^{m} y_i-m\sum_{i=1}^{m}x_i\sum_{i=1}^{m} x_iy_i  }
-{m\sum_{i=1}^{m} x_i^2 - \left(\sum_{i=1}^{m} x_i\right)^2}   \\
-\end{align}</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-\begin{align}
-a_0= &amp;\frac{\sum_{i=1}^{m} x_i^2\sum_{i=1}^{m} y_i-\sum_{i=1}^{m}x_i\sum_{i=1}^{m} x_iy_i  }
-{m\sum_{i=1}^{m} x_i^2 - \left(\sum_{i=1}^{m} x_i\right)^2}   \\
-\end{align}
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>where the coefficients $a_0$ and $a_1$ can be easily obtained</p>
-$$
-a_0 = \frac{\sum_{i=1}^{m} x_i^2\sum_{i=1}^{m}y_i - \sum_{i=1}^{m} x_iy_i \sum_{i=1}^{m} x_i }
-{m\sum_{i=1}^{m} x_i^2 - \left(\sum_{i=1}^{m} x_i\right)^2}\,, \hspace{1.5cm}
-a_1 = \frac{m\sum_{i=1}^{m} x_iy_i - \sum_{i=1}^{m} x_i \sum_{i=1}^{m} y_i }
-{m\sum_{i=1}^{m} x_i^2 - \left(\sum_{i=1}^{m} x_i\right)^2} 
-$$<p>Now, using the error definition one can find the error associated to the approximation made,</p>
-<p>since the coefficients $a_0$ and $a_1$ are already known.</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>From <a href="https://stackoverflow.com/a/43623588/2268280">Least sqaure method in python</a>:</p>
-<blockquote><p>There are many <a href="https://python4mpia.github.io/fitting_data/fitting_data.html">curve fitting</a> functions in scipy and numpy and each is used differently, e.g. <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.leastsq.html">scipy.optimize.leastsq</a> and <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html">scipy.optimize.least_squares</a>. For simplicity, we will use <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html">scipy.optimize.curve_fit</a>, but it is difficult to find an optimized regression curve without selecting reasonable starting parameters. A simple technique will later be demonstrated on selecting starting parameters.</p>
-</blockquote>
-<p>In our first example the starting parameters will be just zeros</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Example-1">Example 1<a class="anchor-link" href="#Example-1"> </a></h2>
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>A body is moving under the influence of an external force, the variation of the position measured for different 
-times are compiled in table 1</p>
-<table>
-<thead><tr>
-<th style="text-align:center">t(s)</th>
-<th style="text-align:center">x(m)</th>
-<th style="text-align:center">v(m/s)         </th>
-</tr>
-</thead>
-<tbody>
-<tr>
-<td style="text-align:center">0</td>
-<td style="text-align:center">2.76</td>
-<td style="text-align:center">33.10</td>
-</tr>
-<tr>
-<td style="text-align:center">1.11</td>
-<td style="text-align:center">29.66</td>
-<td style="text-align:center">21.33</td>
-</tr>
-<tr>
-<td style="text-align:center">2.22</td>
-<td style="text-align:center">46.83</td>
-<td style="text-align:center">16.57</td>
-</tr>
-<tr>
-<td style="text-align:center">3.33</td>
-<td style="text-align:center">44.08</td>
-<td style="text-align:center">-5.04</td>
-</tr>
-<tr>
-<td style="text-align:center">4.44</td>
-<td style="text-align:center">37.26</td>
-<td style="text-align:center">-11.74</td>
-</tr>
-<tr>
-<td style="text-align:center">5.55</td>
-<td style="text-align:center">12.03</td>
-<td style="text-align:center">-27.32</td>
-</tr>
-</tbody>
-</table>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#&#39;x&#39;: [2.76,  29.66,46.83,44.08,37.26,12.03],</span>
-<span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span> <span class="p">{</span><span class="s1">&#39;t&#39;</span><span class="p">:</span> <span class="p">[</span> <span class="mf">0.</span><span class="p">,</span>  <span class="mf">1.11</span><span class="p">,</span>  <span class="mf">2.22</span><span class="p">,</span>  <span class="mf">3.33</span><span class="p">,</span>  <span class="mf">4.44</span><span class="p">,</span> <span class="mf">5.55</span><span class="p">],</span><span class="c1">#&#39;x&#39;: [2.76,  29.66,46.83,44.08,37.26,12.03],</span>
-                  <span class="s1">&#39;v&#39;</span><span class="p">:</span> <span class="p">[</span><span class="mf">33.10</span><span class="p">,</span> <span class="mf">21.33</span><span class="p">,</span> <span class="mf">16.57</span><span class="p">,</span> <span class="o">-</span><span class="mf">5.04</span><span class="p">,</span> <span class="o">-</span><span class="mf">11.74</span><span class="p">,</span> <span class="o">-</span><span class="mf">27.32</span><span class="p">]}</span> <span class="p">)</span>
-<span class="n">df</span><span class="p">[[</span><span class="s1">&#39;t&#39;</span><span class="p">,</span><span class="s1">&#39;v&#39;</span><span class="p">]]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
-
-    .dataframe thead th {
-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>t</th>
-      <th>v</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>0.00</td>
-      <td>33.10</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>1.11</td>
-      <td>21.33</td>
-    </tr>
-    <tr>
-      <th>2</th>
-      <td>2.22</td>
-      <td>16.57</td>
-    </tr>
-    <tr>
-      <th>3</th>
-      <td>3.33</td>
-      <td>-5.04</td>
-    </tr>
-    <tr>
-      <th>4</th>
-      <td>4.44</td>
-      <td>-11.74</td>
-    </tr>
-    <tr>
-      <th>5</th>
-      <td>5.55</td>
-      <td>-27.32</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">t</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">v</span><span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;$t$ [s]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;$v$ [m/s]&#39;</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">15</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/statistics_20_0.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">scipy.optimize</span> <span class="k">as</span> <span class="nn">optimize</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">a1</span><span class="p">,</span><span class="n">a0</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">a1</span><span class="o">*</span><span class="n">t</span><span class="o">+</span><span class="n">a0</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>The starting point is optional</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">starting_parameters</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span><span class="p">,</span><span class="n">Δa</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">curve_fit</span><span class="p">(</span><span class="n">func</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">t</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">v</span><span class="p">,</span><span class="n">p0</span><span class="o">=</span><span class="n">starting_parameters</span><span class="p">)</span>
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-<p>The correlation matrix,$\Delta a$ , give as the error in the determination of the parameters</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;-&#39;</span><span class="o">*</span><span class="mi">20</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">Δa</span><span class="p">)</span>
-<span class="n">σ</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">Δa</span><span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;-&#39;</span><span class="o">*</span><span class="mi">20</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;a1=</span><span class="si">{:.2f}</span><span class="s1">±</span><span class="si">{:.2f}</span><span class="s1">, a0=</span><span class="si">{:.2f}</span><span class="s1">±</span><span class="si">{:.2f}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">σ</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span><span class="n">σ</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="p">)</span>
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-<pre>[-10.88597169  34.69190478]
---------------------
-[[ 0.68398817 -1.89806718]
- [-1.89806718  7.72513347]]
---------------------
-a1=-10.89±0.83, a0=34.69±2.78
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">t_lin</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mf">5.55</span><span class="p">)</span>
-<span class="n">v_model</span><span class="o">=</span><span class="n">func</span><span class="p">(</span><span class="n">t_lin</span><span class="p">,</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-<span class="n">df</span><span class="p">[</span><span class="s1">&#39;v_fitted&#39;</span><span class="p">]</span><span class="o">=</span><span class="n">func</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">t</span><span class="p">,</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
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-<table border="1" class="dataframe">
-  <thead>
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-      <th></th>
-      <th>t</th>
-      <th>v</th>
-      <th>v_fitted</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>0.00</td>
-      <td>33.10</td>
-      <td>34.691905</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>1.11</td>
-      <td>21.33</td>
-      <td>22.608476</td>
-    </tr>
-    <tr>
-      <th>2</th>
-      <td>2.22</td>
-      <td>16.57</td>
-      <td>10.525048</td>
-    </tr>
-    <tr>
-      <th>3</th>
-      <td>3.33</td>
-      <td>-5.04</td>
-      <td>-1.558381</td>
-    </tr>
-    <tr>
-      <th>4</th>
-      <td>4.44</td>
-      <td>-11.74</td>
-      <td>-13.641810</td>
-    </tr>
-    <tr>
-      <th>5</th>
-      <td>5.55</td>
-      <td>-27.32</td>
-      <td>-25.725238</td>
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-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">t</span><span class="p">,</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">df</span><span class="o">.</span><span class="n">t</span><span class="o">+</span><span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span><span class="s1">&#39;g-&#39;</span><span class="p">,</span><span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s2">&quot;Linear adjust&quot;</span><span class="p">)</span>
-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">t</span><span class="o">.</span><span class="n">size</span><span class="p">):</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">df</span><span class="o">.</span><span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="n">df</span><span class="o">.</span><span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">]]),</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">df</span><span class="o">.</span><span class="n">v</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="n">df</span><span class="o">.</span><span class="n">v_fitted</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="p">]),</span><span class="s2">&quot;c-&quot;</span><span class="p">)</span>
-    
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span> <span class="s2">&quot;$t$ [s]&quot;</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">18</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span> <span class="s2">&quot;$v$  [m/s]&quot;</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">18</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Linear data adjust&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
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-<p><strong>Activity</strong>: plot the error bands. Use <a href="https://beta.deepnote.com/project/4b6ea05f-f155-4eae-959c-cd064deaa7b6">https://beta.deepnote.com/project/4b6ea05f-f155-4eae-959c-cd064deaa7b6</a></p>
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-<p><strong>Activity</strong>: Solve the problem with <code>fmin_powell</code> for the fit only (not the error correlation matrix)
-$$
-E = E(a_0,a_1) = \sum_{i=1}^{m}[y_i - (a_1 x_i + a_0)]^2 
-$$</p>
-<p><strong>Solution</strong>:
-Let a=[a_0,a_1]</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">E</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">x</span><span class="o">=</span><span class="n">df</span><span class="o">.</span><span class="n">t</span><span class="o">.</span><span class="n">values</span><span class="p">,</span><span class="n">y</span><span class="o">=</span><span class="n">df</span><span class="o">.</span><span class="n">v</span><span class="o">.</span><span class="n">values</span><span class="p">):</span>
-    <span class="k">return</span> <span class="p">((</span><span class="n">y</span><span class="o">-</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">E</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-</pre></div>
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-<pre>2686.9554000000003</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">E</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
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-<pre>3105.6670999999997</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">E</span><span class="p">([</span><span class="mi">33</span><span class="p">,</span><span class="o">-</span><span class="mi">10</span><span class="p">])</span>
-</pre></div>
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-<pre>79.44340000000005</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">Ea0</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">a1</span><span class="o">=-</span><span class="mi">10</span><span class="p">):</span>
-    <span class="n">EE</span><span class="o">=</span><span class="p">[]</span>
-    <span class="k">for</span> <span class="n">a0</span> <span class="ow">in</span> <span class="n">x</span><span class="p">:</span>
-        <span class="n">EE</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">E</span><span class="p">([</span><span class="n">a0</span><span class="p">,</span><span class="n">a1</span><span class="p">]))</span>
-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">EE</span><span class="p">)</span>
-</pre></div>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Ea0</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-</pre></div>
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-<pre>array([5560.2434, 5203.4434])</pre>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a0</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">50</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">a0</span><span class="p">,</span><span class="n">Ea0</span><span class="p">(</span><span class="n">a0</span><span class="p">,</span><span class="o">-</span><span class="mi">10</span><span class="p">))</span>
-</pre></div>
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-<pre>[&lt;matplotlib.lines.Line2D at 0x7fd10f5e52b0&gt;]</pre>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">optimize</span><span class="o">.</span><span class="n">fmin_powell</span><span class="p">(</span><span class="n">E</span><span class="p">,[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span><span class="n">full_output</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-</pre></div>
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-<pre>Optimization terminated successfully.
-         Current function value: 58.991928
-         Iterations: 5
-         Function evaluations: 169
-</pre>
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-<pre>(array([ 34.69190471, -10.88597167]),
- 58.99192761904762,
- array([[ 9.58580558, -2.62474386],
-        [ 0.03589244,  0.0271459 ]]),
- 5,
- 169,
- 0)</pre>
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-</div>
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-</div>
-</div>
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-<h2 id="How-does-this-work:">How does this work:<a class="anchor-link" href="#How-does-this-work:"> </a></h2><p>From <a href="https://stackoverflow.com/a/43623588/2268280">Leat sqaure method in python</a>:</p>
-<blockquote><p><a href="">curve_fit</a> is one of many optimization functions offered by scipy. Given an initial value, the resulting estimated parameters are iteratively refined so that the resulting curve minimizes the residual error, or difference between the fitted line and sampling data. A better guess reduces the number of iterations and speeds up the result. With these estimated parameters for the fitted curve, one can now calculate the specific coefficients for a particular equation (a final exercise left to the OP).</p>
-</blockquote>
-
-</div>
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Finding adjusting parameters </span>
-<span class="k">def</span> <span class="nf">Linear_least_square</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="p">):</span>
-    
-    <span class="c1">#Finding coefficients </span>
-    <span class="n">length</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="n">square_x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">([</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">xrange</span><span class="p">(</span><span class="n">length</span><span class="p">)])</span>
-    <span class="n">sum_xy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">([</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">xrange</span><span class="p">(</span><span class="n">length</span><span class="p">)])</span>
-    <span class="n">sum_x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="n">sum_y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-    <span class="n">a0</span> <span class="o">=</span> <span class="p">(</span> <span class="n">square_x</span><span class="o">*</span><span class="n">sum_y</span> <span class="o">-</span> <span class="n">sum_xy</span><span class="o">*</span><span class="n">sum_x</span> <span class="p">)</span> <span class="o">/</span> <span class="p">(</span> <span class="n">length</span><span class="o">*</span><span class="n">square_x</span>  <span class="o">-</span> <span class="n">sum_x</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
-    <span class="n">a1</span> <span class="o">=</span> <span class="p">(</span> <span class="n">length</span><span class="o">*</span><span class="n">sum_xy</span> <span class="o">-</span> <span class="n">sum_x</span><span class="o">*</span><span class="n">sum_y</span> <span class="p">)</span> <span class="o">/</span> <span class="p">(</span> <span class="n">length</span><span class="o">*</span><span class="n">square_x</span>  <span class="o">-</span> <span class="n">sum_x</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
-    
-    <span class="c1">#Returning a_0 and a_1 coefficients</span>
-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">a0</span><span class="p">,</span><span class="n">a1</span><span class="p">])</span>
-
-<span class="c1">#Line function adjusting the data set</span>
-<span class="k">def</span> <span class="nf">Line</span><span class="p">(</span><span class="n">a0</span><span class="p">,</span><span class="n">a1</span><span class="p">,</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">a0</span><span class="o">+</span><span class="n">a1</span><span class="o">*</span><span class="n">x</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">xrange</span><span class="o">=</span><span class="nb">range</span>
-<span class="c1">#========================================================</span>
-<span class="c1"># Adjusting to a first order polynomy the data set v </span>
-<span class="c1">#========================================================</span>
-<span class="c1">#Setting figure</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span> <span class="p">)</span>
-
-<span class="c1">#Time</span>
-<span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="mf">0.</span><span class="p">,</span>  <span class="mf">1.11</span><span class="p">,</span>  <span class="mf">2.22</span><span class="p">,</span>  <span class="mf">3.33</span><span class="p">,</span>  <span class="mf">4.44</span><span class="p">,</span> <span class="mf">5.55</span><span class="p">])</span>
-
-<span class="c1">#Velocities measured for every time t[i]</span>
-<span class="n">v</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">33.10</span><span class="p">,</span> <span class="mf">21.33</span><span class="p">,</span> <span class="mf">16.57</span><span class="p">,</span> <span class="o">-</span><span class="mf">5.04</span><span class="p">,</span> <span class="o">-</span><span class="mf">11.74</span><span class="p">,</span> <span class="o">-</span><span class="mf">27.32</span><span class="p">])</span>
-
-<span class="c1">#Making data adjust</span>
-<span class="n">a0</span><span class="p">,</span> <span class="n">a1</span> <span class="o">=</span> <span class="n">Linear_least_square</span><span class="p">(</span> <span class="n">t</span><span class="p">,</span><span class="n">v</span> <span class="p">)</span>
-
-<span class="c1">#Finding error associated to linear approximation</span>
-<span class="n">E</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">([</span> <span class="p">(</span> <span class="n">v</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">Line</span><span class="p">(</span><span class="n">a0</span><span class="p">,</span><span class="n">a1</span><span class="p">,</span><span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="p">)</span><span class="o">**</span><span class="mi">2</span>  <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">xrange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">))])</span>
-
-<span class="c1">#Plotting solution</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">t</span><span class="p">,</span> <span class="n">Line</span><span class="p">(</span><span class="n">a0</span><span class="p">,</span><span class="n">a1</span><span class="p">,</span><span class="n">t</span><span class="p">),</span> <span class="s2">&quot;.-&quot;</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mf">3.</span><span class="p">,</span><span class="n">color</span> <span class="o">=</span> <span class="s2">&quot;green&quot;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s2">&quot;Lineal adjust&quot;</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span> <span class="n">t</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="s2">&quot;.&quot;</span><span class="p">,</span><span class="n">color</span> <span class="o">=</span> <span class="s2">&quot;blue&quot;</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s2">&quot;Data set&quot;</span> <span class="p">)</span>
-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">xrange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)):</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">]]),</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">v</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="n">Line</span><span class="p">(</span><span class="n">a0</span><span class="p">,</span><span class="n">a1</span><span class="p">,</span><span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">])]),</span><span class="s2">&quot;c-&quot;</span><span class="p">)</span>
-    
-<span class="c1">#Format of figure</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span> <span class="s2">&quot;$t(s)$&quot;</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">18</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span> <span class="s2">&quot;$v(m/s)$&quot;</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">18</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span> <span class="p">(</span><span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">t</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Linear data adjust with error </span><span class="si">%f</span><span class="s2">&quot;</span><span class="o">%</span><span class="k">E</span>)
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-</pre></div>
-
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-<h2 id="-----**Activity**-"><font color="red">     **Activity** </font><a class="anchor-link" href="#-----**Activity**-"> </a></h2><p><a href="http://scipy-cookbook.readthedocs.io/items/robust_regression.html">http://scipy-cookbook.readthedocs.io/items/robust_regression.html</a></p>
-
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-<p><font color='red'>    
-Using the next data set of a spring mass system, find the lineal adjust. 
-</font></p>
-<p><img src="https://github.com/restrepo/ComputationalMethods/blob/master/material/figures/datos.png?raw=1"></p>
-
-</div>
-</div>
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-<h2 id="Non-linear-least-square">Non-linear least square<a class="anchor-link" href="#Non-linear-least-square"> </a></h2>
-</div>
-</div>
-</div>
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-<p>In general, it can be used any polynomial order to adjust a data set, since it is satisfied that $n&lt;m-1$,
-with n the order of the polynomial and m the number of points known. Then, we have
-$$
-P_n(x) = a_nx^n + a_{n-1}x^{n-1}+...+a_1x+a_0
-$$</p>
-<p>Using a similar procedure followed in linear least square approximation, it is chose the constants $a_0,...a_n$ to minimize
-the least square error</p>
-$$
-E = \sum_{i=1}^{m} ( y_i - P_n(x_i) )^2
-= \sum_{i=1}^{m} ( y_i - \sum_{j=0}^{n}a_jx_i^j )^2
-$$<p>Expanding the square difference and taking into account that E to be minimized requires that $\partial E/ 
-\partial a_j = 0 $ for each $j=0,1,...n$. Following these arguments, it is found that the n+1 equations needed
-to solve to find the coefficients $a_j$ are</p>
-$$
-\sum_{k=0}^{n} a_k \sum_{i=1}^{m}x_i^{j+k} = \sum_{i=1}^{m}y_ix_i^j
-$$<p>for each $j=0,1,...n$. A better way to show the equations, where m is the data length and n is the polynomial
-order, is</p>
-$$
-a_0\sum_{i=1}^{m}x_i^0 + a_1\sum_{i=1}^{m}x_i^1 + a_2\sum_{i=1}^{m}x_i^2 + ... +  a_n\sum_{i=1}^{m}x_i^n =  \sum_{i=1}^{m}y_i x_i^0 \\
-a_0\sum_{i=1}^{m}x_i^1 + a_1\sum_{i=1}^{m}x_i^2 + a_2\sum_{i=1}^{m}x_i^3 + ... +  a_n\sum_{i=1}^{m}x_i^{n+1} =  \sum_{i=1}^{m}y_i x_i^1\\ 
-\dotsc \\
-a_0\sum_{i=1}^{m}x_i^n + a_1\sum_{i=1}^{m}x_i^{n+1} + a_2\sum_{i=1}^{m}x_i^{n+2} + ... +  a_n\sum_{i=1}^{m}x_i^{2n} =  \sum_{i=1}^{m}y_i x_i^n
-$$<p>Again, the error associated to the approximation can be obtained by initial definition of E. The error can also be defined 
-using a weight function $W_i$ as</p>
-$$
-E = \sum_{i=1}^{m} W_i( y_i - P_n(x_i) )^2
-$$<p>this function $W_i$ can be defined in several ways. If $W_i = \sigma_i$, i.e., the standard deviation per particle, it is necessary to know the probability distribution followed by the experiments. In these cases where it is not known,
-it is usually taken as one.</p>
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-<h2 id="-----**Activity**-"><font color="red">     **Activity** </font><a class="anchor-link" href="#-----**Activity**-"> </a></h2>
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-<p><font color='red'> 
-Adjust the position column data in the table 1 to a second order polynomial. What is the acceleration suffered
-by the body?
-</font></p>
-
-</div>
-</div>
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-</div>
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-<div class="jb_cell">
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">a2</span><span class="p">,</span><span class="n">a1</span><span class="p">,</span><span class="n">a0</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">a2</span><span class="o">*</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">a1</span><span class="o">*</span><span class="n">t</span><span class="o">+</span><span class="n">a0</span>
-<span class="n">a</span><span class="p">,</span><span class="n">kk</span><span class="o">=</span><span class="n">optimize</span><span class="o">.</span><span class="n">curve_fit</span><span class="p">(</span><span class="n">func</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">t</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">v</span><span class="p">)</span>
-<span class="n">t_lin</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mf">5.55</span><span class="p">)</span>
-<span class="n">v2_model</span><span class="o">=</span><span class="n">func</span><span class="p">(</span><span class="n">t_lin</span><span class="p">,</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-<span class="n">df</span><span class="p">[</span><span class="s1">&#39;v2_fitted&#39;</span><span class="p">]</span><span class="o">=</span><span class="n">func</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">t</span><span class="p">,</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="cell border-box-sizing code_cell rendered">
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-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span>
-</pre></div>
-
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-
-<div class="jb_output_wrapper }}">
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-<div class="output_html rendered_html output_subarea output_execute_result">
-<div>
-<style scoped>
-    .dataframe tbody tr th:only-of-type {
-        vertical-align: middle;
-    }
-
-    .dataframe tbody tr th {
-        vertical-align: top;
-    }
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-        text-align: right;
-    }
-</style>
-<table border="1" class="dataframe">
-  <thead>
-    <tr style="text-align: right;">
-      <th></th>
-      <th>t</th>
-      <th>v</th>
-      <th>v_fitted</th>
-      <th>v2_fitted</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <th>0</th>
-      <td>0.00</td>
-      <td>33.10</td>
-      <td>34.691905</td>
-      <td>33.096072</td>
-    </tr>
-    <tr>
-      <th>1</th>
-      <td>1.11</td>
-      <td>21.33</td>
-      <td>22.608476</td>
-      <td>22.927643</td>
-    </tr>
-    <tr>
-      <th>2</th>
-      <td>2.22</td>
-      <td>16.57</td>
-      <td>10.525048</td>
-      <td>11.801714</td>
-    </tr>
-    <tr>
-      <th>3</th>
-      <td>3.33</td>
-      <td>-5.04</td>
-      <td>-1.558381</td>
-      <td>-0.281715</td>
-    </tr>
-    <tr>
-      <th>4</th>
-      <td>4.44</td>
-      <td>-11.74</td>
-      <td>-13.641810</td>
-      <td>-13.322643</td>
-    </tr>
-    <tr>
-      <th>5</th>
-      <td>5.55</td>
-      <td>-27.32</td>
-      <td>-25.725238</td>
-      <td>-27.321071</td>
-    </tr>
-  </tbody>
-</table>
-</div>
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-</div>
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-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">t</span><span class="p">,</span><span class="n">df</span><span class="o">.</span><span class="n">v</span><span class="p">,</span><span class="s1">&#39;b.&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;exp. data&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_lin</span><span class="p">,</span><span class="n">v_model</span><span class="p">,</span><span class="s1">&#39;g-&#39;</span><span class="p">,</span><span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s2">&quot;Linear adjust&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_lin</span><span class="p">,</span><span class="n">v2_model</span><span class="p">,</span><span class="s1">&#39;r--&#39;</span><span class="p">,</span><span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s2">&quot;quadratic adjust&quot;</span><span class="p">)</span>
-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">t</span><span class="o">.</span><span class="n">size</span><span class="p">):</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">df</span><span class="o">.</span><span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="n">df</span><span class="o">.</span><span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">]]),</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">df</span><span class="o">.</span><span class="n">v</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="n">df</span><span class="o">.</span><span class="n">v_fitted</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="p">]),</span><span class="s2">&quot;c-&quot;</span><span class="p">)</span>
-    
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span> <span class="s2">&quot;$t$ [s]&quot;</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">18</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span> <span class="s2">&quot;$v$  [m/s]&quot;</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">18</span> <span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Linear data adjust&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>    
-</pre></div>
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-<h2 id="-----**Activity**-"><font color="red">     **Activity** </font><a class="anchor-link" href="#-----**Activity**-"> </a></h2>
-</div>
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-<p><font color='red'> 
-The air drag for a sphere that is moving at high speeds can be expresed in the following form 
-$$
-f_{drag} = -\frac{1}{2} C \rho A v^2
-$$</p>
-<p>where C is the drag coefficient(0.5 for a sphere), $\rho$ is the air density (1.29kg/$m^3$) 
-and A is the cross-sectional area. 
-Generate points that have a bias of the value obtanied with
-$f_{drag}$ using np.random.random.</p>
-<p>Afterwards, use a second order polynomial to fit the data 
-generated and find the error associated to the approximation. 
-&lt;/font&gt;</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">scipy.optimize</span> <span class="k">as</span> <span class="nn">optimize</span>
-</pre></div>
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-<h2 id="Fit-to-a-non-linear-function">Fit to a non-linear function<a class="anchor-link" href="#Fit-to-a-non-linear-function"> </a></h2>$$
-f(x)=A \operatorname{e}^{cx} + d\,.
-$$<p>Determinar: $A,c,d$</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">T_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">222</span><span class="p">,</span> <span class="mi">284</span><span class="p">,</span> <span class="mf">308.5</span><span class="p">,</span> <span class="mi">333</span><span class="p">,</span> <span class="mi">358</span><span class="p">,</span> <span class="mi">411</span><span class="p">,</span> <span class="mi">477</span><span class="p">,</span> <span class="mi">518</span><span class="p">,</span> <span class="mi">880</span><span class="p">,</span> <span class="mi">1080</span><span class="p">,</span> <span class="mi">1259</span><span class="p">])</span>
-<span class="n">C_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.1282</span><span class="p">,</span> <span class="mf">0.2308</span><span class="p">,</span> <span class="mf">0.2650</span><span class="p">,</span> <span class="mf">0.3120</span> <span class="p">,</span> <span class="mf">0.3547</span><span class="p">,</span> <span class="mf">0.4530</span><span class="p">,</span> <span class="mf">0.5556</span><span class="p">,</span> <span class="mf">0.6154</span><span class="p">,</span> <span class="mf">0.8932</span><span class="p">,</span> <span class="mf">0.9103</span><span class="p">,</span> <span class="mf">0.9316</span><span class="p">])</span>
-
-<span class="n">x_samp</span> <span class="o">=</span> <span class="n">T_values</span>
-<span class="n">y_samp</span> <span class="o">=</span> <span class="n">C_values</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">A</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">d</span>
-</pre></div>
-
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-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># REGRESSION ------------------------------------------------------------------</span>
-<span class="n">p0</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mf">3e-3</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>                                        <span class="c1"># guessed params</span>
-<span class="n">w</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">optimize</span><span class="o">.</span><span class="n">curve_fit</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">x_samp</span><span class="p">,</span> <span class="n">y_samp</span><span class="p">,</span> <span class="n">p0</span><span class="o">=</span><span class="n">p0</span><span class="p">)</span>     
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;E]stimated Parameters&quot;</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span>  
-
-<span class="c1"># Model</span>
-<span class="n">x_lin</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">200</span><span class="p">,</span><span class="mi">1350</span><span class="p">)</span>
-<span class="n">y_model</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">x_lin</span><span class="p">,</span> <span class="o">*</span><span class="n">w</span><span class="p">)</span>
-</pre></div>
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-<pre>E]stimated Parameters [-1.66301087 -0.0026884   1.00995394]
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
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-</div>
-</div>
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-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x_lin</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x_samp</span><span class="o">.</span><span class="n">max</span><span class="p">(),</span> <span class="mi">50</span><span class="p">)</span>                   <span class="c1"># 50 evenly spaced digits between 0 and max</span>
-<span class="c1"># PLOT ------------------------------------------------------------------------</span>
-<span class="c1"># Visualize data and fitted curves</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x_samp</span><span class="p">,</span> <span class="n">y_samp</span><span class="p">,</span> <span class="s2">&quot;ko&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Data&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x_lin</span><span class="p">,</span> <span class="n">y_model</span><span class="p">,</span> <span class="s2">&quot;k--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Fit&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Least squares regression&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper left&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;f(x)&#39;</span><span class="p">)</span>
-</pre></div>
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-<pre>Text(0, 0.5, &#39;f(x)&#39;)</pre>
-</div>
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-<img src="../images/features/statistics_60_1.png"
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">time</span>
-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">100</span><span class="p">):</span>
-    <span class="n">time</span><span class="o">.</span><span class="n">sleep</span><span class="p">(</span><span class="mf">0.1</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">i</span><span class="o">%</span><span class="k">10</span>==0:
-        <span class="nb">print</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">end</span><span class="o">=</span><span class="s1">&#39;</span><span class="se">\r</span><span class="s1">&#39;</span><span class="p">)</span>
-</pre></div>
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-<pre>90
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
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-</div>
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-<h3 id="Example:-exponential-fit">Example: exponential fit<a class="anchor-link" href="#Example:-exponential-fit"> </a></h3>
-</div>
-</div>
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<h4 id="Guess-the-number-of-pages-of-a-book">Guess the number of pages of a book<a class="anchor-link" href="#Guess-the-number-of-pages-of-a-book"> </a></h4><p><a href="https://docs.google.com/forms/d/e/1FAIpQLSeZJ0QII5JcST-M9_JgGYNzX3GahULVVFc31DQnWjJ4SdUQwg/viewform?fbclid=IwAR0wEadM0ZH-HXmp3lkAM3emCDPxs_6F509BS3EkOvldp-NFzbCOkZVSjR4">https://docs.google.com/forms/d/e/1FAIpQLSeZJ0QII5JcST-M9_JgGYNzX3GahULVVFc31DQnWjJ4SdUQwg/viewform?fbclid=IwAR0wEadM0ZH-HXmp3lkAM3emCDPxs_6F509BS3EkOvldp-NFzbCOkZVSjR4</a></p>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-$$
-f(x)=a\exp\left[ -\frac{(x-\mu)^2}{2\sigma^2} \right]
-$$<p>where $a$ is the height of the gaussian, $\mu$ is the mean (expected value), and $\sigma$ es la varianze</p>
-
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
-</pre></div>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;https://docs.google.com/spreadsheets/d/e/2PACX-1vTu_XE2dAiTcjHTfbaVKt7xEl_GnNeF_VYFsIBi5uM-gqBlBRfNHso-X1z3lxV7IW2f9UYKmZkSOYv-/pub?output=csv&#39;</span><span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Guess&#39;</span><span class="p">]</span><span class="o">=</span><span class="n">df</span><span class="o">.</span><span class="n">Guess</span><span class="o">.</span><span class="n">str</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s1">&#39;,&#39;</span><span class="p">,</span><span class="s1">&#39;&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-</pre></div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">bins</span><span class="o">=</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1500</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>See solution: <a href="./gaussian_fit.ipynb">gaussian_fit.ipynb</a></p>
-
-</div>
-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>See fit exponential: <a href="https://github.com/restrepo/COVID-19">https://github.com/restrepo/COVID-19</a></p>
-
-</div>
-</div>
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-<h1 id="Random-Numbers">Random Numbers<a class="anchor-link" href="#Random-Numbers"> </a></h1>
-</div>
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-<p>In nature it is not uncommon finding phenomena that are random intrinsic, this is why it becomes a necessity to produce random numbers in order to model such events. But, let us think about the operations that a computer can do, they are done following certain stablished rules, how then can be generated random numbers?</p>
-<p>This is achieved until certain point, it is only possible produce pseudo numbers, i.e, numbers obtained following some basic rules. At sequence of numbers apparently random but that are going to repeat after some period.</p>
-<p>Now, the most basic way to understand the generation of a pseudo-random number consists in following the next recurrence rule that produces integer random numbers</p>
-$$
-r_{i+1} = (ar_i+b)\%N 
-$$<p>$r_i$ is the seed, $a$ and $b$ and $N$ are coefficients chose. Notice that $\%$ represents the module, then the numbers obtained are going to be smaller than N.</p>
-<p>Now, consider the case when $ a= 3 $, $b = 2$ and $N = 10$ and the initial seed $r_0 = 3$. The new seed is the number obtaine
-in the last step.</p>
-
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%</span><span class="k">pylab</span> inline
-</pre></div>
-
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-<pre>Populating the interactive namespace from numpy and matplotlib
-</pre>
-</div>
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-</div>
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-<pre>/usr/local/lib/python3.5/dist-packages/IPython/core/magics/pylab.py:160: UserWarning: pylab import has clobbered these variables: [&#39;colors&#39;, &#39;random&#39;]
-`%matplotlib` prevents importing * from pylab and numpy
-  &#34;\n`%matplotlib` prevents importing * from pylab and numpy&#34;
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="c1"># Randon number function </span>
-<span class="k">def</span> <span class="nf">random_number</span><span class="p">(</span><span class="n">seed</span><span class="p">):</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">seed</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span><span class="o">%</span><span class="k">N</span>
-
-<span class="c1">#Constant values </span>
-<span class="n">a</span> <span class="o">=</span> <span class="mi">4</span>
-<span class="n">b</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="n">N</span> <span class="o">=</span> <span class="mi">9</span>
-<span class="c1">#Amount of random numbers  </span>
-<span class="n">N_iter</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="n">rnumber</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">N_iter</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<span class="c1">#Initial seed</span>
-<span class="n">rnumber</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">4</span>
-<span class="c1">#rnumber[0] = 10</span>
-
-<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_iter</span><span class="p">):</span> 
-    <span class="n">rnumber</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">random_number</span><span class="p">(</span> <span class="n">rnumber</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-
-<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Random numbers produced using a = </span><span class="si">%d</span><span class="s2">, b = </span><span class="si">%d</span><span class="s2"> and N = </span><span class="si">%d</span><span class="se">\n</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="p">(</span> <span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">N</span> <span class="p">))</span>  
-<span class="nb">print</span> <span class="p">(</span><span class="n">rnumber</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>   
-</pre></div>
-
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-</div>
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-<pre>Random numbers produced using a = 4, b = 1 and N = 9
-
-[8. 6. 7. 2. 0. 1. 5. 3. 4. 8. 6. 7. 2. 0. 1.]
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
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-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">random</span>
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">import</span> <span class="nn">time</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="p">)</span>
-<span class="n">time</span><span class="o">.</span><span class="n">sleep</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-<div class="output">
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-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>1605636036.2581604
-1605636038.2598178
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> <span class="c1">#seed is unix time</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
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-
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-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>0.6982585360672241</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span><span class="o">=</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">4890098</span><span class="p">)</span>
-<span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(),</span>
-<span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(),</span>
-<span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">())</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
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-<div class="jb_output_wrapper }}">
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-<pre>(0.6771981606438447, 0.7282646242796362, 0.12733022922307846)</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
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-<div class="output">
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-<div class="output_area">
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-<pre>array([0.74979577, 0.77025198, 0.1943525 , 0.418922  , 0.91964572])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-</pre></div>
-
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-</div>
-</div>
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-<pre>array([0.4359949 , 0.02592623, 0.54966248, 0.43532239, 0.4203678 ])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Notice that after N random numbers produced the apparently random sequence starts repeating again, i.e., N is the period of the sequence. Then, it is necessary to take N as big as possible but without incuring in an overflow.  What happens when the initial seed is changed?</p>
-<p><img src="https://github.com/restrepo/ComputationalMethods/blob/master/material/figures/random1.png?raw=1"></p>
-<p>To generate random numbers can be used numpy library, specifically the set functions random. There is also another library
-used random, but it contains few functions comparing with numpy. 
-To generate a random number between 0 and 1 initiallizing the seed with the actual time.</p>
-
-</div>
-</div>
-</div>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">()</span>
-<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">()</span>
-</pre></div>
-
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-<pre>0.5246933156265494</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>To generate a number between  and a number A</p>
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">A</span> <span class="o">=</span> <span class="mf">100.</span>
-<span class="n">A</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">((</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">))</span>
-</pre></div>
-
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-</div>
-</div>
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-<div class="output">
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-<pre>array([[13.25980014, 10.85324426, 81.24657358, 73.5488896 , 76.84488754],
-       [34.51142695, 72.82941893, 61.76035147, 98.75100252, 77.34796303],
-       [28.27847299, 14.76777458, 55.78354501, 24.62013376, 36.56582928],
-       [81.08850504, 60.25923238, 71.22826739, 62.52388123, 56.28302001],
-       [89.37094444, 36.14415972, 16.15729293, 31.74634766, 24.41171835]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-</div>
-</div>
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>To generate a random number between a range -B to B</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">B</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="n">B</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">B</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">()</span>
-</pre></div>
-
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-</div>
-</div>
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-<pre>-3.578128990519911</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
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-<div class="text_cell_render border-box-sizing rendered_html">
-<p>The implementation in numpy is with:</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">15</span><span class="p">,</span><span class="mi">15</span><span class="p">,(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
-</pre></div>
-
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-<pre>array([[  5.34784268,  -2.27612914],
-       [  7.8623893 , -13.88564232]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
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-<h2 id="Example-2">Example 2<a class="anchor-link" href="#Example-2"> </a></h2>
-</div>
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-<h3 id="Random-walk">Random walk<a class="anchor-link" href="#Random-walk"> </a></h3><p>Start at the origin and take a 2D random walk. It is chosen values for $\Delta x'$ and $\Delta y'$ in the range [-1,1].
-They are normalized so each step is of unit length.</p>
-
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#Initial positions</span>
-<span class="n">x0</span> <span class="o">=</span> <span class="mf">0.</span>
-<span class="n">y0</span> <span class="o">=</span> <span class="mf">0.</span>
-<span class="n">pos</span> <span class="o">=</span> <span class="p">[</span><span class="n">x0</span><span class="p">,</span><span class="n">y0</span><span class="p">]</span>
-<span class="c1">#Number of random steps</span>
-<span class="n">N_steps</span> <span class="o">=</span> <span class="mi">1000</span>
-<span class="c1">#Number of random walks</span>
-<span class="n">N_walks</span> <span class="o">=</span> <span class="mi">3</span>
-
-
-<span class="k">def</span> <span class="nf">Random_walk</span><span class="p">(</span> <span class="n">pos</span><span class="p">,</span> <span class="n">N</span> <span class="p">):</span>
-    
-    <span class="c1">#Initializing the seed</span>
-    <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">()</span>
-    
-    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span> <span class="n">N</span> <span class="p">)</span>    
-    <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span> <span class="n">N</span> <span class="p">)</span>    
-
-    <span class="c1">#Initial conditions</span>
-    <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">pos</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">y</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">pos</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-    
-    <span class="c1">#Generating random positions</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">N_steps</span><span class="p">):</span>
-        <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span>  <span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span> <span class="p">)</span><span class="o">*</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span>  <span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span> <span class="p">)</span><span class="o">*</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-    
-    <span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-    
-
-<span class="n">colors</span> <span class="o">=</span> <span class="p">(</span><span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="s1">&#39;g&#39;</span><span class="p">,</span> <span class="s1">&#39;c&#39;</span><span class="p">,</span> <span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="s1">&#39;m&#39;</span><span class="p">,</span> <span class="s1">&#39;y&#39;</span><span class="p">,</span> <span class="s1">&#39;k&#39;</span><span class="p">)</span>
-<span class="n">axisNum</span> <span class="o">=</span> <span class="mi">0</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span> <span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span> <span class="p">)</span>
-
-<span class="c1">#Plotting random walks</span>
-<span class="k">for</span> <span class="n">j</span><span class="p">,</span><span class="n">c</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_walks</span><span class="p">),</span><span class="n">colors</span><span class="p">):</span>
-    
-    <span class="n">axisNum</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="o">=</span> <span class="n">Random_walk</span><span class="p">(</span> <span class="n">pos</span><span class="p">,</span> <span class="n">N_steps</span> <span class="p">)</span>
-    <span class="c1">#If N_walks &gt; repeat the colors</span>
-    <span class="n">color</span> <span class="o">=</span> <span class="n">colors</span><span class="p">[</span><span class="n">axisNum</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">colors</span><span class="p">)]</span>  
-    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="s2">&quot;v-&quot;</span><span class="p">,</span> <span class="n">color</span> <span class="o">=</span> <span class="n">color</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span> <span class="mf">0.5</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s2">&quot;walk </span><span class="si">%d</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
-    
-
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Random walks&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>&lt;matplotlib.legend.Legend at 0x7f95f6040e10&gt;</pre>
-</div>
-
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/statistics_91_1.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
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-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="-----"><font color="red">     </font><a class="anchor-link" href="#-----"> </a></h2><p><strong>Activity</strong></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><font color='red'> 
-Radioactive decay: Spontaneous decay is a natural process in which a particle, with no external stimulation, decays into other
-particles. Then, the total amount of particles in a sample decreases with time, so will the number of decays. 
-The probability decay is constant per particle and is given by</p>
-$$
-P = -\lambda  
-$$<p>Determin when radioactive decay looks like exponential decay and when it looks stochastic depending on the initial number of particles $N(t)$ .For this, suposse that the decay rate is 0.3$\times 10^6 s^{-1}$ . Make a logarithmic plot to show the results. 
-&lt;/font&gt;</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">15</span><span class="p">,</span><span class="mi">15</span><span class="p">,(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>array([[ 1.56800444,  5.62105311],
-       [10.43483622, 14.89746624]])</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Espacio de parámetros aleatorios</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">N</span><span class="o">=</span><span class="mi">10000</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="n">N</span><span class="p">),</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="n">N</span><span class="p">),</span><span class="s1">&#39;r.&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>[&lt;matplotlib.lines.Line2D at 0x7f95f5c66a90&gt;]</pre>
-</div>
-
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/statistics_96_1.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Hasta dos ordenes de magnitud</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">N</span><span class="o">=</span><span class="mi">10000</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">100</span><span class="p">,</span><span class="n">N</span><span class="p">),</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">100</span><span class="p">,</span><span class="n">N</span><span class="p">),</span><span class="s1">&#39;r.&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_text output_subarea output_execute_result">
-<pre>[&lt;matplotlib.lines.Line2D at 0x7f95f5c42f98&gt;]</pre>
-</div>
-
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/statistics_98_1.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>Varios ordenes de magnitud: $10^{-4}$ a $10^2$</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">N</span><span class="o">=</span><span class="mi">10000</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">loglog</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mf">1E-4</span><span class="p">,</span><span class="mf">1E2</span><span class="p">,</span><span class="n">N</span><span class="p">),</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mf">1E-4</span><span class="p">,</span><span class="mf">1E2</span><span class="p">,</span><span class="n">N</span><span class="p">),</span><span class="s1">&#39;r.&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>[&lt;matplotlib.lines.Line2D at 0x7f95f5c00940&gt;]</pre>
-</div>
-
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/statistics_100_1.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>$x=10^{\log_{10}(x)}$</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
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-<div class=" highlight hl-ipython3"><pre><span></span><span class="mi">10</span><span class="o">**</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="mf">24.456</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
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-<pre>24.455999999999996</pre>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">N</span><span class="o">=</span><span class="mi">10000</span>
-<span class="n">xmin</span><span class="o">=</span><span class="mf">3E-4</span><span class="p">;</span><span class="n">xmax</span><span class="o">=</span><span class="mf">5E2</span>
-<span class="n">ymin</span><span class="o">=</span><span class="n">xmin</span><span class="p">;</span><span class="n">ymax</span><span class="o">=</span><span class="n">xmax</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">loglog</span><span class="p">(</span>  <span class="mi">10</span><span class="o">**</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">xmin</span><span class="p">),</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">xmax</span><span class="p">),</span><span class="n">N</span><span class="p">),</span>
-             <span class="mi">10</span><span class="o">**</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">ymin</span><span class="p">),</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">ymax</span><span class="p">),</span><span class="n">N</span><span class="p">),</span><span class="s1">&#39;r.&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
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-<div class="output_text output_subarea output_execute_result">
-<pre>[&lt;matplotlib.lines.Line2D at 0x7f95f6008dd8&gt;]</pre>
-</div>
-
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_png output_subarea ">
-<img src="../images/features/statistics_103_1.png"
->
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
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-<pre>38</pre>
-</div>
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-</div>
-</div>
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-</div>
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-</div>
-</div>
-
- 
-
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-    <main class="jupyter-page">
-    <div id="page-info"><div id="page-title">Introduction</div>
-</div>
-    <div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p><a href="https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/index.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>To open in Colab use the previous button or replace in the address bar:</p>
-
-<pre><code>github.com → colab.research.google.com/github</code></pre>
-<h1 id="Computational-Methods-for-Physics-&amp;-Astronomy">Computational Methods for Physics &amp; Astronomy<a class="anchor-link" href="#Computational-Methods-for-Physics-&amp;-Astronomy"> </a></h1><h2 id="Book-version:"><span style="color:red">Book version:</span><a class="anchor-link" href="#Book-version:"> </a></h2><p><a href="https://restrepo.github.io/ComputationalMethods/">https://restrepo.github.io/ComputationalMethods/</a></p>
-<p>by: <strong>Sebastian Bustamante</strong> <em>2014/2015</em> <strong>Diego Restrepo</strong> 2017...</p>
-<p><img src="https://raw.githubusercontent.com/sbustamante/ComputationalMethods/master/material/figures/Collage.png" alt=""></p>
-<p>This course is intended for students of Astronomy and Physics at the Universidad de Antioquia 
-and will cover some numerical methods commonly used in science and specially in physics ans astronomy. These topics will be addressed from a formal context but also keeping 
-a practical and computational approach, illustrating many useful applications in
-problems of physics and astronomy.</p>
-<p>The practical component will be almost entirely developed in <em>Python</em> and 
-slightly less in <em>C</em> (when computational performance is required). 
-However students with knowledge in other programming languages (except
-privative languages like MatLab, Mathematica) are also aimed to use them.</p>
-<p>In this repository it can be found all the related material of the course, 
-including the detailed program, notes and presentations, examples (ipython 
-notebooks) and homeworks. (This repository may be subject to changes continuously 
-as the course advances).</p>
-<h2 id="Instructions-to-use-the-repository">Instructions to use the repository<a class="anchor-link" href="#Instructions-to-use-the-repository"> </a></h2><ul>
-<li>Use the Colab button to open the reposotry in an executable enviroment</li>
-<li>To save changes: <ol>
-<li><code>File</code> → <code>Download .ipynb</code></li>
-<li>Open a repository in GitHub and Upload the file</li>
-<li>To open again in Colab replace in the address bar: <code>github.com → colab.research.google.com/github</code> </li>
-</ol>
-</li>
-</ul>
-<p><a href="https://github.com/sbustamante/ComputationalMethods/raw/master/syllabus/Programa_M%C3%A9todos_Computacionales.pdf"><strong>SYLLABUS</strong></a>:
-detailed description of the program of the course, including a brief motivation and presentation, 
-topics to be covered, evaluation and bibliography.</p>
-<h2 id="Contents">Contents<a class="anchor-link" href="#Contents"> </a></h2><h3 id="1.-Python-(1-week)">1. <strong>Python</strong> <em>(1 week)</em><a class="anchor-link" href="#1.-Python-(1-week)"> </a></h3><p><strong>Topics</strong></p>
-<ul>
-<li><a href="./material/Syncing_fork.ipynb">Git usage</a> <a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/Syncing_fork.ipynb">[View]</a></li>
-</ul>
-<p><em>Processing</em></p>
-<ul>
-<li><a href="./overview-python.ipynb">Overview of python</a> <a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/overview-python.ipynb">[View]</a> <!-- href="" target="_parent">--></li>
-<li><a href="./material/scientific-libraries.ipynb">Implementation of scientific libraries</a><!-- - [Basic scripting](./material/basic-scripting.ipynb)--></li>
-</ul>
-<p><em>Data analysis</em></p>
-<ul>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/Pandas.ipynb">Pandas</a></li>
-</ul>
-<p><em>Visualization</em></p>
-<ul>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/matplotlib.ipynb">Plotting with matplotlib</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/ipython-notebooks.ipynb">Ipython notebooks</a></li>
-</ul>
-<p><strong>Activities</strong></p>
-<ul>
-<li><em>Activity 01:</em> Solve the problems in the section <a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/Basic_exercises.ipynb">Exercises</a>. </li>
-</ul>
-<h3 id="2.-Mathematical-Preliminaries-(1-week)">2. <strong>Mathematical Preliminaries</strong> <em>(1 week)</em><a class="anchor-link" href="#2.-Mathematical-Preliminaries-(1-week)"> </a></h3><p><strong>Topics</strong></p>
-<ul>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/computer-arithmetics.ipynb">Computer arithmetics and round-off methods</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/algorithms-convergence.ipynb">Algorithms and convergence</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/numba.ipynb">Optimization</a></li>
-</ul>
-<h3 id="3.-One-Variable-Equations-(2-weeks)">3. <strong>One Variable Equations</strong> <em>(2 weeks)</em><a class="anchor-link" href="#3.-One-Variable-Equations-(2-weeks)"> </a></h3><p>./one-variable-equations.ipynb
-<strong>Topics</strong></p>
-<ul>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/one-variable-equations.ipynb#Bisection-Method">Bisection method</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/one-variable-equations-fixed-point.ipynb#Fixed-point-Iteration">Fixed-point iteration</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/one-variable-equations-fixed-point.ipynb#Newton-Raphson-Method">Newton-Raphson method</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/one-variable-equations-fixed-point.ipynb#Secant-Method">Secant method</a></li>
-</ul>
-<h3 id="4.-Interpolation-Techniques-(2-weeks)">4. <strong>Interpolation Techniques</strong> <em>(2 weeks)</em><a class="anchor-link" href="#4.-Interpolation-Techniques-(2-weeks)"> </a></h3><p><strong>Topics</strong></p>
-<ul>
-<li><a href="./material/interpolation.ipynb#Linear-Interpolation">Linear interpolation</a> <a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/interpolation.ipynb#Linear-Interpolation">[View]</a></li>
-<li><a href="./material/interpolation.ipynb#Lagrange-Polynomial">Lagrange polynomials</a> <a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/interpolation.ipynb#Lagrange-Polynomial">[View]</a></li>
-<li><a href="./material/interpolation_details.ipynb#Divided-Differences">Divided differences</a> <a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/interpolation_details.ipynb#Divided-Differences">[View]</a></li>
-<li><a href="./material/interpolation_details.ipynb#Hermite-Interpolation">Hermite interpolation</a> <a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/interpolation_details.ipynb#Hermite-Interpolation">[View]</a></li>
-<li><a href="./material/Minimization.ipynb">Minimization</a> <a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/Minimization.ipynb">[View]</a></li>
-</ul>
-<h3 id="5.-Numerical-Calculus-(2-weeks)">5. <strong>Numerical Calculus</strong> <em>(2 weeks)</em><a class="anchor-link" href="#5.-Numerical-Calculus-(2-weeks)"> </a></h3><p><strong>Topics</strong></p>
-<ul>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/numerical-calculus.ipynb#Numerical-Differentiation">Numerical differentiation</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/numerical-calculus-integration.ipynb#Numerical-Integration">Numerical integration</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/numerical-calculus-integration.ipynb#Composite-Numerical-Integration">Composite numerical integration</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/numerical-calculus-integration.ipynb#Adaptive-Quadrature-Methods">Adaptive quadrature methods</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/numerical-calculus-integration.ipynb#Improper-Integrals">Improper integrals</a></li>
-</ul>
-<p><strong>Activities</strong></p>
-<ul>
-<li><em>Activity:</em> <a href="http://nbviewer.jupyter.org/github/sbustamante/ComputationalMethods/blob/master/activities/T-profile-bar.ipynb"> Derivative exercise </a></li>
-</ul>
-<h3 id="6.-Linear-Algebra-(2-weeks)">6. <strong>Linear Algebra</strong> <em>(2 weeks)</em><a class="anchor-link" href="#6.-Linear-Algebra-(2-weeks)"> </a></h3><p><strong>Topics</strong></p>
-<ul>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/linear-algebra.ipynb#Gaussian-Elimination">Gaussian elimination</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/linear-algebra.ipynb#Linear-Systems-of-Equations">Linear systems of equations</a> [<a href="./material/linear-algebra.ipynb#Linear-Systems-of-Equation">Local</a>]</li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/linear-algebra.ipynb#Pivoting-Strategies">Pivoting strategies</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/linear-algebra.ipynb#Matrix-Inversion">Matrix inversion</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/linear-algebra.ipynb#Determinant-of-a-Matrix">Determinant of a Matrix</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/linear-algebra.ipynb#LU-Factorization">LU factorization</a></li>
-</ul>
-<h3 id="7.-Differential-Equations-(2-weeks)">7. <strong>Differential Equations</strong> <em>(2 weeks)</em><a class="anchor-link" href="#7.-Differential-Equations-(2-weeks)"> </a></h3><p><strong>Topics</strong></p>
-<ul>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/differential-equations.ipynb#First-Order-Methods">First order methods</a> [<a href="./differential-equations.ipynb#First-Order-Methods">Local</a>]</li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/differential-equations.ipynb#High-Order-Methods">High order methods</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/differential-equations.ipynb#Two-Point-Boundary-Value-Problems">Two-Point boundary value problems</a></li>
-</ul>
-<h3 id="8.-Statistics-(1-week)">8. <strong>Statistics</strong> <em>(1 week)</em><a class="anchor-link" href="#8.-Statistics-(1-week)"> </a></h3><p><strong>Topics</strong></p>
-<ul>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/statistics.ipynb">Data adjust</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/statistics.ipynb#Random-Numbers">Random Numbers</a></li>
-</ul>
-<h3 id="9.-Extra-material-(1-session)">9. <strong>Extra material</strong> <em>(1 session)</em><a class="anchor-link" href="#9.-Extra-material-(1-session)"> </a></h3><p><strong>Topics</strong></p>
-<ul>
-<li><a href="https://repl.it/">Online IDE</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/Intro_clases.ipynb">Clases</a></li>
-<li><a href="https://github.com/restrepo/ComputationalMethods/blob/master/material/cpp.ipynb">C++</a></li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Bibliography">Bibliography<a class="anchor-link" href="#Bibliography"> </a></h2><p>[1a] Gonzalo Galiano Casas, Esperanza García Gonzalo <a href="https://www.unioviedo.es/compnum/expositive/handbook/metnum.pdf">Numerical Computation</a> - <a href="https://drive.google.com/file/d/1gwHYLx5aBf3oXRhmGeG-n1toTtAUhvfX/view?usp=sharing">GD</a> - <a href="https://www.unioviedo.es/compnum/index.php/english-menu">Web page with notebooks</a><br/>
-[1b] <a href="https://scipy-lectures.org/">Scipy Lecture Notes</a> <a href="https://scipy-lectures.org/_downloads/ScipyLectures-simple.pdf">[PDF]</a><br/>
-[1b] <a href="https://drive.google.com/file/d/0BxoOXsn2EUNIekRUMFVPYXVsSTg/view?usp=sharing">Jensen, Computational_Physics</a>.  <a href="https://github.com/mhjensen">GitHub</a> <br/>
-[1d] <a href="http://astro.physics.ncsu.edu/urca/course_files">Introduction to computational physics</a><br/>
-[1e] Thomas J. Sargent John Stachurski, <a href="https://python-programming.quantecon.org">Python Programming for Quantitative Economics</a> 
-<a href="https://github.com/QuantEcon/lecture-python-programming.notebooks">GitHub</a><br/>
-[1f] Ani Adhikari and John DeNero, <a href="https://www.inferentialthinking.com/chapters/intro.html">Computational and Inferential Thinking</a><br/>
-[1g] Mo Mu, <a href="https://www.math.ust.hk/~mamu/courses/230/course.htm">MATH 3311: Introduction to Numerical Methods</a> <a href="https://drive.google.com/drive/folders/1vv38SS7Zpw8mOHG7Kq_3lZr6o9H4xOYP?usp=sharing">GD</a><br/>
-[1h] Zhiliang Xu, <a href="https://www3.nd.edu/~zxu2/ACMS40390-F16.html">ACMS 40390: Fall 2016 - Numerical Analysis</a> <a href="https://drive.google.com/drive/folders/1Z9Fws1v34PuPdGAp_mBCWmTvZM_Qd6Xk?usp=sharing">GD</a>&lt;/br&gt;
-[1f] Computational Physiscs, University of Toronto <a href="https://computation.physics.utoronto.ca/">https://computation.physics.utoronto.ca/</a> <br/>
-[1g] Simon Sirca, Computational Methods for Physicists <a href="https://drive.google.com/file/d/1Yg19RBSe0ifGCn2J-9J5kiY990PL6h7i/view?usp=sharing">GD</a> <br/>
-[1h] Anthony Scopatz and Kathryn D. Huff, Effective Computation in Physics: Field Guide to Research with Python <a href="http://lilith.fisica.ufmg.br/~dickman/transfers/comp/textos/Effective%20Computation%20in%20Physics%20(Python">PDF</a>.pdf) <a href="https://drive.google.com/file/d/13822qJgXS5HbimBz5czM4gDHMg3p7d02/view?usp=sharing">GD</a> <br/>
-[1i] Tao Pang, An Introduction to Computational Physics <a href="http://lilith.fisica.ufmg.br/~dickman/transfers/comp/textos/Pang%20-%20An%20introduction%20to%20computational%20physics%20(2ed.,%20CUP,%202006">PDF</a>.pdf) <br/>
-[2] <a href="https://drive.google.com/file/d/0BxoOXsn2EUNIejJXVEVRUV9DclU/view?usp=sharing">Landau Paez, A Survey of Computational Physics </a><br/>
-[3] <a href="https://drive.google.com/file/d/0BxoOXsn2EUNIQUdFVkctR2xWRUk/view?usp=sharing">Kiusalaas, Numerical Methods in Engineering with Python</a><br/>
-[4] <a href="https://drive.google.com/file/d/0BxoOXsn2EUNIVHR4T1BKMEVDX2c/view?usp=sharing">Sørenssen Elementary Mechanics Using Python<em> A Modern Course Combining Analytical and Numerical Techniques</em></a><br/>
-[5] <a href="https://drive.google.com/file/d/0BxoOXsn2EUNIZTNsT2RYUTloZzQ/view?usp=sharing">Langtangen_Python_Scripting_for_Computational_Science_3rd_edition</a><br/>
-[6] <a href="http://docs.python-guide.org/en/latest/">Python programming standards</a> Repository: <a href="https://github.com/kennethreitz/python-guide">https://github.com/kennethreitz/python-guide</a>)<br/>
-[7] <a href="http://pages.cs.wisc.edu/~amos/412/lecture-notes/">Lecture notes Course Numerical Analysis (MatLab)</a><br/>
-[8] Jupyter+Python: <a href="http://www.physics.nyu.edu/pine/pymanual/html/">Python in Science</a><br/>
-[9] Jupyter+Python: Python for Computational Science and Engineering <a href="http://www.southampton.ac.uk/~fangohr/teaching/python/book/Python-for-Computational-Science-and-Engineering.pdf">PDF</a> <a href="http://www.southampton.ac.uk/~fangohr/teaching/python/book/ipynb/">Notebooks</a><br/>
-[10] Jupyter+Python: <a href="http://sam-dolan.staff.shef.ac.uk/mas212">Scientific Computing and Simulation </a><br/>
-[11] E. Ayres, <a href="http://bit.ly/CPwPython">Computational Physics With Python</a><br/>
-[12] Hans Petter Langtangen <a href="https://hplgit.github.io/bumpy/doc/pub/bumpy.pdf">A worked example on scientific computing with Python</a><br/>
-[12] By Ani Adhikari and John DeNero <a href="https://www.inferentialthinking.com/">Computational and Inferential Thinking</a><br/>
-<a href="https://www.textbook.ds100.org/">13] By Sam Lau, Joey Gonzalez, and Deb Nolan. DS100 Principles and Techniques of Data Science</a><br/>
-[14] Software Design and Development: The Preliminary Course, – January 1, 2002, by Samuel Davis (Author) <a href="https://drive.google.com/file/d/1mq6Mj15VqM62M1cA24WOk60-t1UsKYyE/view?usp=sharing">[PDF]</a><br/>
-[15] Bradley Voytek, <em>et al</em>, <a href="https://datascienceinpractice.github.io/">Data Science in Practice</a><br/>
-[16] P. DeVries, A First Course in Computational Physics <a href="http://lilith.fisica.ufmg.br/~dickman/transfers/comp/textos/DeVries_A_first_course_in_computational_physics.pdf">PDF</a><br/></p>
-
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-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
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-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
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-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>Populating the interactive namespace from numpy and matplotlib
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
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-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
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-</pre></div>
-
-    </div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
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-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea output_execute_result">
-<h1>Hola mundo</h1>
-</div>
-
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
-    <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">display</span><span class="p">(</span> <span class="n">HTML</span><span class="p">(</span><span class="s1">&#39;&lt;h1&gt;Hola mundo&lt;/h1&gt;&#39;</span><span class="p">)</span> <span class="p">)</span> 
-<span class="n">display</span><span class="p">(</span> <span class="n">Math</span><span class="p">(</span><span class="s1">&#39; \int \sin x\, d x&#39;</span><span class="p">)</span> <span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;multiple and mixed print with display(..)&#39;</span><span class="p">)</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-<div class="output_html rendered_html output_subarea ">
-<h1>Hola mundo</h1>
-</div>
-
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-
-
-<div class="output_latex output_subarea ">
-$$ \int \sin x\, d x$$
-</div>
-
-</div>
-</div>
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>multiple and mixed print with display(..)
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-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<p>LaTeX is fully supported (see: <a href="https://stackoverflow.com/a/44375719/2268280">https://stackoverflow.com/a/44375719/2268280</a>)</p>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Requirements">Requirements<a class="anchor-link" href="#Requirements"> </a></h2><ul>
-<li>Modules installed with pip(3)</li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
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-<span class="n">sympy</span>
-<span class="n">ipywidgets</span>
-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>Writing requirements.txt
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-
-<div class="inner_cell">
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-</pre></div>
-
-    </div>
-</div>
-</div>
-
-<div class="output_wrapper">
-<div class="output">
-
-<div class="jb_output_wrapper }}">
-<div class="output_area">
-
-<div class="output_subarea output_stream output_stdout output_text">
-<pre>Writing postBuild
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
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-<span class="nb">cd</span> JSAnimation
-python3 setup.py install
-python setup.py install
-</pre></div>
-</li>
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-
-</div>
-</div>
-</div>
-</div>
-
-<div class="jb_cell">
-
-<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
-<div class="text_cell_render border-box-sizing rendered_html">
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-<li><a href="https://github.com/BruceSherwood/vpython-jupyter">Vpython-jupyter</a> See <a href="./demo.ipynb">demo.ipynb</a></li>
-<li><a href="https://github.com/restrepo/bullet3">bullet3</a></li>
-<li><a href="https://github.com/robclewley/pydstool">Pydstool</a></li>
-</ul>
-
-</div>
-</div>
-</div>
-</div>
-
- 
-
-
-    </main>
-    
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deleted file mode 100755
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-url: https://restrepo.github.io/ComputationalMethods          # the base hostname & protocol for your site, e.g. http://example.com
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-
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diff --git a/_data/toc.yml b/_data/toc.yml
deleted file mode 100755
index 8b72507..0000000
--- a/_data/toc.yml
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@@ -1,137 +0,0 @@
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-- title: Introduction
-  url: /index
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-
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-- title: Python overview
-  url: /features/overview-python
-  not_numbered: false
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-    url: /features/scientific-libraries
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-  - title: Interactive editing with Jupyter
-    url: /features/ipython-notebooks
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-    
-- title: Mathematical Preliminaries 
-  url: /features/computer-arithmetics
-  not_numbered: false
-  expand_sections: false
-  sections:
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-    url: /features/algorithms-convergence
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-- title: One Variable Equations  
-  url: /features/one-variable-equations
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-- title: Interpolation
-  url: /features/interpolation
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-    url: /features/LagrangePoly
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-  - title: Minimization
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-- title: Numerical calculus
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-  expand_sections: false
-  sections:
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-    url: /features/numerical-calculus-integration
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-  - title: Numerical integration
-    url: /features/numerical-calculus-integration
-    not_numbered: false
-  - title: Numerical integration methods
-    url: /features/numerical-calculus-integration-methods
-    not_numbered: false
-
-- title: Linear algebra
-  url: /features/linear-algebra
-  not_numbered: false
-  expand_sections: false
-  sections:
-  - title: Matriz diagonalization
-    url: /features/linear-algebra-diagonalization
-    not_numbered: false
-  - title: Custum Gaussian elimination
-    url: /features/linear-algebra-extra
-    not_numbered: false
-
-- title: Diferential equations
-  url: /features/differential-equations
-  not_numbered: false
-  expand_sections: false
-  sections:
-  - title: SIR Model
-    url: /features/medium_modelling_part_one
-    not_numbered: false
-
-- title: Statistics
-  url: /features/statistics
-  not_numbered: false
-  expand_sections: false
-
-- title: Clases
-  url: /features/Intro_clases
-  not_numbered: false
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diff --git a/_includes/buttons.html b/_includes/buttons.html
deleted file mode 100755
index 03b36c4..0000000
--- a/_includes/buttons.html
+++ /dev/null
@@ -1,9 +0,0 @@
-<div class="buttons">
-{% include buttons/download.html %}
-{% if page.interact_link %}
-  {% include buttons/thebelab.html %}
-  {% include buttons/nbinteract.html %}
-  {% include buttons/binder.html %}
-  {% include buttons/jupyterhub.html %}
-{% endif %}
-</div>
diff --git a/_includes/buttons/binder.html b/_includes/buttons/binder.html
deleted file mode 100755
index c9577f1..0000000
--- a/_includes/buttons/binder.html
+++ /dev/null
@@ -1,14 +0,0 @@
-{% if site.use_binder_button %}
-
-{% if site.use_jupyterlab %}
-    {% assign binder_interact_prefix="urlpath=lab/tree/" %}
-{% else %}
-    {% assign binder_interact_prefix="filepath=" %}
-{% endif %}
-
-{% capture interact_url_binder %}v2/gh/{{ site.binder_repo_org }}/{{ site.binder_repo_name }}/{{ site.binder_repo_branch }}?{{ binder_interact_prefix }}{{ page.interact_link | url_encode }}{% endcapture %}
-{% capture interact_icon_binder %}{{ site.images_url | relative_url }}/logo_binder.svg{% endcapture %}
-
-<a href="{{ site.binderhub_url }}/{{ interact_url_binder }}"><button class="interact-button" id="interact-button-binder"><img class="interact-button-logo" src="{{ interact_icon_binder }}" alt="Interact" />{{ site.binderhub_interact_text }}</button></a>
-
-{%- endif %}
\ No newline at end of file
diff --git a/_includes/buttons/download.html b/_includes/buttons/download.html
deleted file mode 100755
index 190f011..0000000
--- a/_includes/buttons/download.html
+++ /dev/null
@@ -1,13 +0,0 @@
-{% if site.use_download_button -%}
-<div class="download-buttons-dropdown">
-    <button id="dropdown-button-trigger" class="interact-button"><img src="{{ site.images_url | relative_url }}/download-solid.svg" alt="Download" /></button>
-    <div class="download-buttons">
-        {% if page.interact_link -%}
-        <a href="{{ page.interact_link | relative_url }}" download>
-        <button id="interact-button-download" class="interact-button">{{ page.suffix | capitalize }}</button>
-        </a>
-        {% endif %}
-        <a id="interact-button-print"><button id="interact-button-download" class="interact-button">.pdf</button></a>
-    </div>
-</div>
-{%- endif %}
diff --git a/_includes/buttons/jupyterhub.html b/_includes/buttons/jupyterhub.html
deleted file mode 100755
index 8e17d31..0000000
--- a/_includes/buttons/jupyterhub.html
+++ /dev/null
@@ -1,13 +0,0 @@
-{% if site.use_jupyterhub_button %}
-
-{% if site.use_jupyterlab %}
-    {% assign hub_app="lab" %}
-{% else %}
-    {% assign hub_app="notebook" %}
-{% endif %}
-
-{% capture interact_url_jupyterhub %}hub/user-redirect/git-pull?repo={{ site.binder_repo_base }}/{{ site.binder_repo_org }}/{{ site.binder_repo_name }}&amp;branch={{ site.binder_repo_branch }}&amp;subPath={{ page.interact_link | url_encode }}&amp;app={{ hub_app }}{% endcapture %}
-{% capture interact_icon_jupyterhub %}{{ site.images_url | relative_url }}/logo_jupyterhub.svg{% endcapture %}
-<a href="{{ site.jupyterhub_url }}/{{ interact_url_jupyterhub }}"><button class="interact-button" id="interact-button-jupyterhub"><img class="interact-button-logo" src="{{ interact_icon_jupyterhub }}" alt="Interact" />{{ site.jupyterhub_interact_text }}</button></a>
-
-{% endif %}
diff --git a/_includes/buttons/nbinteract.html b/_includes/buttons/nbinteract.html
deleted file mode 100755
index eaf6e83..0000000
--- a/_includes/buttons/nbinteract.html
+++ /dev/null
@@ -1,3 +0,0 @@
-{% if site.use_show_widgets_button and page.has_widgets -%}
-<button id="interact-button-show-widgets" class="interact-button js-nbinteract-widget">Show Widgets</button>
-{% endif %}
\ No newline at end of file
diff --git a/_includes/buttons/thebelab.html b/_includes/buttons/thebelab.html
deleted file mode 100755
index 44a8322..0000000
--- a/_includes/buttons/thebelab.html
+++ /dev/null
@@ -1,3 +0,0 @@
-{% if site.use_thebelab_button -%}
-<button id="interact-button-thebelab" class="interact-button">{{ site.thebelab_button_text }}</button>
-{% endif %}
\ No newline at end of file
diff --git a/_includes/css_entry.scss b/_includes/css_entry.scss
deleted file mode 100755
index f693426..0000000
--- a/_includes/css_entry.scss
+++ /dev/null
@@ -1,18 +0,0 @@
-@import 'inuitcss/settings/settings.core';
-@import 'settings/settings.global.scss';
-
-@import 'inuitcss/tools/tools.font-size';
-@import 'inuitcss/tools/tools.clearfix';
-@import 'inuitcss/tools/tools.hidden';
-@import 'inuitcss/tools/tools.mq';
-
-@import 'inuitcss/elements/elements.page';
-@import 'inuitcss/elements/elements.headings';
-@import 'inuitcss/elements/elements.images';
-@import 'inuitcss/elements/elements.tables';
-@import 'elements/elements.typography';
-@import 'elements/elements.syntax-highlighting';
-@import 'elements/elements.tables';
-@import 'elements/elements.links';
-
-@import 'components/components.textbook__page';
diff --git a/_includes/fb_tags.html b/_includes/fb_tags.html
deleted file mode 100755
index 1ff5bc3..0000000
--- a/_includes/fb_tags.html
+++ /dev/null
@@ -1,7 +0,0 @@
-<meta property="og:url"         content="{{ page.url | replace:'index.html','' | prepend: site.baseurl | prepend: site.url  | relative_url }}" />
-<meta property="og:type"        content="article" />
-<meta property="og:title"       content="{% if page.title %}{{ page.title | escape }}{% else %}{{ site.title | escape }}{% endif %}" />
-<meta property="og:description" content="{{ page.content | strip_html | strip_newlines | truncate: 160 }}" />
-<meta property="og:image"       content="{{ site.textbook_logo | absolute_url }}" />
-
-<meta name="twitter:card" content="summary">
diff --git a/_includes/footer.html b/_includes/footer.html
deleted file mode 100755
index a42f4ea..0000000
--- a/_includes/footer.html
+++ /dev/null
@@ -1,3 +0,0 @@
-<footer>
-  <p class="footer">{{ site.footer_text }}</p>
-</footer>
diff --git a/_includes/google_analytics.html b/_includes/google_analytics.html
deleted file mode 100755
index 23b6bd1..0000000
--- a/_includes/google_analytics.html
+++ /dev/null
@@ -1,11 +0,0 @@
-{% if site.google_analytics.mytrackingcode %}
-<!-- Global site tag (gtag.js) - Google Analytics -->
-<script async src="https://www.googletagmanager.com/gtag/js?id={{ site.google_analytics.mytrackingcode }}"></script>
-<script>
-  window.dataLayer = window.dataLayer || [];
-  function gtag(){dataLayer.push(arguments);}
-  gtag('js', new Date());
-
-  gtag('config', '{{ site.google_analytics.mytrackingcode }}');
-</script>
-{% endif %}
diff --git a/_includes/head.html b/_includes/head.html
deleted file mode 100755
index e4b63f4..0000000
--- a/_includes/head.html
+++ /dev/null
@@ -1,89 +0,0 @@
-<head>
-  <meta charset="utf-8">
-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
-  <meta name="viewport" content="width=device-width,minimum-scale=1">
-
-  <title>{% if page.title %}{{ page.title | escape }}{% else %}{{ site.title | escape }}{% endif %}</title>
-  <meta name="description" content="{{ page.content | strip_html | strip_newlines | truncate: 160 }}">
-
-  <link rel="canonical" href="{{ page.url | replace:'index.html','' | prepend: site.baseurl | prepend: site.url | relative_url }}">
-  <link rel="alternate" type="application/rss+xml" title="{{ site.title }}" href="{{ "/feed.xml" | prepend: site.baseurl | prepend: site.url | relative_url }}">
-
-  {% include fb_tags.html %}
-
-  <script type="application/ld+json">
-  {% include metadata.json %}
-  </script>
-  <link rel="stylesheet" href="{{ site.css_url | relative_url }}/styles.css">
-
-  <!-- <link rel="manifest" href="/manifest.json"> -->
-  <!-- <link rel="mask-icon" href="/safari-pinned-tab.svg" color="#efae0a"> -->
-  <meta name="msapplication-TileColor" content="#da532c">
-  <meta name="msapplication-TileImage" content="/mstile-144x144.png">
-  <meta name="theme-color" content="#233947">
-
-  <!-- Favicon -->
-  <link rel="shortcut icon" type="image/x-icon" href="{{ site.favicon_path | relative_url }}">
-
-  <!-- MathJax Config -->
-  {% include mathjax.html %}
-
-  <!-- DOM updating function -->
-  <script src="{{ site.js_url | relative_url }}/page/dom-update.js"></script>
-
-  <!-- Selectors for elements on the page -->
-  <script src="{{ site.js_url | relative_url }}/page/documentSelectors.js"></script>
-
-  <!-- Define some javascript variables that will be useful in other javascript -->
-  <script>
-    const site_basename = '{{ site.baseurl | strip / }}';
-  </script>
-
-  <!-- Add AnchorJS to let headers be linked -->
-  <script src="https://cdnjs.cloudflare.com/ajax/libs/anchor-js/4.2.0/anchor.min.js" async></script>
-  <script src="{{ site.js_url | relative_url }}/page/anchors.js" async></script>
-
-  <!-- Include Turbolinks to make page loads fast -->
-  <!-- https://github.com/turbolinks/turbolinks -->
-  <script src="https://cdnjs.cloudflare.com/ajax/libs/turbolinks/5.2.0/turbolinks.js" async></script>
-  <meta name="turbolinks-cache-control" content="no-cache">
-
-  <!-- Load nbinteract for widgets -->
-  {% include js/nbinteract.html %}
-
-  <!-- Load Thebelab for interactive widgets -->
-  {% include js/thebelab.html %}
-
-  <!-- Load the auto-generating TOC (non-async otherwise the TOC won't load w/ turbolinks) -->
-  <script src="https://cdnjs.cloudflare.com/ajax/libs/tocbot/4.8.1/tocbot.min.js" async></script>
-  <script src="{{ site.js_url | relative_url }}/page/tocbot.js"></script>
-
-  <!-- Google analytics -->
-  {% include google_analytics.html %}
-
-  <!-- Clipboard copy button -->
-  <script src="https://cdnjs.cloudflare.com/ajax/libs/clipboard.js/2.0.4/clipboard.min.js" async></script>
-
-  <!-- Load custom website scripts -->
-  <script src="{{ site.js_url | relative_url }}/scripts.js" async></script>
-
-  <!-- Load custom user CSS and JS  -->
-  <script src="{{ site.custom_static_url | relative_url }}/custom.js" async></script>
-  <link rel="stylesheet" href="{{ site.custom_static_url | relative_url }}/custom.css">
-
-  <!-- Update interact links w/ REST param, is defined in includes so we can use templates -->
-  {% include js/interact-update.html %}
-
-  <!-- Lunr search code - will only be executed on the /search page -->
-  <script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.6/lunr.min.js" async></script>
-  <script>{% include search/lunr/lunr-en.js %}</script>
-
-  <!-- Load JS that depends on site variables -->
-  <script src="{{ site.js_url | relative_url }}/page/copy-button.js" async></script>
-
-  <!-- Hide cell code -->
-  <script src="{{ site.js_url | relative_url }}/page/hide-cell.js" async></script>
-
-  <!-- Printing the screen -->
-  {% include js/print.html %}
-</head>
diff --git a/_includes/js/interact-update.html b/_includes/js/interact-update.html
deleted file mode 100755
index e93c5ca..0000000
--- a/_includes/js/interact-update.html
+++ /dev/null
@@ -1,142 +0,0 @@
-{% if site.use_jupyterhub_button or site.use_binder_button %}
-<script>
-/**
-  * To auto-embed hub URLs in interact links if given in a RESTful fashion
- */
-
-function getJsonFromUrl(url) {
-  var query = url.split('?');
-  if (query.length < 2) {
-    // No queries so just return false
-    return false;
-  }
-  query = query[1];
-  // Collect REST params into a dictionary
-  var result = {};
-  query.split("&").forEach(function(part) {
-    var item = part.split("=");
-    result[item[0]] = decodeURIComponent(item[1]);
-  });
-  return result;
-}
-    
-function dict2param(dict) {
-    params = Object.keys(dict).map(function(k) {
-        return encodeURIComponent(k) + '=' + encodeURIComponent(dict[k])
-    });
-    return params.join('&')
-}
-
-// Parse a Binder URL, converting it to the string needed for JupyterHub
-function binder2Jupyterhub(url) {
-  newUrl = {};
-  parts = url.split('v2/gh/')[1];
-  // Grab the base repo information
-  repoinfo = parts.split('?')[0];
-  var [org, repo, ref] = repoinfo.split('/');
-  newUrl['repo'] = ['https://github.com', org, repo].join('/');
-  newUrl['branch'] = ref
-  // Grab extra parameters passed
-  params = getJsonFromUrl(url);
-  if (params['filepath'] !== undefined) {
-    newUrl['subPath'] = params['filepath']
-  }
-  return dict2param(newUrl);
-}
-
-// Filter out potentially unsafe characters to prevent xss
-function safeUrl(url)
-{
-   return String(encodeURIComponent(url))
-            .replace(/&/g, '&amp;')
-            .replace(/"/g, '&quot;')
-            .replace(/'/g, '&#39;')
-            .replace(/</g, '&lt;')
-            .replace(/>/g, '&gt;');
-}
-
-function addParamToInternalLinks(hub) {
-  var links = document.querySelectorAll("a").forEach(function(link) {
-    var href = link.href;
-    // If the link is an internal link...
-    if (href.search("{{ site.url }}") !== -1 || href.startsWith('/') || href.search("127.0.0.1:") !== -1) {
-      // Assume we're an internal link, add the hub param to it
-      var params = getJsonFromUrl(href);
-      if (params !== false) {
-        // We have REST params, so append a new one
-        params['jupyterhub'] = hub;
-      } else {
-        // Create the REST params
-        params = {'jupyterhub': hub};
-      }
-      // Update the link
-      var newHref = href.split('?')[0] + '?' + dict2param(params);
-      link.setAttribute('href', decodeURIComponent(newHref));
-    }
-  });
-  return false;
-}
-
-
-// Update interact links
-function updateInteractLink() {
-    // hack to make this work since it expects a ? in the URL
-    rest = getJsonFromUrl("?" + location.search.substr(1));
-    jupyterHubUrl = rest['jupyterhub'];
-    var hubType = null;
-    var hubUrl = null;
-    if (jupyterHubUrl !== undefined) {
-      hubType = 'jupyterhub';
-      hubUrl = jupyterHubUrl;
-    }
-
-    if (hubType !== null) {
-      // Sanitize the hubUrl
-      hubUrl = safeUrl(hubUrl);
-
-      // Add HTTP text if omitted
-      if (hubUrl.indexOf('http') < 0) {hubUrl = 'http://' + hubUrl;}
-      var interactButtons = document.querySelectorAll("button.interact-button")
-      var lastButton = interactButtons[interactButtons.length-1];
-      var link = lastButton.parentElement;
-
-      // If we've already run this, skip the link updating
-      if (link.nextElementSibling !== null) {
-        return;
-      }
-
-      // Update the link and add context div
-      var href = link.getAttribute('href');
-      if (lastButton.id === 'interact-button-binder') {
-        // If binder links exist, we need to re-work them for jupyterhub
-        if (hubUrl.indexOf('http%3A%2F%2Flocalhost') > -1) {
-          // If localhost, assume we're working from a local Jupyter server and remove `/hub`
-          first = [hubUrl, 'git-sync'].join('/')
-        } else {
-          first = [hubUrl, 'hub', 'user-redirect', 'git-sync'].join('/')
-        }
-        href = first + '?' + binder2Jupyterhub(href);
-      } else {
-        // If interact button isn't binderhub, assume it's jupyterhub
-        // If JupyterHub links, we only need to replace the hub url
-        href = href.replace("{{ site.jupyterhub_url }}", hubUrl);
-        if (hubUrl.indexOf('http%3A%2F%2Flocalhost') > -1) {
-          // Assume we're working from a local Jupyter server and remove `/hub`
-          href = href.replace("/hub/user-redirect", "");
-        }
-      }
-      link.setAttribute('href', decodeURIComponent(href));
-
-      // Add text after interact link saying where we're launching
-      hubUrlNoHttp = decodeURIComponent(hubUrl).replace('http://', '').replace('https://', '');
-      link.insertAdjacentHTML('afterend', '<div class="interact-context">on ' + hubUrlNoHttp + '</div>');
-
-      // Update internal links so we retain the hub url
-      addParamToInternalLinks(hubUrl);
-    }
-}
-
-runWhenDOMLoaded(updateInteractLink)
-document.addEventListener('turbolinks:load', updateInteractLink)
-</script>
-{% endif %}
\ No newline at end of file
diff --git a/_includes/js/nbinteract.html b/_includes/js/nbinteract.html
deleted file mode 100755
index 97772c6..0000000
--- a/_includes/js/nbinteract.html
+++ /dev/null
@@ -1,33 +0,0 @@
-{% if site.use_show_widgets_button and page.has_widgets %}
-<!-- Include nbinteract for interactive widgets -->
-<script src="https://unpkg.com/nbinteract-core" async></script>
-
-<script>
-let interact
-
-const initializeNbinteract = () => {
-  // If NbInteract hasn't loaded, wait one second and try again
-  if (window.NbInteract === undefined) {
-    setTimeout(initializeNbinteract, 1000)
-    return
-  }
-
-  if (interact === undefined) {
-    console.log('Initializing nbinteract...')
-    interact = new window.NbInteract({
-      baseUrl: 'https://mybinder.org',
-      spec: '{{ site.binder_repo_org }}/{{ site.binder_repo_name }}/{{ site.binder_repo_branch }}',
-      provider: 'gh',
-    })
-    window.interact = interact
-  } else {
-    console.log("nbinteract already initialized...")
-  }
-
-  interact.prepare()
-}
-
-// Initialize nbinteract
-initFunction(initializeNbinteract);
-</script>
-{% endif %}
\ No newline at end of file
diff --git a/_includes/js/print.html b/_includes/js/print.html
deleted file mode 100755
index 9cda3ac..0000000
--- a/_includes/js/print.html
+++ /dev/null
@@ -1,32 +0,0 @@
-<!-- Include nbinteract for interactive widgets -->
-<script src="https://printjs-4de6.kxcdn.com/print.min.js" async></script>
-<script>
-printContent = () => {
-    // MathJax displays a second version of any math for assistive devices etc.
-    // This prevents double-rendering in the PDF output.
-    var ignoreAssistList = [];
-    assistives = document.querySelectorAll('.MathJax_Display span.MJX_Assistive_MathML').forEach((element, index) => {
-        var thisId = 'MathJax-assistive-' + index.toString();
-        element.setAttribute('id', thisId);
-        ignoreAssistList.push(thisId)
-    });
-
-    // Print the actual content object
-    printJS({
-        printable: 'textbook_content',
-        type: 'html',
-        css: "{{ site.css_url | relative_url }}/styles.css",
-        style: "#textbook_content {padding-top: 40px};",
-        scanStyles: false,
-        targetStyles: ["*"],
-        ignoreElements: ignoreAssistList,
-        documentTitle: "{{ site.download_page_header }}"
-    })
-};
-
-initPrint = () => {
-    document.querySelector('#interact-button-print').addEventListener('click', printContent)
-}
-
-initFunction(initPrint)
-</script>
diff --git a/_includes/js/thebelab-cell-button.html b/_includes/js/thebelab-cell-button.html
deleted file mode 100755
index 3b9da83..0000000
--- a/_includes/js/thebelab-cell-button.html
+++ /dev/null
@@ -1,27 +0,0 @@
-{% if site.use_thebelab_button -%}
-<script>
-/**
- * Set up thebelab button for code blocks
- */
-
-const thebelabCellButton = id =>
-  `<a id="thebelab-cell-button-${id}" class="btn thebebtn o-tooltip--left" data-tooltip="Interactive Mode">
-    <img src="{{ site.images_url | relative_url }}/edit-button.svg" alt="Start thebelab interactive mode">
-  </a>`
-
-
-const addThebelabButtonToCodeCells =  () => {
-
-  const codeCells = document.querySelectorAll('div.input_area > div.highlight:not(.output) pre')
-  codeCells.forEach((codeCell, index) => {
-    const id = codeCellId(index)
-    codeCell.setAttribute('id', id)
-    if (document.getElementById("thebelab-cell-button-" + id) == null) {
-      codeCell.insertAdjacentHTML('afterend', thebelabCellButton(id));
-    }
-  })
-}
-
-initFunction(addThebelabButtonToCodeCells);
-</script>
-{% endif %}
diff --git a/_includes/js/thebelab-page-config.html b/_includes/js/thebelab-page-config.html
deleted file mode 100755
index 94478c7..0000000
--- a/_includes/js/thebelab-page-config.html
+++ /dev/null
@@ -1,32 +0,0 @@
-<script type="text/x-thebe-config">
-{%- if page.kernel_name %}
-    {% assign kernelName = page.kernel_name %}
-{% else %}
-    {% assign kernelName = "python3" %}
-{% endif -%}
-
-{%- if page.kernel_name %}
-    {% if page.kernel_name contains "python" %}
-        {% assign cm_language="python" %}
-    {% else %}
-        {% assign cm_language={{ page.kernel_name }} %}
-    {% endif %}
-{% else %}
-    {% assign cm_language="python" %}
-{% endif -%}
-{
-    requestKernel: true,
-    binderOptions: {
-    repo: "{{ site.binder_repo_org }}/{{ site.binder_repo_name }}",
-    ref: "{{ site.binder_repo_branch }}",
-    },
-    codeMirrorConfig: {
-    theme: "{{ site.codemirror_theme }}",
-    mode: "{{ cm_language }}"
-    },
-    kernelOptions: {
-    kernelName: "{{ kernelName }}",
-    path: "{{ page.kernel_path }}"
-    }
-}
-</script>
\ No newline at end of file
diff --git a/_includes/js/thebelab.html b/_includes/js/thebelab.html
deleted file mode 100755
index 97b970c..0000000
--- a/_includes/js/thebelab.html
+++ /dev/null
@@ -1,95 +0,0 @@
-<!-- Include Thebelab for interactive code if it's enabled -->
-{% if site.use_thebelab_button %}
-
-<!-- Display Thebelab button in each code cell -->
-{% include js/thebelab-cell-button.html %}
-<script src="https://unpkg.com/thebelab@latest/lib/index.js" async></script>
-<script>
-    /**
-     * Add attributes to Thebelab blocks
-     */
-
-    const initThebelab = () => {
-        const addThebelabToCodeCells = () => {
-            console.log("Adding thebelab to code cells...");
-            // If Thebelab hasn't loaded, wait a bit and try again. This
-            // happens because we load ClipboardJS asynchronously.
-            if (window.thebelab === undefined) {
-                setTimeout(addThebelabToCodeCells, 250)
-            return
-            }
-
-            // If we already detect a Thebelab cell, don't re-run
-            if (document.querySelectorAll('div.thebelab-cell').length > 0) {
-                return;
-            }
-
-            // Find all code cells, replace with Thebelab interactive code cells
-            const codeCells = document.querySelectorAll('.input_area pre')
-            codeCells.forEach((codeCell, index) => {
-                const id = codeCellId(index)
-
-                // Clean up the language to make it work w/ CodeMirror and add it to the cell
-                dataLanguage = "{{ kernelName }}"
-                dataLanguage = detectLanguage(dataLanguage);
-                codeCell.setAttribute('data-language', dataLanguage)
-                codeCell.setAttribute('data-executable', 'true')
-
-                // If the code cell is hidden, show it
-                var inputCheckbox = document.querySelector(`input#hidebtn${codeCell.id}`);
-                if (inputCheckbox !== null) {
-                    setCodeCellVisibility(inputCheckbox, 'visible');
-                }
-            });
-
-            // Remove the event listener from the page so keyboard press doesn't
-            // Change page
-            document.removeEventListener('keydown', initPageNav)
-            keyboardListener = false;
-
-            // Init thebelab
-            thebelab.bootstrap();
-
-            // Remove copy buttons since they won't work anymore
-            const copyAndThebeButtons = document.querySelectorAll('.copybtn, .thebebtn')
-            copyAndThebeButtons.forEach((button, index) => {
-                button.remove();
-            });
-
-            // Remove outputs since they'll be stale
-            const outputs = document.querySelectorAll('.output *, .output')
-            outputs.forEach((output, index) => {
-                output.remove();
-            });
-
-            // Find any cells with an initialization tag and ask ThebeLab to run them when ready
-            var thebeInitCells = document.querySelectorAll('div.tag_thebelab-init');
-            thebeInitCells.forEach((cell) => {
-                console.log("Initializing ThebeLab with cell: " + cell.id);
-                cell.querySelector('.thebelab-run-button').click();
-            });
-        }
-
-        // Add event listener for the function to modify code cells
-        const thebelabButtons = document.querySelectorAll('[id^=thebelab], [id$=thebelab]')
-        thebelabButtons.forEach((thebelabButton,index) => {
-            if (thebelabButton === null) {
-                setTimeout(initThebelab, 250)
-                return
-            };
-            thebelabButton.addEventListener('click', addThebelabToCodeCells);
-        });
-    }
-
-    // Initialize Thebelab
-    initFunction(initThebelab);
-
-// Helper function to munge the language name
-var detectLanguage = (language) => {
-    if (language.indexOf('python') > -1) {
-        language = "python";
-    }
-    return language;
-}
-</script>
-{% endif %}
diff --git a/_includes/mathjax.html b/_includes/mathjax.html
deleted file mode 100755
index f25affe..0000000
--- a/_includes/mathjax.html
+++ /dev/null
@@ -1,29 +0,0 @@
-<!-- Allow inline math using $ and automatically break long math lines -->
-<!-- (mostly) copied from nbconvert configuration -->
-<!-- https://github.com/jupyter/nbconvert/blob/master/nbconvert/templates/html/mathjax.tpl -->
-<script type="text/x-mathjax-config">
-MathJax.Hub.Config({
-    tex2jax: {
-        inlineMath: [ ['$','$'], ["\\(","\\)"] ],
-        displayMath: [ ['$$','$$'], ["\\[","\\]"] ],
-        processEscapes: true,
-        processEnvironments: true
-    },
-    // Center justify equations in code and markdown cells. Elsewhere
-    // we use CSS to left justify single line equations in code cells.
-    displayAlign: 'center',
-    "HTML-CSS": {
-        styles: {'.MathJax_Display': {"margin": 0}},
-        linebreaks: { automatic: true },
-    },
-    {% if site.number_equations %}
-    // Number LaTeX-style equations
-    "TeX": {
-        equationNumbers: {
-          autoNumber: "all"
-        }
-    }
-    {% endif %}
-});
-</script>
-<script src='https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS_HTML' async></script>
diff --git a/_includes/metadata.json b/_includes/metadata.json
deleted file mode 100755
index 089680f..0000000
--- a/_includes/metadata.json
+++ /dev/null
@@ -1,29 +0,0 @@
-{
-  "@context": "http://schema.org",
-  "@type": "NewsArticle",
-  "mainEntityOfPage": "{{ page.url | replace:'index.html','' | prepend: site.baseurl | prepend: site.url }}",
-  "headline": "{% if page.title %}{{ page.title | escape }}{% else %}{{ site.title | escape }}{% endif %}",
-  "datePublished": "{% if page.date %}{{ page.date | date_to_xmlschema }}{% else %}{{ site.time | date_to_xmlschema }}{% endif %}",
-  "dateModified": "{% if page.date %}{{ page.date | date_to_xmlschema }}{% else %}{{ site.time | date_to_xmlschema }}{% endif %}",
-  "description": "{{ page.content | strip_html | strip_newlines | truncate: 160 }}",
-  "author": {
-    "@type": "Person",
-    "name": "{{ site.author }}"
-  },
-  "publisher": {
-    "@type": "Organization",
-    "name": "Data 100 at UC Berkeley",
-    "logo": {
-      "@type": "ImageObject",
-      "url": "{{ site.logo | prepend: site.baseurl | prepend: site.url }}",
-      "width": 60,
-      "height": 60
-    }
-  },
-  "image": {
-    "@type": "ImageObject",
-    "url": "{{ site.logo | prepend: site.baseurl | prepend: site.url }}",
-    "height": 60,
-    "width": 60
-  }
-}
diff --git a/_includes/page-nav.html b/_includes/page-nav.html
deleted file mode 100755
index 58e860e..0000000
--- a/_includes/page-nav.html
+++ /dev/null
@@ -1,37 +0,0 @@
-{% comment %}
-Only the URLs from the TOC are used here. The title for
-prev/next is pulled from the respective page's metadata.
-We loop through the "build" collection to determine the
-page title based on the *current* page's next/prev URL.
-{% endcomment %}
-<nav class="c-page__nav">
-  {% if page.prev_page %}
-    {% for build_page in site.build %}
-      {% if build_page.url == page.prev_page.url %}
-        {% assign prev_title = build_page.title %}
-      {% endif %}
-    {%- endfor %}
-    {% if prev_title | size > 40 %}
-      {% assign prev_title = prev_title | truncate: 37 %}
-    {% endif %}
-
-    <a id="js-page__nav__prev" class="c-page__nav__prev" href="{{ page.prev_page.url | relative_url }}">
-      〈 <span class="u-margin-right-tiny"></span> {{ prev_title }}
-    </a>
-  {% endif %}
-
-  {% if page.next_page %}
-    {% for build_page in site.build %}
-      {% if build_page.url == page.next_page.url %}
-        {% assign next_title = build_page.title %}
-      {% endif %}
-    {%- endfor %}
-
-    {% if next_title | size > 40 %}
-      {% assign next_title = next_title | truncate: 37 %}
-    {% endif %}
-    <a id="js-page__nav__next" class="c-page__nav__next" href="{{ page.next_page.url | relative_url }}">
-      {{ next_title }} <span class="u-margin-right-tiny"></span> 〉
-    </a>
-  {% endif %}
-</nav>
diff --git a/_includes/search/lunr/lunr-en.js b/_includes/search/lunr/lunr-en.js
deleted file mode 100755
index b21ace9..0000000
--- a/_includes/search/lunr/lunr-en.js
+++ /dev/null
@@ -1,84 +0,0 @@
-var initQuery = function() {
-  // See if we have a search box
-  var searchInput = document.querySelector('input#lunr_search');
-  if (searchInput === null) {
-    return;
-  }
-
-  // Function to parse our lunr cache
-  var idx = lunr(function () {
-    this.field('title')
-    this.field('excerpt')
-    this.field('categories')
-    this.field('tags')
-    this.ref('id')
-
-    this.pipeline.remove(lunr.trimmer)
-
-    for (var item in store) {
-      this.add({
-        title: store[item].title,
-        excerpt: store[item].excerpt,
-        categories: store[item].categories,
-        tags: store[item].tags,
-        id: item
-      })
-    }
-  });
-
-  // Run search upon keyup
-  searchInput.addEventListener('keyup', function () {
-    var resultdiv = document.querySelector('#results');
-    var query = document.querySelector("input#lunr_search").value.toLowerCase();
-    var result =
-      idx.query(function (q) {
-        query.split(lunr.tokenizer.separator).forEach(function (term) {
-          q.term(term, { boost: 100 })
-          if(query.lastIndexOf(" ") != query.length-1){
-            q.term(term, {  usePipeline: false, wildcard: lunr.Query.wildcard.TRAILING, boost: 10 })
-          }
-          if (term != ""){
-            q.term(term, {  usePipeline: false, editDistance: 1, boost: 1 })
-          }
-        })
-      });
-
-      // Empty the results div
-      while (resultdiv.firstChild) {
-        resultdiv.removeChild(resultdiv.firstChild);
-      }
-
-    resultdiv.insertAdjacentHTML('afterbegin', '<p class="results__found">'+result.length+' Result(s) found</p>');
-    for (var item in result) {
-      var ref = result[item].ref;
-      if(store[ref].teaser){
-        var searchitem =
-          '<div class="list__item">'+
-            '<article class="archive__item" itemscope itemtype="https://schema.org/CreativeWork">'+
-              '<h2 class="archive__item-title" itemprop="headline">'+
-                '<a href="'+store[ref].url+'" rel="permalink">'+store[ref].title+'</a>'+
-              '</h2>'+
-              '<div class="archive__item-teaser">'+
-                '<img src="'+store[ref].teaser+'" alt="">'+
-              '</div>'+
-              '<p class="archive__item-excerpt" itemprop="description">'+store[ref].excerpt.split(" ").splice(0,20).join(" ")+'...</p>'+
-            '</article>'+
-          '</div>';
-      }
-      else{
-    	  var searchitem =
-          '<div class="list__item">'+
-            '<article class="archive__item" itemscope itemtype="https://schema.org/CreativeWork">'+
-              '<h2 class="archive__item-title" itemprop="headline">'+
-                '<a href="'+store[ref].url+'" rel="permalink">'+store[ref].title+'</a>'+
-              '</h2>'+
-              '<p class="archive__item-excerpt" itemprop="description">'+store[ref].excerpt.split(" ").splice(0,20).join(" ")+'...</p>'+
-            '</article>'+
-          '</div>';
-      }
-      resultdiv.insertAdjacentHTML('beforeend', searchitem);
-    }
-  });
-};
-
-initFunction(initQuery);
diff --git a/_includes/search/lunr/lunr-store.js b/_includes/search/lunr/lunr-store.js
deleted file mode 100755
index ffa41a0..0000000
--- a/_includes/search/lunr/lunr-store.js
+++ /dev/null
@@ -1,28 +0,0 @@
-var store = [
-  {%- for c in site.collections -%}
-    {%- if forloop.last -%}
-      {%- assign l = true -%}
-    {%- endif -%}
-    {%- assign docs = c.docs | where_exp:'doc','doc.search != false' -%}
-    {%- for doc in docs -%}
-      {%- if doc.header.teaser -%}
-        {%- capture teaser -%}{{ doc.header.teaser }}{%- endcapture -%}
-      {%- else -%}
-        {%- assign teaser = site.teaser -%}
-      {%- endif -%}
-      {
-        "title": {{ doc.title | jsonify }},
-        {% assign truncateWords=site.search_max_words_in_content %}
-        "excerpt": {{ doc.search | jsonify }},
-        "categories": {{ doc.categories | jsonify }},
-        "tags": {{ doc.tags | jsonify }},
-        "url": {{ doc.url | absolute_url | jsonify }},
-        "teaser":
-          {%- if teaser contains "://" -%}
-            {{ teaser | jsonify }}
-          {%- else -%}
-            {{ teaser | absolute_url | jsonify }}
-          {%- endif -%}
-      }{%- unless forloop.last and l -%},{%- endunless -%}
-    {%- endfor -%}
-  {%- endfor -%}]
\ No newline at end of file
diff --git a/_includes/sidebar.html b/_includes/sidebar.html
deleted file mode 100755
index 192aa69..0000000
--- a/_includes/sidebar.html
+++ /dev/null
@@ -1,157 +0,0 @@
-{% comment %}
-Partial for the textbook sidebar. Renders each chapter and its sections from
-_data/toc.yml .
-
-Much of the logic here is to add active classes to the currently active
-section. The currently active section / chapter should be highlighted in the
-sidebar.
-
-If a chapter or any of its sections are the current page, we should display the
-chapter's sections. Otherwise, we hide the sections to keep the sidebar short.
-
-We also prefix the sidebar entries with the chapter/section number. We assume
-a 1-level nesting; we will label 1.2, but not 1.2.1.
-{% endcomment %}
-
-{% assign chapter_num = 1 %}
-
-<nav id="js-sidebar" class="c-textbook__sidebar">
-  {% if site.textbook_logo %}<a href="{{ site.textbook_logo_link }}"><img src="{{ site.textbook_logo | relative_url }}" class="textbook_logo" id="sidebar-logo" alt="textbook logo" data-turbolinks-permanent/></a>{% endif %}
-  <h2 class="c-sidebar__title">{{ site.title }}</h2>
-  <ul class="c-sidebar__chapters">
-    {% for chapter in site.data.toc %}
-      {% comment %}
-      If the entry is a divider, render a divider and move to next entry.
-      {% endcomment %}
-      {% if chapter.divider %}
-        <li class="c-sidebar__divider"></li>
-        {% continue %}
-      {% elsif chapter.header %}
-        <li><h2 class="c-sidebar__title">{{ chapter.header }}</li>
-        {% continue %}
-      {% endif %}
-
-      {% comment %}
-      Find the title for this chapter
-      {% endcomment %}
-      {% assign chapter_url_html = chapter.url | append: '.html' %}
-      {% for build_page in site.build %}
-        {% if build_page.url == chapter_url_html %}
-          {% assign chapter_title = build_page.title %}
-        {% endif %}
-      {%- endfor %}
-      {% if chapter.external == true %}{% assign chapter_title = chapter.title  %}{% endif %}
-
-      {% assign topUrl = chapter.url | relative_url %}
-      {% unless chapter.external == true %}
-        {% assign topUrl = topUrl | append: '.html'  %}
-      {% endunless %}
-      <li class="c-sidebar__chapter" data-url="{{ chapter.url }}">
-        <a class="c-sidebar__entry"
-          href="{{ topUrl }}"
-        >
-          {% unless chapter.not_numbered or site.number_toc_chapters != true %}
-            {{ chapter_num }}.
-          {% endunless %}
-          {{ chapter_title }}
-        </a>
-      </li>
-
-      {% comment %}
-      Flags for whether we hide all inactive sections or sub-sections
-      {% endcomment %}
-      {% if site.collapse_inactive_chapters == true %}
-        {% assign sectionsHideClass = " u-hidden-visually" %}
-      {% else %}
-        {% assign sectionsHideClass = "" %}
-      {% endif %}
-
-      {% if site.collapse_inactive_sections == true %}
-        {% assign subSectionsHideClass = " u-hidden-visually" %}
-      {% else %}
-        {% assign subSectionsHideClass = "" %}
-      {% endif %}
-
-      {% comment %}Per-chapter overrides from TOC data{% endcomment %}
-      {% if chapter.expand_sections == true %}
-        {% assign sectionsHideClass = "" %}
-      {% endif %}
-
-      {% if chapter.sections %}
-        {% comment %}
-        By default, all sections are hidden. We show a chapter's sections
-        if the chapter or any of its sections are the current page.
-        {% endcomment %}
-
-        {% assign section_num = 1 %}
-
-        <ul class="c-sidebar__sections{{ sectionsHideClass }}">
-          {% for section in chapter.sections %}
-            {% comment %}Per-section overrides from TOC data{% endcomment %}
-            {% if section.expand_subsections == true %}
-              {% assign subSectionsHideClass = "" %}
-            {% endif %}
-
-            {% comment %}
-            Find the title for this chapter
-            {% endcomment %}
-            {% assign section_url_html = section.url | append: '.html' %}
-            {% for build_page in site.build %}
-              {% if build_page.url == section_url_html %}
-                {% assign section_title = build_page.title %}
-              {% endif %}
-            {%- endfor %}
-            {% if section.external == true %}{% assign section_title = section.title  %}{% endif %}
-
-            <li class="c-sidebar__section" data-url="{{ section.url }}">
-              <a class="c-sidebar__entry"
-                href="{{ section.url  | relative_url | append: '.html'}}"
-              >
-                {% unless chapter.not_numbered or section.not_numbered or site.number_toc_chapters != true %}
-                  {{ chapter_num }}.{{ section_num }}
-                {% endunless %}
-                {{ section_title }}
-              </a>
-            </li>
-            {% if section.subsections %}
-              {% assign subsection_num = 1 %}
-              <ul class='c-sidebar__subsections{{ subSectionsHideClass }}'>
-              {% for subsection in section.subsections %}
-                {% comment %}
-                Find the title for this sub-section
-                {% endcomment %}
-                {% assign subsection_url_html = subsection.url | append: '.html' %}
-                {% for build_page in site.build %}
-                  {% if build_page.url == subsection_url_html %}
-                    {% assign subsection_title = build_page.title %}
-                  {% endif %}
-                {%- endfor %}
-                {% if subsection.external == true %}{% assign subsection_title = subsection.title  %}{% endif %}
-                <li class="c-sidebar__subsection" data-url="{{ subsection.url }}">
-                  <a class="c-sidebar__entry"
-                    href="{{ subsection.url | relative_url | append: '.html'}}"
-                  >
-                    {% unless chapter.not_numbered or section.not_numbered or subsection.not_numbered or site.number_toc_chapters != true %}
-                      {{ chapter_num }}.{{ section_num }}.{{ subsection_num }}
-                      {% assign subsection_num = subsection_num | plus: 1 %}
-                    {% endunless %}
-                    {{ subsection_title }}
-                  </a>
-                </li>
-              {% endfor %}
-              </ul>
-            {% endif %}
-            {% unless chapter.not_numbered or section.not_numbered %}
-              {% assign section_num = section_num | plus: 1 %}
-            {% endunless %}
-          {% endfor %}
-        </ul>
-      {% endif %}
-
-      {% unless chapter.not_numbered %}
-        {% assign chapter_num = chapter_num | plus: 1 %}
-      {% endunless %}
-    {% endfor %}
-  </ul>
-  <p class="sidebar_footer">{{ site.sidebar_footer_text }}</p>
-</nav>
diff --git a/_includes/topbar.html b/_includes/topbar.html
deleted file mode 100755
index 3a26729..0000000
--- a/_includes/topbar.html
+++ /dev/null
@@ -1,23 +0,0 @@
-<div class="c-topbar" id="top-navbar">
-  <!-- We show the sidebar by default so we use .is-active -->
-  <div class="c-topbar__buttons">
-    <button
-      id="js-sidebar-toggle"
-      class="hamburger hamburger--arrowalt is-active"
-    >
-      <span class="hamburger-box">
-        <span class="hamburger-inner"></span>
-      </span>
-    </button>
-    {% include buttons.html %}
-  </div>
-  <!-- Empty sidebar placeholder that we'll auto-fill with javascript -->
-  <aside class="sidebar__right">
-    <header><h4 class="nav__title"><img src="{{ site.images_url | relative_url }}/list-solid.svg" alt="Search" /> {{ page.toc_label | default: site.data.ui-text[site.locale].toc_label }}</h4></header>
-    <nav class="onthispage">
-    </nav>
-  </aside>
-  <a href="{{ '/search.html' | relative_url }}" class="topbar-right-button" id="search-button">
-    <img src="{{ site.images_url | relative_url }}/search-solid.svg" alt="Search" />
-  </a>
-</div>

From 74164159684ed87394e2343e6661f191abe28213 Mon Sep 17 00:00:00 2001
From: Diego Restrepo <restrepo@udea.edu.co>
Date: Mon, 6 Feb 2023 19:55:52 -0500
Subject: [PATCH 2/9] clean

---
 _layouts/default.html                         |  27 --
 .../components/_components.book__layout.scss  | 151 ------
 .../components/_components.book__topbar.scss  |  67 ---
 .../_components.interact-button.scss          |  60 ---
 .../components/_components.page__footer.scss  |   7 -
 _sass/components/_components.page__nav.scss   |  19 -
 .../_components.page__onthispage.scss         | 100 ----
 _sass/components/_components.search.scss      |  97 ----
 _sass/components/_components.thebelab.scss    |  33 --
 _sass/hamburgers/_base.scss                   |  69 ---
 _sass/hamburgers/hamburgers.scss              | 118 -----
 _sass/hamburgers/types/_arrowalt.scss         |  36 --
 _sass/main.scss                               |  29 --
 .../_objects.thebelab-in-cell-button.scss     |  25 -
 .../components/_components.hidecells.scss     |  75 ---
 _sass/page/components/_components.page.scss   | 307 ------------
 .../components/_components.sidebar-right.scss | 123 -----
 _sass/page/elements/_elements.links.scss      |  29 --
 .../_elements.syntax-highlighting.scss        |  83 ----
 _sass/page/elements/_elements.tables.scss     |  28 --
 _sass/page/elements/_elements.typography.scss |  63 ---
 _sass/page/elements/_elements.variables.scss  |  26 -
 .../generic/_generic.phone-scrolling.scss     |   7 -
 .../inuitcss/elements/_elements.headings.scss |  45 --
 .../inuitcss/elements/_elements.images.scss   |  39 --
 .../inuitcss/elements/_elements.page.scss     |  19 -
 .../inuitcss/elements/_elements.tables.scss   |  11 -
 .../inuitcss/generic/_generic.box-sizing.scss |  22 -
 .../inuitcss/generic/_generic.normalize.scss  | 447 ------------------
 .../page/inuitcss/generic/_generic.reset.scss |  56 ---
 .../inuitcss/generic/_generic.shared.scss     |  33 --
 .../page/inuitcss/objects/_objects.block.scss |  61 ---
 _sass/page/inuitcss/objects/_objects.box.scss |  48 --
 .../page/inuitcss/objects/_objects.crop.scss  | 165 -------
 .../page/inuitcss/objects/_objects.flag.scss  | 231 ---------
 .../inuitcss/objects/_objects.layout.scss     | 308 ------------
 .../inuitcss/objects/_objects.list-bare.scss  |  20 -
 .../objects/_objects.list-inline.scss         |  16 -
 .../page/inuitcss/objects/_objects.media.scss | 144 ------
 .../page/inuitcss/objects/_objects.pack.scss  |  96 ----
 .../page/inuitcss/objects/_objects.ratio.scss | 117 -----
 .../page/inuitcss/objects/_objects.table.scss |  76 ---
 .../inuitcss/objects/_objects.wrapper.scss    |  13 -
 .../settings/_example.settings.config.scss    |  24 -
 .../settings/_example.settings.global.scss    |  13 -
 .../inuitcss/settings/_settings.core.scss     |  92 ----
 .../page/inuitcss/tools/_tools.clearfix.scss  |  20 -
 .../page/inuitcss/tools/_tools.font-size.scss |  72 ---
 _sass/page/inuitcss/tools/_tools.hidden.scss  |  18 -
 _sass/page/inuitcss/tools/_tools.mq.scss      | 351 --------------
 .../utilities/_utilities.clearfix.scss        |  11 -
 .../utilities/_utilities.headings.scss        |  36 --
 .../inuitcss/utilities/_utilities.hide.scss   |  21 -
 .../inuitcss/utilities/_utilities.print.scss  |  88 ----
 .../_utilities.responsive-spacings.scss       | 124 -----
 .../utilities/_utilities.spacings.scss        |  60 ---
 .../inuitcss/utilities/_utilities.widths.scss | 184 -------
 _sass/page/main.scss                          | 102 ----
 _sass/page/objects/_objects.copy-button.scss  |  20 -
 _sass/page/objects/_objects.tooltip.scss      |  54 ---
 _sass/page/settings/settings.global.scss      |  25 -
 assets/css/styles.scss                        |   4 -
 assets/custom/custom.css                      |   5 -
 assets/custom/custom.js                       |   1 -
 assets/html/index.html                        |  27 --
 assets/html/search_form.html                  |  15 -
 assets/images/download-solid.svg              |  13 -
 assets/images/edit-button.svg                 |  13 -
 assets/images/list-solid.svg                  |  14 -
 assets/images/logo_binder.svg                 |  19 -
 assets/images/logo_jupyterhub.svg             |   1 -
 assets/images/search-solid.svg                |  12 -
 assets/js/page/anchors.js                     |  12 -
 assets/js/page/copy-button.js                 |  61 ---
 assets/js/page/documentSelectors.js           |  35 --
 assets/js/page/dom-update.js                  |  17 -
 assets/js/page/hide-cell.js                   |  86 ----
 assets/js/page/tocbot.js                      |  38 --
 assets/js/scripts.js                          | 257 ----------
 content/LICENSE.md                            |   5 -
 content/features/Intro_clases.ipynb           |   1 -
 content/features/LagrangePoly.ipynb           |   1 -
 content/features/LagrangePolynomial.py        |   1 -
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diff --git a/_layouts/default.html b/_layouts/default.html
deleted file mode 100755
index 5a49bb4..0000000
--- a/_layouts/default.html
+++ /dev/null
@@ -1,27 +0,0 @@
-<!DOCTYPE html>
-<html lang="en">
-  {% include head.html %}
-  <body>
-    <!-- Include the ThebeLab config so it gets reloaded on each page -->
-    {% include js/thebelab-page-config.html %}
-
-    <!-- .js-show-sidebar shows sidebar by default -->
-    <div id="js-textbook" class="c-textbook {% if site.show_sidebar %}js-show-sidebar{% endif %}">
-      {% include sidebar.html %}
-      {% if page.search_page != true %}
-      {% endif %}
-      {% include topbar.html %}
-      <main class="c-textbook__page" tabindex="-1">
-            <div class="c-textbook__content" id="textbook_content">
-              {{ content }}
-            </div>
-            <div class="c-textbook__footer" id="textbook_footer">
-              {% include page-nav.html %}
-              {% include footer.html %}
-            </div>
-
-        </div>
-      </main>
-    </div>
-  </body>
-</html>
diff --git a/_sass/components/_components.book__layout.scss b/_sass/components/_components.book__layout.scss
deleted file mode 100755
index 7caa4a4..0000000
--- a/_sass/components/_components.book__layout.scss
+++ /dev/null
@@ -1,151 +0,0 @@
-/**
- * The website contains two main components: the sidebar and the textbook page.
- * This file specifies the layout and includes classes to show/hide the sidebar
- * on small screens.
- *
- * The actual styling for the sidebar and page are located in their respective
- * component SCSS files. This file manages the layout and width only.
- *
- * By default, the sidebar is not visible.
- *
- * [1]: The entire page is positioned relative so that when the page overflows
- *   (e.g. sidebar open on small screens) the user can't scroll left/right.
- * [2]: The sidebar and the textbook page are positioned absolute so that we
- *   can use translate() on the textbook page to reveal the sidebar.
- * [3]: Setting the background color hides the sidebar when it's behind the
- *   page (otherwise the page is transparent).
- *
- * When the sidebar is visible:
- *
- * [4]: Shift the textbook page over to the left. On small screens, the page
- *   will overflow since the sidebar takes up most of the screen.
- * [5]: On larger screens, the page and sidebar have enough room to read them
- *   simultaneously, so make sure that the page doesn't overflow.
- */
-
-$left-sidebar-width: 300px;
-$page-max-width: 880px;
-$right-sidebar-width: 220px;
-$topbar-height: 60px;
-
-.c-textbook {
-  /* [1] */
-  position: relative;
-  height: 100vh;
-  overflow: hidden;
-  margin: 0 0 0 auto;
-}
-
-.c-topbar {
-  background-color: $book-background-color;
-  position: fixed;
-  top: 0;
-  height: $topbar-height;
-  width: 100%;
-  left: 0;
-  padding: $spacing-unit-small $spacing-unit-small 0 $spacing-unit-med * 2;
-  z-index: 1;
-  transition: top 250ms, transform 250ms ease; // For animations
-}
-
-@include mq($until: tablet) {
-  .c-topbar.hidetop {
-    // At desktop, we stop hiding the navbar
-    top: -250px;
-  }
-}
-
-.c-textbook__sidebar,
-.c-textbook__page {
-  /* [2] */
-  height: 100vh;
-  overflow: auto;
-  position: fixed;
-  background-color: $book-background-color; /* [3] */
-}
-
-.c-textbook__sidebar {
-  width: $left-sidebar-width;
-  top: 0;
-  left: 0;
-}
-
-.c-textbook__page {
-
-  width: $textbook-page-width;
-  transition: transform 250ms ease;
-  left: 0;
-  padding: 0 $spacing-unit $spacing-unit-small $spacing-unit-small * 3;
-  overflow-x: visible;
-
-  &:focus {
-    /* [2] */
-    outline: none;
-  }
-}
-
-.sidebar__right {
-  // By default we hide the sidebar
-  display: none;
-
-  // Spacing for the sidebar
-  width: $right-sidebar-width - $spacing-unit-small; // To account for the small margin on the right
-  position: relative;
-  float: right;
-  z-index: 1; // Keep sidebar under page content
-
-  @include mq($from: tablet) {
-    // Show right TOC at laptop size
-    display: block;
-  }
-}
-
-.js-show-sidebar {
-  .c-textbook__page, .c-topbar {
-    /* [4] */
-    transform: translate($left-sidebar-width, 0);
-
-    @include mq($from: tablet) {
-      /* [5] */
-      width: calc(100% - #{$left-sidebar-width});
-    }
-  }
-}
-
-.c-textbook__content {
-  clear: both;
-  padding-top: $topbar-height * 1.5;
-  width: 95%;
-}
-
-.c-textbook__content, .c-textbook__footer {
-  max-width: $page-max-width;
-}
-
-.c-page__nav {
-  display: flex;
-  justify-content: space-between;
-  align-items: center;
-  padding-top: 30px;
-}
-
-// Make sure that the bottom content has the same width as non-sidebar content
-.footer, .c-page__nav {
-  @include mq($from: laptop) {
-    width: $textbook-page-with-sidebar-width;
-  }
-}
-
-// Scrollbar width
-::-webkit-scrollbar {
-  width: 5px;
-  background: #f1f1f1;
-}
-
-::-webkit-scrollbar-thumb {
-  background: #c1c1c1;
-}
-
-main, nav {
-  scrollbar-width: thin;
-}
\ No newline at end of file
diff --git a/_sass/components/_components.book__topbar.scss b/_sass/components/_components.book__topbar.scss
deleted file mode 100755
index d9e0ca5..0000000
--- a/_sass/components/_components.book__topbar.scss
+++ /dev/null
@@ -1,67 +0,0 @@
-.c-topbar__label {
-  @include inuit-font-size(12px);
-  display: inline-block;
-  margin-left: $spacing-unit-tiny;
-  vertical-align: middle;
-  text-transform: uppercase;
-}
-
-.c-topbar {
-  .hamburger, .buttons {
-    float: left;
-  }
-
-  #js-sidebar-toggle {
-    margin-right: 5px;
-    padding-top: 4px;
-  }
-
-  span.hamburger-box {
-    width: 40px;
-    height: 30px;
-    padding-left: 10px;
-  }
-
-  .c-topbar__buttons {
-    @include mq($from: tablet) {
-      width: calc(100% - #{$right-sidebar-width} - 20px)
-    }
-  }
-
-  .topbar-right-button {
-    display: block;
-    float: right;
-    padding: 0 1rem;
-
-    img {
-      width: 20px;
-      margin-top: 4px;
-    }
-  }
-}
-
-// Download buttons
-
-.download-buttons {
-  display: none;
-  position: absolute;
-
-  button {
-    min-width: 100px !important;
-    border: 1px white solid !important;
-    border-radius: 0 !important;
-  }
-}
-
-.download-buttons-dropdown {
-  position: relative;
-  display: inline-block;
-
-  &:hover div.download-buttons {
-    display: block;
-  }
-
-  img {
-    height: 18px;
-  }
-}
diff --git a/_sass/components/_components.interact-button.scss b/_sass/components/_components.interact-button.scss
deleted file mode 100755
index c552fa5..0000000
--- a/_sass/components/_components.interact-button.scss
+++ /dev/null
@@ -1,60 +0,0 @@
-/**
- * Stylings for Interact and Show Widget buttons.
- *
- * [1]: We abuse CSS selector specificity here since the buttons at the top of
- *   the notebook might have both .interact-button and .js=nbinteract-widget.
- * [2]: We want the top buttons to be large.
- * [3]: However, a .js=nbinteract-widget appearing alone midway through the
- *   notebook should be small.
- *
- */
-
-$color-interact-button: #5a5a5a !default;
-
-%interact-button {
-  @include inuit-font-size(14px);
-  background-color: $color-interact-button;
-  border-radius: 3px;
-  border: none;
-  color: white;
-  cursor: pointer;
-  display: inline-block;
-  font-weight: 700;
-  /* [2] */
-  padding: $spacing-unit-tiny $spacing-unit-med;
-  text-decoration: none;
-
-  &:hover,
-  &:focus {
-    text-decoration: none;
-  }
-}
-
-.interact-button-logo {
-  height: 1.35em;
-  padding-right: 10px;
-  margin-left: -5px;
-}
-
-.buttons {
-  margin-bottom: $spacing-unit;
-
-  /* [1] */
-  .interact-button {
-    @extend %interact-button;
-  }
-}
-
-.js-nbinteract-widget {
-  @extend %interact-button;
-
-  /* [3] */
-  padding: $spacing-unit-tiny $spacing-unit;
-  margin-bottom: $spacing-unit-small;
-}
-
-// If the interact button link is changed with a REST param
-div.interact-context {
-  display: inline;
-  padding-left: 1em;
-}
\ No newline at end of file
diff --git a/_sass/components/_components.page__footer.scss b/_sass/components/_components.page__footer.scss
deleted file mode 100755
index 9a4d5c4..0000000
--- a/_sass/components/_components.page__footer.scss
+++ /dev/null
@@ -1,7 +0,0 @@
-.footer {
-    text-align: center;
-    font-size: 14px;
-    padding: 20px;
-    opacity: 0.7;
-    margin-bottom: 0px;
-}
diff --git a/_sass/components/_components.page__nav.scss b/_sass/components/_components.page__nav.scss
deleted file mode 100755
index 69db85e..0000000
--- a/_sass/components/_components.page__nav.scss
+++ /dev/null
@@ -1,19 +0,0 @@
-/**
- * Styling for the Next Page / Previous Page links at the bottom of textbook
- * pages.
- */
-
-$color-nav-links: rgba(0, 140, 255, 0.7);
-
-.c-page__nav__prev,
-.c-page__nav__next {
-  flex: 1;
-  color: $color-nav-links;
-  border: 1px solid $color-nav-links;
-  border-radius: 3px;
-  padding: $spacing-unit-small 0;
-}
-
-.c-page__nav__next {
-  text-align: right;
-}
diff --git a/_sass/components/_components.page__onthispage.scss b/_sass/components/_components.page__onthispage.scss
deleted file mode 100755
index 548ee2d..0000000
--- a/_sass/components/_components.page__onthispage.scss
+++ /dev/null
@@ -1,100 +0,0 @@
-/**
- * Styling for the onthispage elements.
- *
- * [1]: The sidebar is implemented as ul and li elements so we need to remove
- *   the bullets and margins. Also make chapter fonts a bit bigger.
- * [2]: The entries are <a> tags so we need to remove the default styling.
- * [3]: The sidebar divider is just an empty element with a border.
- * [4]: The current section needs a higher specificity to override the :hover
- *   selectors used previously.
- * [5]: The logo displayed above the sidebar
- * [6]: The footer at the bottom of the sidebar
- */
-$color-sidebar-bg: rgba(255, 255, 255, 0) !default;
-$color-sidebar-entry: #364149 !default;
-$color-sidebar-entry--active: $color-links !default;
-$color-sidebar-divider: #bbb !default;
-
-.c-textbook__sidebar {
-  background-color: $color-sidebar-bg;
-  padding: $spacing-unit-small;
-
-  @include inuit-font-size(14px);
-  border-right: 1px solid rgba(0, 0, 0, 0.07);
-  opacity: 0.6;
-  -webkit-transition: opacity 0.2s ease-in-out;
-  transition: opacity 0.2s ease-in-out;
-
-  &:hover {
-    opacity: 1;
-  }
-}
-
-/* [1] */
-.c-sidebar__chapters {
-  list-style: none;
-  margin-left: 0;
-  margin-bottom: 0;
-}
-
-li.c-sidebar__chapter > a {
-  font-size: 1.2em;
-}
-
-/* [1] */
-.c-sidebar__sections, .c-sidebar__subsections {
-  list-style: none;
-  margin-bottom: 0;
-}
-
-.c-sidebar__sections {
-  margin-left: $spacing-unit-small;
-}
-
-.c-sidebar__subsections {
-  margin-left: 20px;
-}
-
-/* [2] */
-.c-sidebar__entry {
-  display: block;
-
-  padding: $spacing-unit-tiny;
-
-  color: $color-sidebar-entry;
-  text-decoration: none;
-
-  &:hover {
-    text-decoration: underline;
-  }
-
-  &:visited {
-    color: $color-sidebar-entry;
-  }
-}
-
-/* [4] */
-.c-sidebar__entry--active.c-sidebar__entry--active {
-  color: $color-sidebar-entry--active;
-}
-
-/* [3] */
-.c-sidebar__divider {
-  border-top: 1px solid $color-sidebar-divider;
-  margin: $spacing-unit-tiny;
-}
-
-/* [5] */
-img.textbook_logo {
-  margin-top: 20px;
-  max-height: 100px;
-  margin: 0px auto 20px auto;
-  display: block;
-}
-
-/* [6] */
-p.sidebar_footer {
-  text-align: center;
-  padding: 10px 20px 0px 0px;
-  font-size: .9em;
-}
\ No newline at end of file
diff --git a/_sass/components/_components.search.scss b/_sass/components/_components.search.scss
deleted file mode 100755
index 0f3ae89..0000000
--- a/_sass/components/_components.search.scss
+++ /dev/null
@@ -1,97 +0,0 @@
-/* ==========================================================================
-   SEARCH
-   ========================================================================== */
-  // Taken from https://github.com/mmistakes/minimal-mistakes
-  //  Variables
-  $large: 1024px !default;
-  $x-large: 1280px !default;
-  $type-size-1: 2.441em !default;
-  $type-size-2: 1.953em !default;
-  $type-size-3: 1.563em !default;
-  $type-size-6: 0.75em !default;
-  $intro-transition: intro 0.3s both !default;
-
-  // Rules
-  .layout--search {
-    .archive__item-teaser {
-      margin-bottom: 0.25em;
-    }
-  }
-
-  .search__toggle {
-    margin-left: 1rem;
-    margin-right: 1rem;
-    border: 0;
-    outline: none;
-    color: #393e46;
-    background-color: transparent;
-    cursor: pointer;
-    -webkit-transition: 0.2s;
-    transition: 0.2s;
-
-    &:hover {
-      color: #000;
-    }
-  }
-
-  .search-icon {
-    width: 100%;
-    height: 100%;
-  }
-
-  .search-content {
-    //display: none;
-    //visibility: hidden;
-    padding-top: 1em;
-    padding-bottom: 1em;
-
-    &__inner-wrap {
-      width: 100%;
-      margin-left: auto;
-      margin-right: auto;
-      padding-left: 1em;
-      padding-right: 1em;
-      -webkit-animation: $intro-transition;
-      animation: $intro-transition;
-      -webkit-animation-delay: 0.15s;
-      animation-delay: 0.15s;
-
-      .search-input {
-        display: block;
-        margin-bottom: 0;
-        padding: 0;
-        border: none;
-        outline: none;
-        box-shadow: none;
-        background-color: transparent;
-        font-size: $type-size-3;
-        }
-    }
-
-    &.is--visible {
-      display: block;
-      visibility: visible;
-
-      &::after {
-        content: "";
-        display: block;
-      }
-    }
-
-    .results__found {
-      margin-top: 0.5em;
-      font-size: $type-size-6;
-    }
-
-    .archive__item {
-      margin-bottom: 2em;
-    }
-
-    .archive__item-title {
-      margin-top: 0;
-    }
-
-    .archive__item-excerpt {
-      margin-bottom: 0;
-    }
-  }
\ No newline at end of file
diff --git a/_sass/components/_components.thebelab.scss b/_sass/components/_components.thebelab.scss
deleted file mode 100755
index 2bc6859..0000000
--- a/_sass/components/_components.thebelab.scss
+++ /dev/null
@@ -1,33 +0,0 @@
-
-.thebelab-cell {
-    // To ensure that thebelab cells are always the top of the Z-stack
-    position: relative;
-    z-index: 999;
-}
-
-.thebelab-button {
-    z-index: 999;
-    display: inline-block;
-    padding: 0.35em 1.2em;
-    margin: 0px 1px;
-    border-radius: 0.12em;
-    box-sizing:  border-box;
-    text-decoration: none;
-    font-family: 'Roboto', sans-serif;
-    font-weight: 300;
-    text-align: center;
-    transition: all 0.2s;
-    background-color: #dddddd;
-    border: 0.05em solid white;
-    color: #000000;
-}
-
-.thebelab-button:hover{
-    border: 0.05em solid black;
-    background-color: #fcfcfc;
-}
-
-
-div.jp-OutputArea-output {
-    padding: 5px;
-}
diff --git a/_sass/hamburgers/_base.scss b/_sass/hamburgers/_base.scss
deleted file mode 100755
index 15f4b2d..0000000
--- a/_sass/hamburgers/_base.scss
+++ /dev/null
@@ -1,69 +0,0 @@
-// Hamburger
-// ==================================================
-.hamburger {
-  padding: $hamburger-padding-y $hamburger-padding-x;
-  display: inline-block;
-  cursor: pointer;
-
-  transition-property: opacity, filter;
-  transition-duration: $hamburger-hover-transition-duration;
-  transition-timing-function: $hamburger-hover-transition-timing-function;
-
-  // Normalize (<button>)
-  font: inherit;
-  color: inherit;
-  text-transform: none;
-  background-color: transparent;
-  border: 0;
-  margin: 0;
-  overflow: visible;
-
-  &:hover {
-    @if $hamburger-hover-use-filter == true {
-      filter: $hamburger-hover-filter;
-    } @else {
-      opacity: $hamburger-hover-opacity;
-    }
-  }
-}
-
-.hamburger-box {
-  width: $hamburger-layer-width;
-  height: $hamburger-layer-height * 3 + $hamburger-layer-spacing * 2;
-  display: inline-block;
-  position: relative;
-  vertical-align: middle;
-}
-
-.hamburger-inner {
-  display: block;
-  top: 50%;
-  margin-top: $hamburger-layer-height / -2;
-
-  &,
-  &::before,
-  &::after {
-    width: $hamburger-layer-width;
-    height: $hamburger-layer-height;
-    background-color: $hamburger-layer-color;
-    border-radius: $hamburger-layer-border-radius;
-    position: absolute;
-    transition-property: transform;
-    transition-duration: 0.15s;
-    transition-timing-function: ease;
-  }
-
-  &::before,
-  &::after {
-    content: '';
-    display: block;
-  }
-
-  &::before {
-    top: ($hamburger-layer-spacing + $hamburger-layer-height) * -1;
-  }
-
-  &::after {
-    bottom: ($hamburger-layer-spacing + $hamburger-layer-height) * -1;
-  }
-}
diff --git a/_sass/hamburgers/hamburgers.scss b/_sass/hamburgers/hamburgers.scss
deleted file mode 100755
index 240963c..0000000
--- a/_sass/hamburgers/hamburgers.scss
+++ /dev/null
@@ -1,118 +0,0 @@
-@charset "UTF-8";
-/*!
- * Hamburgers
- * @description Tasty CSS-animated hamburgers
- * @author Jonathan Suh @jonsuh
- * @site https://jonsuh.com/hamburgers
- * @link https://github.com/jonsuh/hamburgers
- */
-
-// Settings
-// ==================================================
-$hamburger-padding-x: 15px !default;
-$hamburger-padding-y: 15px !default;
-$hamburger-layer-width: 40px !default;
-$hamburger-layer-height: 4px !default;
-$hamburger-layer-spacing: 6px !default;
-$hamburger-layer-color: #000 !default;
-$hamburger-layer-border-radius: 4px !default;
-$hamburger-hover-opacity: 0.7 !default;
-$hamburger-hover-transition-duration: 0.15s !default;
-$hamburger-hover-transition-timing-function: linear !default;
-
-// To use CSS filters as the hover effect instead of opacity,
-// set $hamburger-hover-use-filter as true and
-// change the value of $hamburger-hover-filter accordingly.
-$hamburger-hover-use-filter: false !default;
-$hamburger-hover-filter: opacity(50%) !default;
-
-// Types (Remove or comment out what you don’t need)
-// ==================================================
-$hamburger-types: (
-    3dx,
-    3dx-r,
-    3dy,
-    3dy-r,
-    3dxy,
-    3dxy-r,
-    arrow,
-    arrow-r,
-    arrowalt,
-    arrowalt-r,
-    arrowturn,
-    arrowturn-r,
-    boring,
-    collapse,
-    collapse-r,
-    elastic,
-    elastic-r,
-    emphatic,
-    emphatic-r,
-    minus,
-    slider,
-    slider-r,
-    spin,
-    spin-r,
-    spring,
-    spring-r,
-    stand,
-    stand-r,
-    squeeze,
-    vortex,
-    vortex-r
-  )
-  !default;
-
-// Base Hamburger (We need this)
-// ==================================================
-@import 'base';
-
-// Hamburger types
-// ==================================================
-// @import "types/3dx";
-// @import "types/3dx-r";
-// @import "types/3dy";
-// @import "types/3dy-r";
-// @import "types/3dxy";
-// @import "types/3dxy-r";
-// @import "types/arrow";
-// @import "types/arrow-r";
-@import 'types/arrowalt';
-// @import "types/arrowalt-r";
-// @import "types/arrowturn";
-// @import "types/arrowturn-r";
-// @import "types/boring";
-// @import "types/collapse";
-// @import "types/collapse-r";
-// @import "types/elastic";
-// @import "types/elastic-r";
-// @import "types/emphatic";
-// @import "types/emphatic-r";
-// @import "types/minus";
-// @import "types/slider";
-// @import "types/slider-r";
-// @import "types/spin";
-// @import "types/spin-r";
-// @import "types/spring";
-// @import "types/spring-r";
-// @import "types/stand";
-// @import "types/stand-r";
-// @import "types/squeeze";
-// @import "types/vortex";
-// @import "types/vortex-r";
-
-// ==================================================
-// Cooking up additional types:
-//
-// The Sass for each hamburger type should be nested
-// inside an @if directive to check whether or not
-// it exists in $hamburger-types so only the CSS for
-// included types are generated.
-//
-// e.g. hamburgers/types/_new-type.scss
-//
-// @if index($hamburger-types, new-type) {
-//   .hamburger--new-type {
-//     ...
-//   }
-// }
diff --git a/_sass/hamburgers/types/_arrowalt.scss b/_sass/hamburgers/types/_arrowalt.scss
deleted file mode 100755
index 6c5f2e2..0000000
--- a/_sass/hamburgers/types/_arrowalt.scss
+++ /dev/null
@@ -1,36 +0,0 @@
-@if index($hamburger-types, arrowalt) {
-  /*
-   * Arrow Alt
-   */
-  .hamburger--arrowalt {
-    .hamburger-inner {
-      &::before {
-        transition: top 0.1s 0.1s ease,
-                    transform 0.1s cubic-bezier(0.165, 0.84, 0.44, 1);
-      }
-
-      &::after {
-        transition: bottom 0.1s 0.1s ease,
-                    transform 0.1s cubic-bezier(0.165, 0.84, 0.44, 1);
-      }
-    }
-
-    &.is-active {
-      .hamburger-inner {
-        &::before {
-          top: 0;
-          transform: translate3d($hamburger-layer-width * -0.2, $hamburger-layer-width * -0.25, 0) rotate(-45deg) scale(0.7, 1);
-          transition: top 0.1s ease,
-                      transform 0.1s 0.1s cubic-bezier(0.895, 0.03, 0.685, 0.22);
-        }
-
-        &::after {
-          bottom: 0;
-          transform: translate3d($hamburger-layer-width * -0.2, $hamburger-layer-width * 0.25, 0) rotate(45deg) scale(0.7, 1);
-          transition: bottom 0.1s ease,
-                      transform 0.1s 0.1s cubic-bezier(0.895, 0.03, 0.685, 0.22);
-        }
-      }
-    }
-  }
-}
diff --git a/_sass/main.scss b/_sass/main.scss
deleted file mode 100755
index 907a459..0000000
--- a/_sass/main.scss
+++ /dev/null
@@ -1,29 +0,0 @@
-/**
-  Most of Jupyter Book's CSS is defined in the single-page CSS
-  folder. This CSS adds some book-specific CSS rules on top of it.
- */
-
- // Base page CSS
-@import 'page/main';
-
-// OBJECTS (some of these override defaults)
-$inuit-wrapper-width: 800px;
-$hamburger-padding-x: 0;
-$hamburger-padding-y: 8px;
-$hamburger-layer-width: 20px;
-$hamburger-layer-height: 3px;
-$hamburger-layer-spacing: 3px;
-$hamburger-layer-color: $color-text;
-$hamburger-types: (arrowalt);
-@import 'hamburgers/hamburgers';
-@import 'objects/objects.thebelab-in-cell-button';
-
-// COMPONENTS
-@import 'components/components.book__layout';
-@import 'components/components.book__topbar';
-@import 'components/components.page__onthispage';
-@import 'components/components.page__nav';
-@import 'components/components.page__footer';
-@import 'components/components.interact-button';
-@import 'components/components.thebelab';
-@import 'components/components.search';
diff --git a/_sass/objects/_objects.thebelab-in-cell-button.scss b/_sass/objects/_objects.thebelab-in-cell-button.scss
deleted file mode 100755
index c23dea8..0000000
--- a/_sass/objects/_objects.thebelab-in-cell-button.scss
+++ /dev/null
@@ -1,25 +0,0 @@
-/* in-cell thebelab buttons */
-.thebebtn {
-  position: absolute;
-  top: 0;
-  right: $spacing-unit;
-
-  margin: $spacing-unit-tiny/2 $spacing-unit-tiny;
-  width: $spacing-unit;
-  height: $spacing-unit;
-  padding: 0 4px 2px;
-
-  i {
-    font-style: normal;
-    color: #777777
-  }
-}
-
-.input_area .highlight {
-  position: relative;
-}
-
-// Hide code buttons
-div.jb_input div.input {
-  position: relative;
-}
diff --git a/_sass/page/components/_components.hidecells.scss b/_sass/page/components/_components.hidecells.scss
deleted file mode 100755
index 998729a..0000000
--- a/_sass/page/components/_components.hidecells.scss
+++ /dev/null
@@ -1,75 +0,0 @@
-// Mixins and variables
-@mixin transition($in) {
-	transition:$in;
-	-webkit-transition:$in;
-	-moz-transition:$in;
-	-o-transition:$in;
-	-ms-transition:$in;
-}
-
-@mixin transform($in) {
-	transform:$in;
-	-webkit-transform:$in;
-	-moz-transform:$in;
-	-o-transform:$in;
-	-ms-transform:$in;
-}
-
-$plusminus-height: 2.5px;
-$plusminus-anim-length: .25s;
-
-// All hidden elements
-.hidden {
-    visibility: hidden;
-    opacity: 0;
-    height: 10px;
-    padding: 0px !important;
-}
-
-// Plusminus buttons
-// Adapted from https://codepen.io/FluidOfInsanity/pen/EyQGgw
-
-input[type="checkbox"] {
-    display: none;
-}
-
-.plusminus {
-    display: block;
-    position: absolute;
-    top: 9px;
-    right: -30px;
-    padding: .3em;
-
-    width: 20px;
-    height: 20px;
-    background: #d4d4d4;
-    border-radius: 25px;
-    @include transition(0.25s);
-}
-
-div.cell:hover .plusminus {
-    background: rgb(122, 130, 136);
-    @include transition(0.25s);
-}
-
-.plusminus span {
-    display: block;
-    position: absolute;
-    border-radius: 3px;
-    background: #f2f2f2;
-    @include transition(all $plusminus-anim-length ease);
-
-    margin: 0% 15%;
-    height: $plusminus-height;
-    width: 70%;
-
-    /*- half the width*/
-    left: 0px;
-    bottom:0px;
-    right:0px;
-    top: calc(50% - #{$plusminus-height} / 2);
-}
-
-input:checked ~ .plusminus span.pm_v {
-    @include transform(rotate(-90deg));
-}
diff --git a/_sass/page/components/_components.page.scss b/_sass/page/components/_components.page.scss
deleted file mode 100755
index d86a770..0000000
--- a/_sass/page/components/_components.page.scss
+++ /dev/null
@@ -1,307 +0,0 @@
-/**
- * Styling for textbook pages. Jupyter notebooks generate their own HTML markup
- * which we style under the #ipython-notebook selector.
- *
- * Most of the textbook content is styled using this component.
- *
- * [1]: Within-textbook__page layout
- * [2]: Since we use JS to focus the page on load, hide the outline to avoid
- *   visual cruft.
- *
- * Notebook styling:
- *
- * [4]: Some tables are too wide for the page on smaller screens so we let the
- *   user scroll horizontally.
- * [5]: Inline code snippets (usually variable names) should have a gray bg.
- */
-
-/**
- * [1] Within-textbook__page layout
-*/
-
-div.jb_cell {
-  width: 100%;
-  position: relative;
-
-  @include mq($from: laptop) {
-    width: $textbook-page-with-sidebar-width;
-  }
-
-  // Add some empty space before headers for anchor links
-  h1, h2, h3, h4, h5 {
-    &:before {
-      display: block;
-      content: " ";
-      margin-top: -80px;
-      height: 80px;
-      visibility: hidden;
-      pointer-events: none;
-    }
-  }
-}
-
-#page-title {
-  font-size: 2.2em;
-  font-weight: lighter;
-  line-height: 1em;
-}
-
-#page-author {
-  font-size: 1.4em;
-  font-style: italic;
-  font-weight: lighter;
-}
-
-#page-info {
-  margin-bottom: 20px;
-  max-width: $textbook-page-with-sidebar-width;
-}
-
-// Right margin
-div.jb_cell {
-  &.tag_popout {
-    .cell {
-      border-left: 3px solid #c3c3c3;
-      padding-left: 5%
-    }
-
-    // On laptops, pop out to the right instead of in-line
-    @include mq($from: laptop) {
-      z-index: 999;
-      width: calc(100% - #{$textbook-page-with-sidebar-width});
-      font-size: .8em;
-      position: relative;
-      float: right;
-      padding-left: $spacing-unit-large * .8;
-      clear: both;
-
-      h1, h2, h3, h4, h5, h6 {
-          margin: .5em 0px 0px 0px;
-      }
-
-      // This prevents *sequential* popouts from creating a bunch of white-space when stacked
-      & + div.tag_popout {
-        height: 0px;
-      }
-
-      // Remove the border when we pop-out to the right
-      .cell {
-        border-left: none;
-        padding-left: 0;
-      }
-    }
-  }
-
-  &.tag_full_width {
-      width: 100%;
-      div.input {
-          // Input sections of cells should stay the same width, only outputs are wide
-          width: 70%;
-      }
-
-    img {
-      max-width: 100%;
-    }
-  }
-    
-  &.tag_output_scroll div.output_wrapper {
-      max-height: 24em;
-      overflow-y: auto;
-  }
-}
-
-
-
-/**
- * Jupyter notebook styling
- */
-
-div.highlighter-rouge {
-  // Ensures that items within this div can be positioned absolutely
-  position: relative;
-}
-
-div.inner_cell > div > div > pre,
-div.inner_cell > div > pre
-div.output_wrapper pre {
-  background-color: $color-light-gray;
-  font-size: 0.9em;
-  margin-bottom: $spacing-unit-small;
-  padding: $spacing-unit-small;
-  overflow-x: auto;
-  border: 1px solid $color-dark-gray;
-}
-
-// If we've got intentionally non-highlighted markup, make sure that the lines wrap
-pre code.language-no-highlight {
-  white-space: pre-wrap;
-}
-
-div.output_wrapper {
-  position: relative;
-  margin-bottom: $spacing-unit-small;
-}
-
-div.output_wrapper pre {
-  border: 1px solid $color-light-gray !important;
-  background-color: #fcfcfc !important;
-}
-
-.output_html {
-  /* [4] */
-  overflow-x: auto;
-}
-
-/* [5] */
-code {
-  padding: 4px;
-  background: #F7F7F7;
-  font-size: inherit;
-}
-
-pre.highlight code {
-  display: block;
-}
-
-h1, h2, h3, h4, h5, h6 {
-  code {
-    font-size: inherit;
-  }
-}
-
-/* Outputs */
-.output_stream,
-.output_data_text,
-.output_traceback_line {
-  margin-left: 2% !important;
-  border: none !important;
-  border-radius: 4px !important;
-  background-color: #fafafa !important;
-  box-shadow: none !important;
-}
-
-.output_stream:before,
-.output_data_text:before,
-.output_traceback_line:before {
-  content: none !important;
-}
-
-.output_text pre {
-  background-color: #f8f8fb !important;
-}
-
-.output_html,
-.output_png::before {
-  padding: 1em 1.5em !important;
-}
-
-// Images that are on their own should be centered
-// For text cells, only apply to images that are in their own <p>
-.output_png img,
-div.text_cell_render > p > img,
-div.text_cell_render > p > a > img {
-  max-width: 500px;
-  display: block;
-  margin-left: auto;
-  margin-right: auto;
-}
-
-.output_png img {
-  width: 100% !important;
-}
-
-.hl-ipython3 pre::before,
-.output_subarea pre::before,
-.output_html::before,
-.output_png::before {
-  color: #ccc !important;
-  float: left !important;
-  margin-bottom: 0.25em !important;
-  width: 100% !important;
-}
-
-/**
-  * nbinteract styling
-  */
-.cell.nbinteract-left {
-  width: 50%;
-  float: left;
-}
-
-.cell.nbinteract-right {
-  width: 50%;
-  float: right;
-}
-
-.cell.nbinteract-hide_in > .input {
-  display: none;
-}
-
-.cell.nbinteract-hide_out > .output_wrapper {
-  display: none;
-}
-
-.cell:after {
-  content: '';
-  display: table;
-  clear: both;
-}
-
-div.output_subarea {
-  max-width: initial;
-}
-
-.jp-OutputPrompt {
-  display: none;
-}
-
-// Quotation cells
-
-// Default
-blockquote {
-  margin: $spacing-unit-small;
-  border-left: 4px solid $color-dark-gray;
-  padding: $spacing-unit-tiny $spacing-unit-med;
-
-  p {
-    margin-bottom: $spacing-unit-small;
-  }
-
-  ul, ol {
-    margin-bottom: $spacing-unit-tiny;
-  }
-
-  // This should only trigger if the last child of a blockquote is an ul/ol
-  > :last-child li {
-    text-align: right;
-    list-style-type: none;
-    margin: none;
-    padding-right: 5%;
-
-    &:before {
-      content: "—";
-      padding-right: 10px;
-    }
-  }
-}
-
-// Epigraph adds new styling
-div.jb_cell.tag_epigraph blockquote {
-
-  font-style: italic;
-  font-size: 1.2em;
-  width: 90%;
-  margin-left: auto;
-  margin-right: auto;
-  border-left: none;
-
-  ul {
-    font-style: normal;
-  }
-}
-
-// MathJax blocks
-
-.MathJax_Display {
-  padding: 20px 0px;
-}
\ No newline at end of file
diff --git a/_sass/page/components/_components.sidebar-right.scss b/_sass/page/components/_components.sidebar-right.scss
deleted file mode 100755
index f582450..0000000
--- a/_sass/page/components/_components.sidebar-right.scss
+++ /dev/null
@@ -1,123 +0,0 @@
-/**
- * Styling for within-page navigation.
- *
- * [1]: When the screen is large enough to hold the sidebar, place the sidebar
- *   offset a bit off the right side of the screen.
- *
- * [2]: When the screen is large enough that the page content itself starts
- *   having margins, use a semi-hack to position the sidebar just the right
- *   offset from the center of the screen to be in the correct spot. We're
- *   aiming to have the sidebar positioned just to the right of the page
- *   content which means we take the page width without the sidebar, divide it
- *   by 2, and add padding.
- */
-
-/* [2] */
-.sidebar__right > * {
-    direction: ltr;
-}
-
-aside.sidebar__right {
-    // Positioning
-    overflow-x: hidden;
-    background-color: $book-background-color;
-    overflow-y: auto;
-    max-height: 90vh; // Required for scrolling to work properly
-    scrollbar-width: thin;
-    direction: rtl;
-
-    // Font and look
-    font-family: -apple-system, BlinkMacSystemFont, 'Roboto', 'Segoe UI',
-        'Helvetica Neue', 'Lucida Grande', Arial, sans-serif;
-    color: #7a8288;
-    border-left: 1px solid #c3c3c3;
-    text-transform: uppercase;
-    letter-spacing: 1px;
-
-    // Icon
-    img {
-        width: 15px;
-        padding-bottom: 2px;
-    }
-
-    // Show/hide navbar underneath
-    nav {
-        transition: opacity .25s ease-in-out, height .25s ease-in-out;
-        -moz-transition: opacity .25s ease-in-out, height .25s ease-in-out;
-        -webkit-transition: opacity .25s ease-in-out, height .25s ease-in-out;
-        overflow-y: hidden;
-        opacity: 0;
-        height: 0px;
-
-        @include mq($from: desktop) {
-            opacity: 100;
-            height: auto;
-        }
-
-        @include mq($from: laptop) {
-            &.no_sidebar_content {
-              opacity: 100;
-              height: auto;
-            }
-        }
-    }
-
-    &:hover nav {
-        opacity: 100;
-        height: auto;
-    }
-}
-
-
-h4.nav__title {
-    color: #7a8288;
-    margin: 0;
-    padding: 0.5rem 1rem;
-    font-family: -apple-system, BlinkMacSystemFont, 'Roboto', 'Segoe UI',
-        'Helvetica Neue', 'Lucida Grande', Arial, sans-serif;
-    font-size: 0.8em;
-    font-weight: bold;
-}
-
-ul.toc__menu {
-    margin: 0;
-    padding: 0;
-    width: 100%;
-    list-style: none;
-    font-size: 0.8rem;
-}
-
-ul.toc__menu a {
-    display: block;
-    padding: 0.25rem .75rem;
-    color: #898c8f;
-    font-size: 0.8em;
-    font-weight: bold;
-    line-height: 1.5;
-}
-
-.toc__menu ul {
-    margin-left: 0px;
-}
-
-.toc__menu li ul {
-    li {
-        list-style-type: none;
-        padding-left: 18px;
-    }
-
-    a {
-        font-weight: normal;
-        padding: 0.25rem .5rem;
-    }
-}
-
-// Active sidebar entries
-nav.onthispage li.active a {
-    color: #0077d8;
-}
-
-li.active {
-    border-left: 1px solid #0077d8;
-    margin-left: -1px;
-}
diff --git a/_sass/page/elements/_elements.links.scss b/_sass/page/elements/_elements.links.scss
deleted file mode 100755
index 0d36c21..0000000
--- a/_sass/page/elements/_elements.links.scss
+++ /dev/null
@@ -1,29 +0,0 @@
-/**
- * Styling for links
- */
-
-a,
-a:visited {
-  color: $color-links;
-  text-decoration: none;
-}
-
-// Anchor links
-// Uncomment this after https://github.com/jupyter/nbconvert/pull/1101 is released
-// main.jupyter-page {
-//   a.anchor-link {
-//     visibility: hidden;
-//     font-size: 15px;
-//     padding-left: 5px;
-
-//     i {
-//       font-style: normal;
-//     }
-//   }
-
-//   h2, h3, h4, h5 {
-//     &:hover a.anchor-link {
-//       visibility: visible;
-//     }
-//   }
-// }
\ No newline at end of file
diff --git a/_sass/page/elements/_elements.syntax-highlighting.scss b/_sass/page/elements/_elements.syntax-highlighting.scss
deleted file mode 100755
index 14abebc..0000000
--- a/_sass/page/elements/_elements.syntax-highlighting.scss
+++ /dev/null
@@ -1,83 +0,0 @@
-/**
- * Syntax highlighting for code cells
- */
-
-pre.highlight {
-    overflow-x: auto;
-    padding: .5em;
-}
-.highlight {
-
-    .c     { color: #998; font-style: italic } // Comment
-    .err   { color: #a61717; background-color: #e3d2d2 } // Error
-    .k     { font-weight: bold } // Keyword
-    .o     { font-weight: bold } // Operator
-    .cm    { color: #998; font-style: italic } // Comment.Multiline
-    .cp    { color: #999; font-weight: bold } // Comment.Preproc
-    .c1    { color: #998; font-style: italic } // Comment.Single
-    .cs    { color: #999; font-weight: bold; font-style: italic } // Comment.Special
-    .gd    { color: #000; background-color: #fdd } // Generic.Deleted
-    .gd .x { color: #000; background-color: #faa } // Generic.Deleted.Specific
-    .ge    { font-style: italic } // Generic.Emph
-    .gr    { color: #a00 } // Generic.Error
-    .gh    { color: #999 } // Generic.Heading
-    .gi    { color: #000; background-color: #dfd } // Generic.Inserted
-    .gi .x { color: #000; background-color: #afa } // Generic.Inserted.Specific
-    .go    { color: #888 } // Generic.Output
-    .gp    { color: #555 } // Generic.Prompt
-    .gs    { font-weight: bold } // Generic.Strong
-    .gu    { color: #aaa } // Generic.Subheading
-    .gt    { color: #a00 } // Generic.Traceback
-    .kc    { font-weight: bold } // Keyword.Constant
-    .kd    { font-weight: bold } // Keyword.Declaration
-    .kp    { font-weight: bold } // Keyword.Pseudo
-    .kr    { font-weight: bold } // Keyword.Reserved
-    .kt    { color: #458; font-weight: bold } // Keyword.Type
-    .m     { color: #099 } // Literal.Number
-    .s     { color: #d14 } // Literal.String
-    .na    { color: #008080 } // Name.Attribute
-    .nb    { color: #0086B3 } // Name.Builtin
-    .nc    { color: #458; font-weight: bold } // Name.Class
-    .no    { color: #008080 } // Name.Constant
-    .ni    { color: #800080 } // Name.Entity
-    .ne    { color: #900; font-weight: bold } // Name.Exception
-    .nf    { color: #900; font-weight: bold } // Name.Function
-    .nn    { color: #555 } // Name.Namespace
-    .nt    { color: #000080 } // Name.Tag
-    .nv    { color: #008080 } // Name.Variable
-    .ow    { font-weight: bold } // Operator.Word
-    .w     { color: #bbb } // Text.Whitespace
-    .mf    { color: #099 } // Literal.Number.Float
-    .mh    { color: #099 } // Literal.Number.Hex
-    .mi    { color: #099 } // Literal.Number.Integer
-    .mo    { color: #099 } // Literal.Number.Oct
-    .sb    { color: #d14 } // Literal.String.Backtick
-    .sc    { color: #d14 } // Literal.String.Char
-    .sd    { color: #d14 } // Literal.String.Doc
-    .s2    { color: #d14 } // Literal.String.Double
-    .se    { color: #d14 } // Literal.String.Escape
-    .sh    { color: #d14 } // Literal.String.Heredoc
-    .si    { color: #d14 } // Literal.String.Interpol
-    .sx    { color: #d14 } // Literal.String.Other
-    .sr    { color: #009926 } // Literal.String.Regex
-    .s1    { color: #d14 } // Literal.String.Single
-    .ss    { color: #990073 } // Literal.String.Symbol
-    .bp    { color: #999 } // Name.Builtin.Pseudo
-    .vc    { color: #008080 } // Name.Variable.Class
-    .vg    { color: #008080 } // Name.Variable.Global
-    .vi    { color: #008080 } // Name.Variable.Instance
-    .il    { color: #099 } // Literal.Number.Integer.Long
-}
-
-/* ANSI coloring for text_regex.ipynb */
-.ansi-bold {
-    font-weight: bold;
-}
-
-.ansi-yellow-bg {
-    background-color: #DDB62B;
-}
-
-.ansi-black-intense-fg {
-    color: #282C36;
-}
diff --git a/_sass/page/elements/_elements.tables.scss b/_sass/page/elements/_elements.tables.scss
deleted file mode 100755
index 5a7db04..0000000
--- a/_sass/page/elements/_elements.tables.scss
+++ /dev/null
@@ -1,28 +0,0 @@
-/**
- * Styling for tables. In the textbook, almost all tables display the contents
- * of a pandas DataFrame.
- *
- * [1]: Some tables are very small, so we shouldn't force them to be full-width
- * [2]: Add striping to table rows to make them easier to read.
- */
-
-table {
-  /* [1] */
-  max-width: 100%;
-  width: initial;
-
-  border-collapse: collapse;
-  border-spacing: 0;
-}
-
-table td,
-table th {
-  padding: $spacing-unit-tiny $spacing-unit-small;
-  border: 1px solid $color-dark-gray;
-  text-align: right;
-}
-
-/* [2] */
-table tr:nth-child(2n) {
-  background-color: $color-light-gray;
-}
diff --git a/_sass/page/elements/_elements.typography.scss b/_sass/page/elements/_elements.typography.scss
deleted file mode 100755
index 5248f29..0000000
--- a/_sass/page/elements/_elements.typography.scss
+++ /dev/null
@@ -1,63 +0,0 @@
-/**
- * Site-wide typography
- */
-
-body {
-  // Use system font stack:
-  // https://css-tricks.com/snippets/css/system-font-stack/
-  font-family: $global-font-family  !important;
-  $color: $color-text;
-}
-
-/**
- * Code-like elements.
- */
-pre,
-code,
-kbd,
-samp {
-  font-family: $monospace;
-  font-style: normal;
-}
-
-/**
- * Header styling
- */
-$header-font-family: $sans-serif !default;
-
-h1,
-h2,
-h3,
-h4,
-h5,
-h6 {
-  margin: 2em 0 0.5em;
-  line-height: 1.2;
-  font-family: $header-font-family !important;
-  font-weight: bold;
-}
-
-h1 {
-  margin-top: 0;
-  font-size: $type-size-3;
-}
-
-h2 {
-  font-size: $type-size-4;
-}
-
-h3 {
-  font-size: $type-size-5;
-}
-
-h4 {
-  font-size: $type-size-6;
-}
-
-h5 {
-  font-size: $type-size-6;
-}
-
-h6 {
-  font-size: $type-size-6;
-}
\ No newline at end of file
diff --git a/_sass/page/elements/_elements.variables.scss b/_sass/page/elements/_elements.variables.scss
deleted file mode 100755
index 6bbdead..0000000
--- a/_sass/page/elements/_elements.variables.scss
+++ /dev/null
@@ -1,26 +0,0 @@
-/**
- * Variables
- */
-
-$serif: Georgia, Times, serif !default;
-$sans-serif: -apple-system, BlinkMacSystemFont, "Roboto", "Segoe UI",
-  "Helvetica Neue", "Lucida Grande", Arial, sans-serif !default;
-$monospace: Monaco, Consolas, "Lucida Console", monospace !default;
-
-$global-font-family: $sans-serif !default;
-$header-font-family: $sans-serif !default;
-$caption-font-family: $serif !default;
-
-/* type scale */
-$type-size-1: 2.441em !default; // ~39.056px
-$type-size-2: 1.953em !default; // ~31.248px
-$type-size-3: 1.563em !default; // ~25.008px
-$type-size-4: 1.25em !default; // ~20px
-$type-size-5: 1em !default; // ~16px
-$type-size-6: 0.75em !default; // ~12px
-$type-size-7: 0.6875em !default; // ~11px
-$type-size-8: 0.625em !default; // ~10px
- 
-/* Left-margin (applied if screen is big enough) */
-$left-site-margin: 20% !default;
-$site-margin-min-width: 1700px !default;
diff --git a/_sass/page/generic/_generic.phone-scrolling.scss b/_sass/page/generic/_generic.phone-scrolling.scss
deleted file mode 100755
index 61a02d1..0000000
--- a/_sass/page/generic/_generic.phone-scrolling.scss
+++ /dev/null
@@ -1,7 +0,0 @@
-/**
- * Makes scrolling smooth for phones
- */
-
-* {
-  -webkit-overflow-scrolling: touch;
-}
diff --git a/_sass/page/inuitcss/elements/_elements.headings.scss b/_sass/page/inuitcss/elements/_elements.headings.scss
deleted file mode 100755
index 21d8e75..0000000
--- a/_sass/page/inuitcss/elements/_elements.headings.scss
+++ /dev/null
@@ -1,45 +0,0 @@
-/* ==========================================================================
-   #HEADINGS
-   ========================================================================== */
-
-/**
- * Simple default styles for headings 1 through 6. Anything more opinionated
- * than simple font-size changes should likely be applied via classes (see:
- * http://csswizardry.com/2016/02/managing-typography-on-large-apps/).
- */
-
-// We have all of our heading font sizes defined here. Passing these pixel
-// values into our `inuit-font-size()` mixin will generate a rem-based
-// `font-size` with a pixel fallback, as well as generating a `line-height` that
-// will sit on our baseline grid.
-
-$inuit-font-size-h1:  36px !default;
-$inuit-font-size-h2:  28px !default;
-$inuit-font-size-h3:  24px !default;
-$inuit-font-size-h4:  20px !default;
-$inuit-font-size-h5:  18px !default;
-$inuit-font-size-h6:  16px !default;
-
-h1 {
-  @include inuit-font-size($inuit-font-size-h1);
-}
-
-h2 {
-  @include inuit-font-size($inuit-font-size-h2);
-}
-
-h3 {
-  @include inuit-font-size($inuit-font-size-h3);
-}
-
-h4 {
-  @include inuit-font-size($inuit-font-size-h4);
-}
-
-h5 {
-  @include inuit-font-size($inuit-font-size-h5);
-}
-
-h6 {
-  @include inuit-font-size($inuit-font-size-h6);
-}
diff --git a/_sass/page/inuitcss/elements/_elements.images.scss b/_sass/page/inuitcss/elements/_elements.images.scss
deleted file mode 100755
index fa21829..0000000
--- a/_sass/page/inuitcss/elements/_elements.images.scss
+++ /dev/null
@@ -1,39 +0,0 @@
-/* ==========================================================================
-   #IMAGES
-   ========================================================================== */
-
-/**
- * 1. Fluid images for responsive purposes.
- * 2. Offset `alt` text from surrounding copy.
- * 3. Setting `vertical-align` removes the whitespace that appears under `img`
- *    elements when they are dropped into a page as-is. Safer alternative to
- *    using `display: block;`.
- */
-
-img {
-  max-width: 100%; /* [1] */
-  font-style: italic; /* [2] */
-  vertical-align: middle; /* [3] */
-}
-
-
-
-// In case you don't have control over generated `width` and `height` attributes
-// on `<img>` elements in your markup, but still want the images to be fluid,
-// set this to `false`.
-
-$inuit-static-images: true !default;
-
-  @if ($inuit-static-images == true) {
-
-  /**
-   * If a `width` and/or `height` attribute has been explicitly defined, let’s
-   * not make the image fluid.
-   */
-
-  img[width],
-  img[height] {
-    max-width: none;
-  }
-
-}
diff --git a/_sass/page/inuitcss/elements/_elements.page.scss b/_sass/page/inuitcss/elements/_elements.page.scss
deleted file mode 100755
index 6bfa073..0000000
--- a/_sass/page/inuitcss/elements/_elements.page.scss
+++ /dev/null
@@ -1,19 +0,0 @@
-/* ==========================================================================
-   #PAGE
-   ========================================================================== */
-
-/**
- * Simple page-level setup.
- *
- * 1. Set the default `font-size` and `line-height` for the entire project,
- *    sourced from our default variables. The `font-size` is calculated to exist
- *    in ems, the `line-height` is calculated to exist unitlessly.
- * 2. Ensure the page always fills at least the entire height of the viewport.
- */
-
-html,
-body {
-  font-size: ($inuit-global-font-size / 16px) * 1em; /* [1] */
-  line-height: $inuit-global-line-height / $inuit-global-font-size; /* [1] */
-  min-height: 100%; /* [2] */
-}
diff --git a/_sass/page/inuitcss/elements/_elements.tables.scss b/_sass/page/inuitcss/elements/_elements.tables.scss
deleted file mode 100755
index 6256616..0000000
--- a/_sass/page/inuitcss/elements/_elements.tables.scss
+++ /dev/null
@@ -1,11 +0,0 @@
-/* ==========================================================================
-   #TABLES
-   ========================================================================== */
-
-/**
- * 1. Ensure tables fill up as much space as possible.
- */
-
-table {
-  width: 100%; /* [1] */
-}
diff --git a/_sass/page/inuitcss/generic/_generic.box-sizing.scss b/_sass/page/inuitcss/generic/_generic.box-sizing.scss
deleted file mode 100755
index 5779ff1..0000000
--- a/_sass/page/inuitcss/generic/_generic.box-sizing.scss
+++ /dev/null
@@ -1,22 +0,0 @@
-/* ==========================================================================
-   #BOX-SIZING
-   ========================================================================== */
-
-/**
- * More sensible default box-sizing:
- * css-tricks.com/inheriting-box-sizing-probably-slightly-better-best-practice
- */
-
-html {
-  box-sizing: border-box;
-}
-
-* {
-
-  &,
-  &:before,
-  &:after {
-    box-sizing: inherit;
-  }
-
-}
diff --git a/_sass/page/inuitcss/generic/_generic.normalize.scss b/_sass/page/inuitcss/generic/_generic.normalize.scss
deleted file mode 100755
index fa4e73d..0000000
--- a/_sass/page/inuitcss/generic/_generic.normalize.scss
+++ /dev/null
@@ -1,447 +0,0 @@
-/*! normalize.css v7.0.0 | MIT License | github.com/necolas/normalize.css */
-
-/* Document
-   ========================================================================== */
-
-/**
- * 1. Correct the line height in all browsers.
- * 2. Prevent adjustments of font size after orientation changes in
- *    IE on Windows Phone and in iOS.
- */
-
-html {
-  line-height: 1.15; /* 1 */
-  -ms-text-size-adjust: 100%; /* 2 */
-  -webkit-text-size-adjust: 100%; /* 2 */
-}
-
-/* Sections
-   ========================================================================== */
-
-/**
- * Remove the margin in all browsers (opinionated).
- */
-
-body {
-  margin: 0;
-}
-
-/**
- * Add the correct display in IE 9-.
- */
-
-article,
-aside,
-footer,
-header,
-nav,
-section {
-  display: block;
-}
-
-/**
- * Correct the font size and margin on `h1` elements within `section` and
- * `article` contexts in Chrome, Firefox, and Safari.
- */
-
-h1 {
-  font-size: 2em;
-  margin: 0.67em 0;
-}
-
-/* Grouping content
-   ========================================================================== */
-
-/**
- * Add the correct display in IE 9-.
- * 1. Add the correct display in IE.
- */
-
-figcaption,
-figure,
-main { /* 1 */
-  display: block;
-}
-
-/**
- * Add the correct margin in IE 8.
- */
-
-figure {
-  margin: 1em 40px;
-}
-
-/**
- * 1. Add the correct box sizing in Firefox.
- * 2. Show the overflow in Edge and IE.
- */
-
-hr {
-  box-sizing: content-box; /* 1 */
-  height: 0; /* 1 */
-  overflow: visible; /* 2 */
-}
-
-/**
- * 1. Correct the inheritance and scaling of font size in all browsers.
- * 2. Correct the odd `em` font sizing in all browsers.
- */
-
-pre {
-  font-family: monospace, monospace; /* 1 */
-  font-size: 1em; /* 2 */
-}
-
-/* Text-level semantics
-   ========================================================================== */
-
-/**
- * 1. Remove the gray background on active links in IE 10.
- * 2. Remove gaps in links underline in iOS 8+ and Safari 8+.
- */
-
-a {
-  background-color: transparent; /* 1 */
-  -webkit-text-decoration-skip: objects; /* 2 */
-}
-
-/**
- * 1. Remove the bottom border in Chrome 57- and Firefox 39-.
- * 2. Add the correct text decoration in Chrome, Edge, IE, Opera, and Safari.
- */
-
-abbr[title] {
-  border-bottom: none; /* 1 */
-  text-decoration: underline; /* 2 */
-  text-decoration: underline dotted; /* 2 */
-}
-
-/**
- * Prevent the duplicate application of `bolder` by the next rule in Safari 6.
- */
-
-b,
-strong {
-  font-weight: inherit;
-}
-
-/**
- * Add the correct font weight in Chrome, Edge, and Safari.
- */
-
-b,
-strong {
-  font-weight: bolder;
-}
-
-/**
- * 1. Correct the inheritance and scaling of font size in all browsers.
- * 2. Correct the odd `em` font sizing in all browsers.
- */
-
-code,
-kbd,
-samp {
-  font-family: monospace, monospace; /* 1 */
-  font-size: 1em; /* 2 */
-}
-
-/**
- * Add the correct font style in Android 4.3-.
- */
-
-dfn {
-  font-style: italic;
-}
-
-/**
- * Add the correct background and color in IE 9-.
- */
-
-mark {
-  background-color: #ff0;
-  color: #000;
-}
-
-/**
- * Add the correct font size in all browsers.
- */
-
-small {
-  font-size: 80%;
-}
-
-/**
- * Prevent `sub` and `sup` elements from affecting the line height in
- * all browsers.
- */
-
-sub,
-sup {
-  font-size: 75%;
-  line-height: 0;
-  position: relative;
-  vertical-align: baseline;
-}
-
-sub {
-  bottom: -0.25em;
-}
-
-sup {
-  top: -0.5em;
-}
-
-/* Embedded content
-   ========================================================================== */
-
-/**
- * Add the correct display in IE 9-.
- */
-
-audio,
-video {
-  display: inline-block;
-}
-
-/**
- * Add the correct display in iOS 4-7.
- */
-
-audio:not([controls]) {
-  display: none;
-  height: 0;
-}
-
-/**
- * Remove the border on images inside links in IE 10-.
- */
-
-img {
-  border-style: none;
-}
-
-/**
- * Hide the overflow in IE.
- */
-
-svg:not(:root) {
-  overflow: hidden;
-}
-
-/* Forms
-   ========================================================================== */
-
-/**
- * 1. Change the font styles in all browsers (opinionated).
- * 2. Remove the margin in Firefox and Safari.
- */
-
-button,
-input,
-optgroup,
-select,
-textarea {
-  font-family: sans-serif; /* 1 */
-  font-size: 100%; /* 1 */
-  line-height: 1.15; /* 1 */
-  margin: 0; /* 2 */
-}
-
-/**
- * Show the overflow in IE.
- * 1. Show the overflow in Edge.
- */
-
-button,
-input { /* 1 */
-  overflow: visible;
-}
-
-/**
- * Remove the inheritance of text transform in Edge, Firefox, and IE.
- * 1. Remove the inheritance of text transform in Firefox.
- */
-
-button,
-select { /* 1 */
-  text-transform: none;
-}
-
-/**
- * 1. Prevent a WebKit bug where (2) destroys native `audio` and `video`
- *    controls in Android 4.
- * 2. Correct the inability to style clickable types in iOS and Safari.
- */
-
-button,
-html [type="button"], /* 1 */
-[type="reset"],
-[type="submit"] {
-  -webkit-appearance: button; /* 2 */
-}
-
-/**
- * Remove the inner border and padding in Firefox.
- */
-
-button::-moz-focus-inner,
-[type="button"]::-moz-focus-inner,
-[type="reset"]::-moz-focus-inner,
-[type="submit"]::-moz-focus-inner {
-  border-style: none;
-  padding: 0;
-}
-
-/**
- * Restore the focus styles unset by the previous rule.
- */
-
-button:-moz-focusring,
-[type="button"]:-moz-focusring,
-[type="reset"]:-moz-focusring,
-[type="submit"]:-moz-focusring {
-  outline: 1px dotted ButtonText;
-}
-
-/**
- * Correct the padding in Firefox.
- */
-
-fieldset {
-  padding: 0.35em 0.75em 0.625em;
-}
-
-/**
- * 1. Correct the text wrapping in Edge and IE.
- * 2. Correct the color inheritance from `fieldset` elements in IE.
- * 3. Remove the padding so developers are not caught out when they zero out
- *    `fieldset` elements in all browsers.
- */
-
-legend {
-  box-sizing: border-box; /* 1 */
-  color: inherit; /* 2 */
-  display: table; /* 1 */
-  max-width: 100%; /* 1 */
-  padding: 0; /* 3 */
-  white-space: normal; /* 1 */
-}
-
-/**
- * 1. Add the correct display in IE 9-.
- * 2. Add the correct vertical alignment in Chrome, Firefox, and Opera.
- */
-
-progress {
-  display: inline-block; /* 1 */
-  vertical-align: baseline; /* 2 */
-}
-
-/**
- * Remove the default vertical scrollbar in IE.
- */
-
-textarea {
-  overflow: auto;
-}
-
-/**
- * 1. Add the correct box sizing in IE 10-.
- * 2. Remove the padding in IE 10-.
- */
-
-[type="checkbox"],
-[type="radio"] {
-  box-sizing: border-box; /* 1 */
-  padding: 0; /* 2 */
-}
-
-/**
- * Correct the cursor style of increment and decrement buttons in Chrome.
- */
-
-[type="number"]::-webkit-inner-spin-button,
-[type="number"]::-webkit-outer-spin-button {
-  height: auto;
-}
-
-/**
- * 1. Correct the odd appearance in Chrome and Safari.
- * 2. Correct the outline style in Safari.
- */
-
-[type="search"] {
-  -webkit-appearance: textfield; /* 1 */
-  outline-offset: -2px; /* 2 */
-}
-
-/**
- * Remove the inner padding and cancel buttons in Chrome and Safari on macOS.
- */
-
-[type="search"]::-webkit-search-cancel-button,
-[type="search"]::-webkit-search-decoration {
-  -webkit-appearance: none;
-}
-
-/**
- * 1. Correct the inability to style clickable types in iOS and Safari.
- * 2. Change font properties to `inherit` in Safari.
- */
-
-::-webkit-file-upload-button {
-  -webkit-appearance: button; /* 1 */
-  font: inherit; /* 2 */
-}
-
-/* Interactive
-   ========================================================================== */
-
-/*
- * Add the correct display in IE 9-.
- * 1. Add the correct display in Edge, IE, and Firefox.
- */
-
-details, /* 1 */
-menu {
-  display: block;
-}
-
-/*
- * Add the correct display in all browsers.
- */
-
-summary {
-  display: list-item;
-}
-
-/* Scripting
-   ========================================================================== */
-
-/**
- * Add the correct display in IE 9-.
- */
-
-canvas {
-  display: inline-block;
-}
-
-/**
- * Add the correct display in IE.
- */
-
-template {
-  display: none;
-}
-
-/* Hidden
-   ========================================================================== */
-
-/**
- * Add the correct display in IE 10-.
- */
-
-[hidden] {
-  display: none;
-}
diff --git a/_sass/page/inuitcss/generic/_generic.reset.scss b/_sass/page/inuitcss/generic/_generic.reset.scss
deleted file mode 100755
index 728f4a7..0000000
--- a/_sass/page/inuitcss/generic/_generic.reset.scss
+++ /dev/null
@@ -1,56 +0,0 @@
-/* ==========================================================================
-   #RESET
-   ========================================================================== */
-
-/**
- * A very simple reset that sits on top of Normalize.css.
- */
-
-body,
-h1, h2, h3, h4, h5, h6,
-blockquote, p, pre,
-dl, dd, ol, ul,
-figure,
-hr,
-fieldset, legend {
-  margin:  0;
-  padding: 0;
-}
-
-
-
-/**
- * Remove trailing margins from nested lists.
- */
-
-li > {
-
-  ol,
-  ul {
-    margin-bottom: 0;
-  }
-
-}
-
-
-
-/**
- * Remove default table spacing.
- */
-
-table {
-  border-collapse: collapse;
-  border-spacing: 0;
-}
-
-
-
-/**
- * 1. Reset Chrome and Firefox behaviour which sets a `min-width: min-content;`
- *    on fieldsets.
- */
-
-fieldset {
-  min-width: 0; /* [1] */
-  border: 0;
-}
diff --git a/_sass/page/inuitcss/generic/_generic.shared.scss b/_sass/page/inuitcss/generic/_generic.shared.scss
deleted file mode 100755
index 5585316..0000000
--- a/_sass/page/inuitcss/generic/_generic.shared.scss
+++ /dev/null
@@ -1,33 +0,0 @@
-/* ==========================================================================
-   #SHARED
-   ========================================================================== */
-
-/**
- * Shared declarations for certain elements.
- */
-
-/**
- * Always declare margins in the same direction:
- * csswizardry.com/2012/06/single-direction-margin-declarations
- */
-
-address,
-h1, h2, h3, h4, h5, h6,
-blockquote, p, pre,
-dl, ol, ul,
-figure,
-hr,
-table,
-fieldset {
-  margin-bottom: $inuit-global-spacing-unit;
-}
-
-
-
-/**
- * Consistent indentation for lists.
- */
-
-dd, ol, ul {
-  margin-left: $inuit-global-spacing-unit;
-}
diff --git a/_sass/page/inuitcss/objects/_objects.block.scss b/_sass/page/inuitcss/objects/_objects.block.scss
deleted file mode 100755
index ffe1bea..0000000
--- a/_sass/page/inuitcss/objects/_objects.block.scss
+++ /dev/null
@@ -1,61 +0,0 @@
-/* ==========================================================================
-   #BLOCK
-   ========================================================================== */
-
-/**
- * Stacked image-with-text object. A simple abstraction to cover a very commonly
- * occurring design pattern.
- */
-
-.o-block {
-  display: block;
-  text-align: center;
-}
-
-  .o-block__img {
-    margin-bottom: $inuit-global-spacing-unit;
-
-
-    /* Size variants
-       ====================================================================== */
-
-    .o-block--flush > & {
-      margin-bottom: 0;
-    }
-
-    .o-block--tiny > & {
-      margin-bottom: $inuit-global-spacing-unit-tiny;
-    }
-
-    .o-block--small > & {
-      margin-bottom: $inuit-global-spacing-unit-small;
-    }
-
-    .o-block--large > & {
-      margin-bottom: $inuit-global-spacing-unit-large;
-    }
-
-    .o-block--huge > & {
-      margin-bottom: $inuit-global-spacing-unit-huge;
-    }
-
-  }
-
-  .o-block__body {
-    display: block;
-  }
-
-
-
-
-
-/* Alignment variants
-   ========================================================================== */
-
-.o-block--right {
-  text-align: right;
-}
-
-.o-block--left {
-  text-align: left;
-}
diff --git a/_sass/page/inuitcss/objects/_objects.box.scss b/_sass/page/inuitcss/objects/_objects.box.scss
deleted file mode 100755
index 9ce7496..0000000
--- a/_sass/page/inuitcss/objects/_objects.box.scss
+++ /dev/null
@@ -1,48 +0,0 @@
-/* ==========================================================================
-   #BOX
-   ========================================================================== */
-
-/**
- * The box object simply boxes off content. Extend with cosmetic styles in the
- * Components layer.
- *
- * 1. So we can apply the `.o-box` class to naturally-inline elements.
- */
-
-.o-box {
-  @include inuit-clearfix();
-  display: block; /* [1] */
-  padding: $inuit-global-spacing-unit;
-
-  > :last-child {
-    margin-bottom: 0;
-  }
-
-}
-
-
-
-
-
-/* Size variants
-   ========================================================================== */
-
-.o-box--flush {
-  padding: 0;
-}
-
-.o-box--tiny {
-  padding: $inuit-global-spacing-unit-tiny;
-}
-
-.o-box--small {
-  padding: $inuit-global-spacing-unit-small;
-}
-
-.o-box--large {
-  padding: $inuit-global-spacing-unit-large;
-}
-
-.o-box--huge {
-  padding: $inuit-global-spacing-unit-huge;
-}
diff --git a/_sass/page/inuitcss/objects/_objects.crop.scss b/_sass/page/inuitcss/objects/_objects.crop.scss
deleted file mode 100755
index 7a80b96..0000000
--- a/_sass/page/inuitcss/objects/_objects.crop.scss
+++ /dev/null
@@ -1,165 +0,0 @@
-/* ==========================================================================
-   #CROP
-   ========================================================================== */
-
-// A list of cropping ratios that get generated as modifier classes.
-// You should predefine it with only the ratios and names your project needs.
-//
-// The map keys are the strings used in the generated class names, and they can
-// follow any convention, as long as they are properly escaped strings. i.e.:
-//
-//   $inuit-crops: (
-//     "2\\:1"         : (2:1),
-//     "4-by-3"        : (4:3),
-//     "full-hd"       : (16:9),
-//     "card-image"    : (2:3),
-//     "golden-ratio"  : (1.618:1) -> non-integers are okay
-//   ) !default;
-
-$inuit-crops: (
-  "2\\:1"   : (2:1),
-  "4\\:3"   : (4:3),
-  "16\\:9"  : (16:9)
-) !default;
-
-
-
-/**
- * Provide a cropping container in order to display media (usually images)
- * cropped to certain ratios.
- *
- * 1. Set up a positioning context in which the image can sit.
- * 2. This is the crucial part: where the cropping happens.
- */
-
-.o-crop {
-  position: relative; /* [1] */
-  display: block;
-  overflow: hidden; /* [2] */
-}
-
-  /**
-   * Apply this class to the content (usually `img`) that needs cropping.
-   *
-   * 1. Image’s default positioning is top-left in the cropping box.
-   * 2. Make sure the media doesn’t stop itself too soon.
-   */
-
-  .o-crop__content {
-    position: absolute;
-    top:  0; /* [1] */
-    left: 0; /* [1] */
-    max-width: none; /* [2] */
-  }
-
-
-
-  /**
-   * We can position the media in different locations within the cropping area.
-   */
-
-  .o-crop__content--left-top {
-    left: 0;
-  }
-
-  .o-crop__content--left-center {
-    top: 50%;
-    transform: translateY(-50%);
-  }
-
-  .o-crop__content--left-bottom {
-    top: auto;
-    bottom: 0;
-  }
-
-  .o-crop__content--right-top {
-    right: 0;
-    left: auto;
-  }
-
-  .o-crop__content--right-center {
-    top: 50%;
-    right: 0;
-    left: auto;
-    transform: translateY(-50%);
-  }
-
-  .o-crop__content--right-bottom {
-    top: auto;
-    right: 0;
-    bottom: 0;
-    left: auto;
-  }
-
-  .o-crop__content--center-top {
-    left: 50%;
-    transform: translateX(-50%);
-  }
-
-  .o-crop__content--center,
-  .o-crop__content--center-center {
-    top: 50%;
-    left: 50%;
-    transform: translate(-50%, -50%);
-  }
-
-  .o-crop__content--center-bottom {
-    top: auto;
-    bottom: 0;
-    left: 50%;
-    transform: translateX(-50%);
-  }
-
-
-
-
-
-/* Crop-ratio variants
-   ========================================================================== */
-
-/**
- * Generate a series of crop classes to be used like so:
- *
- *   <div class="o-crop  o-crop--golden-ratio">
- *
- */
-
-@each $crop-name, $crop-value in $inuit-crops {
-
-  @each $antecedent, $consequent in $crop-value {
-
-    @if (type-of($antecedent) != number) {
-      @error "`#{$antecedent}` needs to be a number.";
-    }
-
-    @if (type-of($consequent) != number) {
-      @error "`#{$consequent}` needs to be a number.";
-    }
-
-    .o-crop--#{$crop-name} {
-      padding-bottom: ($consequent/$antecedent) * 100%;
-    }
-
-  }
-
-}
-
-
-
-
-
-/* Fill modifier
-   ========================================================================== */
-
-/**
- * Content stretches to fill it's container while maintaining aspect-ratio.
- */
-
-.o-crop--fill {
-
-  > .o-crop__content {
-    min-height: 100%;
-    min-width: 100%;
-  }
-
-}
diff --git a/_sass/page/inuitcss/objects/_objects.flag.scss b/_sass/page/inuitcss/objects/_objects.flag.scss
deleted file mode 100755
index 8c8ecc7..0000000
--- a/_sass/page/inuitcss/objects/_objects.flag.scss
+++ /dev/null
@@ -1,231 +0,0 @@
-/* ==========================================================================
-   #FLAG
-   ========================================================================== */
-
-/**
- * The flag object is a design pattern similar to the media object, however it
- * utilises `display: table[-cell];` to give us control over the vertical
- * alignments of the text and image.
- *
- * http://csswizardry.com/2013/05/the-flag-object/
- *
- * 1. Allows us to control vertical alignments.
- * 2. Force the object to be the full width of its parent. Combined with [1],
- *    this makes the object behave in a quasi-`display: block;` manner.
- * 3. Reset inherited `border-spacing` declarations.
- */
-
-.o-flag {
-  display: table; /* [1] */
-  width: 100%; /* [2] */
-  border-spacing: 0; /* [3] */
-}
-
-  /**
-   * Items within a flag object. There should only ever be one of each.
-   *
-   * 1. Default to aligning content to their middles.
-   */
-
-  .o-flag__img,
-  .o-flag__body {
-    display: table-cell;
-    vertical-align: middle; /* [1] */
-  }
-
-  /**
-   * Flag images have a space between them and the body of the object.
-   *
-   * 1. Force `.flag__img` to take up as little space as possible:
-   *    https://pixelsvsbytes.com/2012/02/this-css-layout-grid-is-no-holy-grail/
-   */
-
-  .o-flag__img {
-    width: 1px; /* [1] */
-    padding-right: $inuit-global-spacing-unit;
-
-    /**
-     * 1. Fixes problem with images disappearing.
-     *
-     *    The direct child selector '>' needs to remain in order for nested flag
-     *    objects to not inherit their parent’s formatting. In case the image tag
-     *    is wrapped into another tag, e.g. an anchor for linking reasons, it will
-     *    disappear. In that case try wrapping the whole o-flag__img object into
-     *    an anchor tag.
-     *
-     *    E.g.:
-     *
-     *      <a href="/">
-     *        <div class="o-flag__img">
-     *          <img src="./link/to/image.jpg" alt="image alt text">
-     *        </div>
-     *      </a>
-     */
-
-    > img {
-      max-width: none; /* [1] */
-    }
-
-  }
-
-  /**
-   * The container for the main content of the flag object.
-   *
-   * 1. Forces the `.flag__body` to take up all remaining space.
-   */
-
-  .o-flag__body {
-    width: auto; /* [1] */
-
-    &,
-    > :last-child {
-      margin-bottom: 0;
-    }
-
-  }
-
-
-
-
-
-/* Size variants
-   ========================================================================== */
-
-.o-flag--flush {
-
-  > .o-flag__img {
-    padding-right: 0;
-    padding-left:  0;
-  }
-
-}
-
-
-.o-flag--tiny {
-
-  > .o-flag__img {
-    padding-right: $inuit-global-spacing-unit-tiny;
-  }
-
-  &.o-flag--reverse {
-
-    > .o-flag__img {
-      padding-right: 0;
-      padding-left: $inuit-global-spacing-unit-tiny;
-    }
-
-  }
-
-}
-
-
-.o-flag--small {
-
-  > .o-flag__img {
-    padding-right: $inuit-global-spacing-unit-small;
-  }
-
-  &.o-flag--reverse {
-
-    > .o-flag__img {
-      padding-right: 0;
-      padding-left: $inuit-global-spacing-unit-small;
-    }
-
-  }
-
-}
-
-
-.o-flag--large {
-
-  > .o-flag__img {
-    padding-right: $inuit-global-spacing-unit-large;
-  }
-
-  &.o-flag--reverse {
-
-    > .o-flag__img {
-      padding-right: 0;
-      padding-left: $inuit-global-spacing-unit-large;
-    }
-
-  }
-
-}
-
-
-.o-flag--huge {
-
-  > .o-flag__img {
-    padding-right: $inuit-global-spacing-unit-huge;
-  }
-
-  &.o-flag--reverse {
-
-    > .o-flag__img {
-      padding-right: 0;
-      padding-left: $inuit-global-spacing-unit-huge;
-    }
-
-  }
-
-}
-
-
-
-
-
-/* Reversed flag
-   ========================================================================== */
-
-/**
- * 1. Swap the rendered direction of the object…
- * 2. …and reset it.
- * 3. Reassign margins to the correct sides.
- */
-
-.o-flag--reverse {
-  direction: rtl; /* [1] */
-
-  > .o-flag__img,
-  > .o-flag__body {
-    direction: ltr; /* [2] */
-  }
-
-  > .o-flag__img {
-    padding-right: 0; /* [3] */
-    padding-left: $inuit-global-spacing-unit; /* [3] */
-  }
-
-}
-
-
-
-
-
-/* Alignment variants
-   ========================================================================== */
-
-/**
- * Vertically align the image- and body-content differently. Defaults to middle.
- */
-
-.o-flag--top {
-
-  > .o-flag__img,
-  > .o-flag__body {
-    vertical-align: top;
-  }
-
-}
-
-
-.o-flag--bottom {
-
-  > .o-flag__img,
-  > .o-flag__body {
-    vertical-align: bottom;
-  }
-
-}
diff --git a/_sass/page/inuitcss/objects/_objects.layout.scss b/_sass/page/inuitcss/objects/_objects.layout.scss
deleted file mode 100755
index 9843385..0000000
--- a/_sass/page/inuitcss/objects/_objects.layout.scss
+++ /dev/null
@@ -1,308 +0,0 @@
-/* ==========================================================================
-   #LAYOUT
-   ========================================================================== */
-
-/**
- * Grid-like layout system.
- *
- * The layout object provides us with a column-style layout system. This file
- * contains the basic structural elements, but classes should be complemented
- * with width utilities, for example:
- *
- *   <div class="o-layout">
- *     <div class="o-layout__item  u-1/2">
- *     </div>
- *     <div class="o-layout__item  u-1/2">
- *     </div>
- *   </div>
- *
- * The above will create a two-column structure in which each column will
- * fluidly fill half of the width of the parent. We can have more complex
- * systems:
- *
- *   <div class="o-layout">
- *     <div class="o-layout__item  u-1/1  u-1/3@medium">
- *     </div>
- *     <div class="o-layout__item  u-1/2  u-1/3@medium">
- *     </div>
- *     <div class="o-layout__item  u-1/2  u-1/3@medium">
- *     </div>
- *   </div>
- *
- * The above will create a system in which the first item will be 100% width
- * until we enter our medium breakpoint, when it will become 33.333% width. The
- * second and third items will be 50% of their parent, until they also become
- * 33.333% width at the medium breakpoint.
- *
- * We can also manipulate entire layout systems by adding a series of modifiers
- * to the `.o-layout` block. For example:
- *
- *   <div class="o-layout  o-layout--reverse">
- *
- * This will reverse the displayed order of the system so that it runs in the
- * opposite order to our source, effectively flipping the system over.
- *
- *   <div class="o-layout  o-layout--[right|center]">
- *
- * This will cause the system to fill up from either the centre or the right
- * hand side. Default behaviour is to fill up the layout system from the left.
- *
- * There are plenty more options available to us: explore them below.
- */
-
-// By default we use the `font-size: 0;` trick to remove whitespace between
-// items. Set this to true in order to use a markup-based strategy like
-// commenting out whitespace or minifying HTML.
-$inuit-use-markup-fix: false !default;
-
-
-
-
-
-/* Default/mandatory classes
-   ========================================================================== */
-
-/**
- * 1. Allows us to use the layout object on any type of element.
- * 2. We need to defensively reset any box-model properties.
- * 3. Use the negative margin trick for multi-row grids:
- *    http://csswizardry.com/2011/08/building-better-grid-systems/
- */
-
-.o-layout {
-  display: block; /* [1] */
-  margin:  0; /* [2] */
-  padding: 0; /* [2] */
-  list-style: none; /* [1] */
-  margin-left: -$inuit-global-spacing-unit; /* [3] */
-
-  @if ($inuit-use-markup-fix == false) {
-    font-size: 0;
-  }
-
-}
-
-  /**
-   * 1. Required in order to combine fluid widths with fixed gutters.
-   * 2. Allows us to manipulate grids vertically, with text-level properties,
-   *    etc.
-   * 3. Default item alignment is with the tops of each other, like most
-   *    traditional grid/layout systems.
-   * 4. By default, all layout items are full-width (mobile first).
-   * 5. Gutters provided by left padding:
-   *    http://csswizardry.com/2011/08/building-better-grid-systems/
-   * 6. Fallback for old IEs not supporting `rem` values.
-   */
-
-  .o-layout__item {
-    box-sizing: border-box; /* [1] */
-    display: inline-block; /* [2] */
-    vertical-align: top; /* [3] */
-    width: 100%; /* [4] */
-    padding-left: $inuit-global-spacing-unit; /* [5] */
-
-    @if ($inuit-use-markup-fix == false) {
-      font-size: $inuit-global-font-size; /* [6] */
-      font-size: 1rem;
-    }
-
-  }
-
-
-
-
-
-/* Gutter size modifiers
-   ========================================================================== */
-
-.o-layout--flush {
-  margin-left: 0;
-
-  > .o-layout__item {
-    padding-left: 0;
-  }
-
-}
-
-
-.o-layout--tiny {
-  margin-left: -$inuit-global-spacing-unit-tiny;
-
-  > .o-layout__item {
-    padding-left: $inuit-global-spacing-unit-tiny;
-  }
-
-}
-
-
-.o-layout--small {
-  margin-left: -$inuit-global-spacing-unit-small;
-
-  > .o-layout__item {
-    padding-left: $inuit-global-spacing-unit-small;
-  }
-
-}
-
-
-.o-layout--large {
-  margin-left: -$inuit-global-spacing-unit-large;
-
-  > .o-layout__item {
-    padding-left: $inuit-global-spacing-unit-large;
-  }
-
-}
-
-
-.o-layout--huge {
-  margin-left: -$inuit-global-spacing-unit-huge;
-
-  > .o-layout__item {
-    padding-left: $inuit-global-spacing-unit-huge;
-  }
-
-}
-
-
-
-
-
-/* Vertical alignment modifiers
-   ========================================================================== */
-
-/**
- * Align all grid items to the middles of each other.
- */
-
-.o-layout--middle {
-
-  > .o-layout__item {
-    vertical-align: middle;
-  }
-
-}
-
-
-/**
- * Align all grid items to the bottoms of each other.
- */
-
-.o-layout--bottom {
-
-  > .o-layout__item {
-    vertical-align: bottom;
-  }
-
-}
-
-
-/**
- * Stretch all grid items of each row to have an equal-height.
- * Please be aware that this modifier class doesn’t take any effect in IE9 and
- * below and other older browsers due to the lack of `display: flex` support.
- */
-
-.o-layout--stretch {
-  display: flex;
-  flex-wrap: wrap;
-
-  > .o-layout__item {
-    display: flex;
-  }
-
-  &.o-layout--center {
-    justify-content: center;
-  }
-
-  &.o-layout--right {
-    justify-content: flex-end;
-  }
-
-  &.o-layout--left {
-    justify-content: flex-start;
-  }
-
-}
-
-
-
-
-
-/* Fill order modifiers
-   ========================================================================== */
-
-/**
- * Fill up the layout system from the centre.
- */
-
-.o-layout--center {
-  text-align: center;
-
-  > .o-layout__item {
-    text-align: left;
-  }
-
-}
-
-
-/**
- * Fill up the layout system from the right-hand side.
- */
-
-.o-layout--right {
-  text-align: right;
-
-  > .o-layout__item {
-    text-align: left;
-  }
-
-}
-
-
-/**
- * Fill up the layout system from the left-hand side. This will likely only be
- * needed when using in conjunction with `.o-layout--reverse`.
- */
-
-.o-layout--left {
-  text-align: left;
-
-  > .o-layout__item {
-    text-align: left;
-  }
-
-}
-
-
-/**
- * Reverse the rendered order of the grid system.
- */
-
-.o-layout--reverse {
-  direction: rtl;
-
-  > .o-layout__item {
-    direction: ltr;
-  }
-
-}
-
-
-
-
-
-/* Auto-widths modifier
-   ========================================================================== */
-
-/**
- * Cause layout items to take up a non-explicit amount of width.
- */
-
-.o-layout--auto {
-
-  > .o-layout__item {
-    width: auto;
-  }
-
-}
diff --git a/_sass/page/inuitcss/objects/_objects.list-bare.scss b/_sass/page/inuitcss/objects/_objects.list-bare.scss
deleted file mode 100755
index 372e1d6..0000000
--- a/_sass/page/inuitcss/objects/_objects.list-bare.scss
+++ /dev/null
@@ -1,20 +0,0 @@
-/* ==========================================================================
-   #LIST-BARE
-   ========================================================================== */
-
-/**
- * Strip list-like appearance from lists by removing their bullets and any
- * indentation.
- *
- * Note: Declaring the item class might not be necessary everywhere,
- * but is for example in <dl> lists for the <dd> children.
- */
-
-.o-list-bare {
-  list-style: none;
-  margin-left: 0;
-}
-
-  .o-list-bare__item {
-    margin-left: 0;
-  }
diff --git a/_sass/page/inuitcss/objects/_objects.list-inline.scss b/_sass/page/inuitcss/objects/_objects.list-inline.scss
deleted file mode 100755
index 8b4fda4..0000000
--- a/_sass/page/inuitcss/objects/_objects.list-inline.scss
+++ /dev/null
@@ -1,16 +0,0 @@
-/* ==========================================================================
-   #LIST-INLINE
-   ========================================================================== */
-
-/**
- * The list-inline object simply displays a list of items in one line.
- */
-
-.o-list-inline {
-  margin-left: 0;
-  list-style: none;
-}
-
-  .o-list-inline__item {
-    display: inline-block;
-  }
diff --git a/_sass/page/inuitcss/objects/_objects.media.scss b/_sass/page/inuitcss/objects/_objects.media.scss
deleted file mode 100755
index 43c7e5b..0000000
--- a/_sass/page/inuitcss/objects/_objects.media.scss
+++ /dev/null
@@ -1,144 +0,0 @@
-/* ==========================================================================
-   #MEDIA
-   ========================================================================== */
-
-/**
- * Place any image- and text-like content side-by-side, as per:
- * http://www.stubbornella.org/content/2010/06/25/the-media-object-saves-hundreds-of-lines-of-code
- */
-
-.o-media {
-  @include inuit-clearfix();
-  display: block;
-}
-
-  .o-media__img {
-    float: left;
-    margin-right: $inuit-global-spacing-unit;
-
-    > img {
-      display: block;
-    }
-
-  }
-
-  .o-media__body {
-    overflow: hidden;
-    display: block;
-
-    &,
-    > :last-child {
-      margin-bottom: 0;
-    }
-
-  }
-
-
-
-
-
-/* Size variants
-   ========================================================================== */
-
-/**
- * Modify the amount of space between our image and our text. We also have
- * reversible options for all available sizes.
- */
-
-.o-media--flush {
-
-  > .o-media__img {
-    margin-right: 0;
-    margin-left: 0;
- }
-
-}
-
-
-.o-media--tiny {
-
-  > .o-media__img {
-    margin-right: $inuit-global-spacing-unit-tiny;
-  }
-
-  &.o-media--reverse {
-
-    > .o-media__img {
-      margin-right: 0;
-      margin-left: $inuit-global-spacing-unit-tiny;
-    }
-
-  }
-
-}
-
-
-.o-media--small {
-
-  > .o-media__img {
-    margin-right: $inuit-global-spacing-unit-small;
-  }
-
-  &.o-media--reverse {
-
-    > .o-media__img {
-      margin-right: 0;
-      margin-left: $inuit-global-spacing-unit-small;
-    }
-
-  }
-
-}
-
-
-.o-media--large {
-
-  > .o-media__img {
-    margin-right: $inuit-global-spacing-unit-large;
-  }
-
-  &.o-media--reverse {
-
-    > .o-media__img {
-      margin-right: 0;
-      margin-left: $inuit-global-spacing-unit-large;
-    }
-
-  }
-
-}
-
-
-.o-media--huge {
-
-  > .o-media__img {
-    margin-right: $inuit-global-spacing-unit-huge;
-  }
-
-  &.o-media--reverse {
-
-    > .o-media__img {
-      margin-right: 0;
-      margin-left: $inuit-global-spacing-unit-huge;
-    }
-
-  }
-
-}
-
-
-
-
-
-/* Reversed media objects
-   ========================================================================== */
-
-.o-media--reverse {
-
-  > .o-media__img {
-    float: right;
-    margin-right: 0;
-    margin-left: $inuit-global-spacing-unit;
-  }
-
-}
diff --git a/_sass/page/inuitcss/objects/_objects.pack.scss b/_sass/page/inuitcss/objects/_objects.pack.scss
deleted file mode 100755
index 06864dd..0000000
--- a/_sass/page/inuitcss/objects/_objects.pack.scss
+++ /dev/null
@@ -1,96 +0,0 @@
-/* ==========================================================================
-   #PACK
-   ========================================================================== */
-
-/**
- * The pack object simply causes any number of elements pack up horizontally to
- * automatically fill an equal, fluid width of their parent.
- *
- * 1. Fill all available space.
- * 2. Remove any leftover styling from lists.
- * 3. Cause children to be automatically equally sized.
- */
-
-.o-pack {
-  width: 100%; /* [1] */
-  margin-left: 0; /* [2] */
-  display: table;
-  table-layout: fixed; /* [3] */
-}
-
-  /**
-   * 1. Cause children to adopt table-like structure.
-   * 2. Default item alignment is with the tops of each other.
-   */
-
-  .o-pack__item {
-    display: table-cell; /* [1] */
-    vertical-align: top; /* [2] */
-
-
-    /* Vertical alignment variants
-       ====================================================================== */
-
-    .o-pack--middle > & {
-      vertical-align: middle;
-    }
-
-    .o-pack--bottom > & {
-      vertical-align: bottom;
-    }
-
-  }
-
-
-
-
-
-/* Unequal-width items
-   ========================================================================== */
-
-.o-pack--auto {
-  table-layout: auto;
-}
-
-
-
-
-
-/* Size variants
-   ========================================================================== */
-
-.o-pack--tiny {
-  border-spacing: $inuit-global-spacing-unit-tiny;
-}
-
-.o-pack--small {
-  border-spacing: $inuit-global-spacing-unit-small;
-}
-
-.o-pack--default {
-  border-spacing: $inuit-global-spacing-unit;
-}
-
-.o-pack--large {
-  border-spacing: $inuit-global-spacing-unit-large;
-}
-
-.o-pack--huge {
-  border-spacing: $inuit-global-spacing-unit-huge;
-}
-
-
-
-
-
-/* Reversed order packs
-   ========================================================================== */
-
-.o-pack--reverse {
-  direction: rtl;
-
-  > .o-pack__item {
-    direction: ltr;
-  }
-
-}
diff --git a/_sass/page/inuitcss/objects/_objects.ratio.scss b/_sass/page/inuitcss/objects/_objects.ratio.scss
deleted file mode 100755
index 216038b..0000000
--- a/_sass/page/inuitcss/objects/_objects.ratio.scss
+++ /dev/null
@@ -1,117 +0,0 @@
-/* ==========================================================================
-   #RATIO
-   ========================================================================== */
-
-// A list of aspect ratios that get generated as modifier classes.
-// You should predefine it with only the ratios and names your project needs.
-//
-// The map keys are the strings used in the generated class names, and they can
-// follow any convention, as long as they are properly escaped strings. i.e.:
-//
-//   $inuit-ratios: (
-//     "2\\:1"         : (2:1),
-//     "4-by-3"        : (4:3),
-//     "full-hd"       : (16:9),
-//     "card-image"    : (2:3),
-//     "golden-ratio"  : (1.618:1) -> non-integers are okay
-//   ) !default;
-
-$inuit-ratios: (
-  "2\\:1"   : (2:1),
-  "4\\:3"   : (4:3),
-  "16\\:9"  : (16:9)
-) !default;
-
-
-
-/**
- * Create ratio-bound content blocks, to keep media (e.g. images, videos) in
- * their correct aspect ratios.
- *
- * http://alistapart.com/article/creating-intrinsic-ratios-for-video
- *
- * 1. Default is a 1:1 ratio (i.e. a perfect square).
- */
-
-.o-ratio {
-  position: relative;
-  display: block;
-
-  &:before {
-    content: "";
-    display: block;
-    width: 100%;
-    padding-bottom: 100%; /* [1] */
-  }
-
-}
-
-  .o-ratio__content,
-  .o-ratio > iframe,
-  .o-ratio > embed,
-  .o-ratio > object {
-    position: absolute;
-    top:    0;
-    bottom: 0;
-    left:   0;
-    height: 100%;
-    width:  100%;
-  }
-
-
-
-
-
-/* Ratio variants.
-   ========================================================================== */
-
-/**
- * Generate a series of ratio classes to be used like so:
- *
- *   <div class="o-ratio  o-ratio--golden-ratio">
- *
- */
-
-@each $ratio-name, $ratio-value in $inuit-ratios {
-
-  @each $antecedent, $consequent in $ratio-value {
-
-    @if (type-of($antecedent) != number) {
-      @error "`#{$antecedent}` needs to be a number.";
-    }
-
-    @if (type-of($consequent) != number) {
-      @error "`#{$consequent}` needs to be a number.";
-    }
-
-    .o-ratio--#{$ratio-name}:before {
-      padding-bottom: ($consequent/$antecedent) * 100%;
-    }
-
-  }
-
-}
-
-
-
-
-
-/* Contain modifier.
-   ========================================================================== */
-
-/**
- * Only works with image content.
- * Contains the image to the boundaries, without cropping or stretching it.
- */
-
-.o-ratio--img-contain {
-
-  > .o-ratio__content:before {
-    height: auto;
-    margin: auto;
-    max-height: 100%;
-    max-width: 100%;
-    width: auto;
-  }
-
-}
diff --git a/_sass/page/inuitcss/objects/_objects.table.scss b/_sass/page/inuitcss/objects/_objects.table.scss
deleted file mode 100755
index 4cb5e75..0000000
--- a/_sass/page/inuitcss/objects/_objects.table.scss
+++ /dev/null
@@ -1,76 +0,0 @@
-/* ==========================================================================
-   #TABLE
-   ========================================================================== */
-
-/**
- * A simple object for manipulating the structure of HTML `table`s.
- */
-
-.o-table {
-  width: 100%;
-}
-
-
-
-
-
-/* Equal-width table cells
-   ========================================================================== */
-
-/**
- * `table-layout: fixed` forces all cells within a table to occupy the same
- * width as each other. This also has performance benefits: because the browser
- * does not need to (re)calculate cell dimensions based on content it discovers,
- * the table can be rendered very quickly. Further reading:
- * https://developer.mozilla.org/en-US/docs/Web/CSS/table-layout#Values
- */
-
-.o-table--fixed {
-  table-layout: fixed;
-}
-
-
-
-
-
-/* Size variants
-   ========================================================================== */
-
-.o-table--tiny {
-
-  th,
-  td {
-    padding: $inuit-global-spacing-unit-tiny;
-  }
-
-}
-
-
-.o-table--small {
-
-  th,
-  td {
-    padding: $inuit-global-spacing-unit-small;
-  }
-
-}
-
-
-.o-table--large {
-
-  th,
-  td {
-    padding: $inuit-global-spacing-unit-large;
-  }
-
-}
-
-
-.o-table--huge {
-
-  th,
-  td {
-    padding: $inuit-global-spacing-unit-huge;
-  }
-
-}
diff --git a/_sass/page/inuitcss/objects/_objects.wrapper.scss b/_sass/page/inuitcss/objects/_objects.wrapper.scss
deleted file mode 100755
index 11b46d5..0000000
--- a/_sass/page/inuitcss/objects/_objects.wrapper.scss
+++ /dev/null
@@ -1,13 +0,0 @@
-/* ==========================================================================
-   #WRAPPER
-   ========================================================================== */
-
-/**
- * Page-level constraining and wrapping elements.
- */
-
-$inuit-wrapper-width: 1200px !default;
-
-@if (type-of($inuit-wrapper-width) != number) {
-  @error "`#{$inuit-wrapper-width}` needs to be a number.";
-}
diff --git a/_sass/page/inuitcss/settings/_example.settings.config.scss b/_sass/page/inuitcss/settings/_example.settings.config.scss
deleted file mode 100755
index 803b8eb..0000000
--- a/_sass/page/inuitcss/settings/_example.settings.config.scss
+++ /dev/null
@@ -1,24 +0,0 @@
-///* ========================================================================
-//   #CONFIG
-//   ======================================================================== */
-
-// A map of global config settings. Define any project-level configuration,
-// feature switches, etc. in here.
-
-$inuit-config: (
-  env: dev,
-  healthcheck: false,
-  debug: true,
-);
-
-// You can access data in this map using the following function:
-//
-// inuit-config(<key>)
-//
-// Example usage:
-//
-// @if (inuit-config(debug) == true) { ...  }
-
-@function inuit-config($key) {
-  @return map-get($inuit-config, $key);
-}
diff --git a/_sass/page/inuitcss/settings/_example.settings.global.scss b/_sass/page/inuitcss/settings/_example.settings.global.scss
deleted file mode 100755
index 80ce708..0000000
--- a/_sass/page/inuitcss/settings/_example.settings.global.scss
+++ /dev/null
@@ -1,13 +0,0 @@
-///* ========================================================================
-//   #GLOBAL
-//   ======================================================================== */
-
-// The global settings file contains any project-wide variables; things that
-// need to be made available to the entire codebase.
-
-
-
-// Standardise some UI treatments.
-
-$global-radius: 3px !default;
-$global-transition: all 300ms ease-in-out !default;
diff --git a/_sass/page/inuitcss/settings/_settings.core.scss b/_sass/page/inuitcss/settings/_settings.core.scss
deleted file mode 100755
index 730e595..0000000
--- a/_sass/page/inuitcss/settings/_settings.core.scss
+++ /dev/null
@@ -1,92 +0,0 @@
-///* ========================================================================
-//   #CORE
-//   ======================================================================== */
-
-// This core file sets up inuitcss’ most important setup variables. They
-// underpin a lot of how the framework functions and should be modified and
-// preconfigured with caution.
-
-
-
-// Baseline grid lines height.
-// Every spacing metric should be based on this.
-
-$inuit-global-baseline:     6px !default;
-
-
-
-// How many grid lines should our spacing unit variants span?
-// Each value should be an unitless integer.
-
-$inuit-global-spacing-unit-factor-tiny:   1 !default;   // 6px
-$inuit-global-spacing-unit-factor-small:  2 !default;   // 12px
-$inuit-global-spacing-unit-factor:        4 !default;   // 24px
-$inuit-global-spacing-unit-factor-large:  8 !default;   // 48px
-$inuit-global-spacing-unit-factor-huge:   16 !default;  // 96px
-
-
-
-// Spacing values are determined based on your project’s global baseline grid.
-// It is not recommended that you modify these following variables
-// (it can break your vertical rhythm), but if you need to, you can.
-
-$inuit-global-spacing-unit:       $inuit-global-baseline * $inuit-global-spacing-unit-factor !default;
-$inuit-global-spacing-unit-tiny:  $inuit-global-baseline * $inuit-global-spacing-unit-factor-tiny !default;
-$inuit-global-spacing-unit-small: $inuit-global-baseline * $inuit-global-spacing-unit-factor-small !default;
-$inuit-global-spacing-unit-large: $inuit-global-baseline * $inuit-global-spacing-unit-factor-large !default;
-$inuit-global-spacing-unit-huge:  $inuit-global-baseline * $inuit-global-spacing-unit-factor-huge !default;
-
-
-
-// Base typographical styles.
-
-$inuit-global-font-size:    16px !default;
-$inuit-global-line-height:  $inuit-global-spacing-unit !default;
-
-
-
-
-
-// Check that the chosen font rules are pixel numbers.
-
-@each $_inuit-font-globals in
-      $inuit-global-font-size
-      $inuit-global-line-height {
-
-  @if (type-of($_inuit-font-globals) == number) {
-
-    @if (unit($_inuit-font-globals) != "px") {
-      @error "`#{$_inuit-font-globals}` needs to be a pixel value.";
-    }
-
-  } @else {
-    @error "`#{$_inuit-font-globals}` needs to be a number.";
-  }
-
-}
-
-
-
-// Check that the chosen size factors are unitless, integer numbers.
-
-@each $_inuit-spacing-unit in
-      $inuit-global-spacing-unit-factor-tiny
-      $inuit-global-spacing-unit-factor-small
-      $inuit-global-spacing-unit-factor-large
-      $inuit-global-spacing-unit-factor-huge {
-
-  @if (type-of($_inuit-spacing-unit) == number) {
-
-    @if (unitless($_inuit-spacing-unit) == false) {
-      @error "`#{$_inuit-spacing-unit}` needs to be unitless.";
-    }
-
-    @if ($_inuit-spacing-unit != ceil($_inuit-spacing-unit)) {
-      @error "`#{$_inuit-spacing-unit}` needs to be an integer.";
-    }
-
-  } @else {
-    @error "`#{$_inuit-spacing-unit}` needs to be a number.";
-  }
-
-}
diff --git a/_sass/page/inuitcss/tools/_tools.clearfix.scss b/_sass/page/inuitcss/tools/_tools.clearfix.scss
deleted file mode 100755
index a348ac6..0000000
--- a/_sass/page/inuitcss/tools/_tools.clearfix.scss
+++ /dev/null
@@ -1,20 +0,0 @@
-///* ========================================================================
-//   #CLEARFIX
-//   ======================================================================== */
-
-// Mixin to drop micro clearfix into a selector. Further reading:
-// http://www.cssmojo.com/the-very-latest-clearfix-reloaded/
-//
-// .usage {
-//   @include inuit-clearfix();
-// }
-
-@mixin inuit-clearfix() {
-
-  &:after {
-    content: "" !important;
-    display: block !important;
-    clear: both !important;
-  }
-
-}
diff --git a/_sass/page/inuitcss/tools/_tools.font-size.scss b/_sass/page/inuitcss/tools/_tools.font-size.scss
deleted file mode 100755
index 290731a..0000000
--- a/_sass/page/inuitcss/tools/_tools.font-size.scss
+++ /dev/null
@@ -1,72 +0,0 @@
-///* ========================================================================
-//   #FONT-SIZE
-//   ======================================================================== */
-
-// Generates a rem font-size (with pixel fallback) and a baseline-compatible
-// unitless line-height from a pixel font-size value. Basic usage is simply:
-//
-//   @include inuit-font-size(18px);
-//
-// You can force a specific line-height by passing it as the second argument:
-//
-//   @include inuit-font-size(16px, 1);
-//
-// You can also modify the line-height by increments, while staying in the
-// baseline grid, by setting the `$modifier` parameter. It takes a positive
-// or negative integer, and it will add or remove "lines" to the  generated
-// line-height. This is the recomended way to do it, unless you really need
-// an absolute value. i.e.:
-//
-//   // add 2 lines:
-//   @include inuit-font-size(24px, $modifier: +2);
-//
-//   // subtract 1 line:
-//   @include inuit-font-size(24px, $modifier: -1);
-
-@mixin inuit-font-size($font-size, $line-height: auto, $modifier: 0, $important: false) {
-
-  @if (type-of($font-size) == number) {
-    @if (unit($font-size) != "px") {
-      @error "`#{$font-size}` needs to be a pixel value.";
-    }
-  } @else {
-    @error "`#{$font-size}` needs to be a number.";
-  }
-
-  @if ($important == true) {
-    $important: !important;
-  } @else if ($important == false) {
-    $important: null;
-  } @else {
-    @error "`#{$important}` needs to be `true` or `false`.";
-  }
-
-  // We provide a `px` fallback for old IEs not supporting `rem` values.
-  font-size: $font-size $important;
-  font-size: ($font-size / $inuit-global-font-size) * 1rem $important;
-
-  @if ($line-height == "auto") {
-
-    // Define how many grid lines each text line should span.
-    // By default, we set it to the minimum number of lines necessary
-    // in order to contain the defined font-size, +1 for some breathing room.
-    // This can be modified with the `$modifier` parameter.
-    $lines: ceil($font-size / $inuit-global-baseline) + $modifier + 1;
-    $line-height: $lines * $inuit-global-baseline;
-
-    line-height: ($line-height / $font-size) $important;
-  }
-
-  @else {
-
-    @if (type-of($line-height) == number or $line-height == "inherit" or $line-height == "normal") {
-      line-height: $line-height $important;
-    }
-
-    @else if ($line-height != 'none' and $line-height != false) {
-      @error "D’oh! `#{$line-height}` is not a valid value for `$line-height`."
-    }
-
-  }
-
-}
diff --git a/_sass/page/inuitcss/tools/_tools.hidden.scss b/_sass/page/inuitcss/tools/_tools.hidden.scss
deleted file mode 100755
index 8fe2491..0000000
--- a/_sass/page/inuitcss/tools/_tools.hidden.scss
+++ /dev/null
@@ -1,18 +0,0 @@
-///* ========================================================================
-//   #HIDDEN-VISUALLY
-//   ======================================================================== */
-
-// Mixin to quickly apply accessible hiding to elements.
-
-@mixin inuit-hidden-visually() {
-  border: 0 !important;
-  clip: rect(0 0 0 0) !important;
-  clip-path: inset(50%) !important;
-  height: 1px !important;
-  margin: -1px !important;
-  overflow: hidden !important;
-  padding: 0 !important;
-  position: absolute !important;
-  white-space: nowrap !important;
-  width: 1px !important;
-}
diff --git a/_sass/page/inuitcss/tools/_tools.mq.scss b/_sass/page/inuitcss/tools/_tools.mq.scss
deleted file mode 100755
index 51904a4..0000000
--- a/_sass/page/inuitcss/tools/_tools.mq.scss
+++ /dev/null
@@ -1,351 +0,0 @@
-@charset "UTF-8"; // Fixes an issue where Ruby locale is not set properly
-                  // See https://github.com/sass-mq/sass-mq/pull/10
-
-/// Base font size on the `<body>` element
-/// @type Number (unit)
-$mq-base-font-size: 16px !default;
-
-/// Responsive mode
-///
-/// Set to `false` to enable support for browsers that do not support @media queries,
-/// (IE <= 8, Firefox <= 3, Opera <= 9)
-///
-/// You could create a stylesheet served exclusively to older browsers,
-/// where @media queries are rasterized
-///
-/// @example scss
-///  // old-ie.scss
-///  $mq-responsive: false;
-///  @import 'main'; // @media queries in this file will be rasterized up to $mq-static-breakpoint
-///                   // larger breakpoints will be ignored
-///
-/// @type Boolean
-/// @link https://github.com/sass-mq/sass-mq#responsive-mode-off Disabled responsive mode documentation
-$mq-responsive: true !default;
-
-/// Breakpoint list
-///
-/// Name your breakpoints in a way that creates a ubiquitous language
-/// across team members. It will improve communication between
-/// stakeholders, designers, developers, and testers.
-///
-/// @type Map
-/// @link https://github.com/sass-mq/sass-mq#seeing-the-currently-active-breakpoint Full documentation and examples
-$mq-breakpoints: (
-    mobile:  375px,
-    tablet:  768px,
-    laptop:  1024px,
-    laptop_mid:  1200px,
-    desktop: 1440px,
-    desktop_wide: 1650px,
-    wide:    2560px
-) !default;
-
-/// Static breakpoint (for fixed-width layouts)
-///
-/// Define the breakpoint from $mq-breakpoints that should
-/// be used as the target width for the fixed-width layout
-/// (i.e. when $mq-responsive is set to 'false') in a old-ie.scss
-///
-/// @example scss
-///  // tablet-only.scss
-///  //
-///  // Ignore all styles above tablet breakpoint,
-///  // and fix the styles (e.g. layout) at tablet width
-///  $mq-responsive: false;
-///  $mq-static-breakpoint: tablet;
-///  @import 'main'; // @media queries in this file will be rasterized up to tablet
-///                   // larger breakpoints will be ignored
-///
-/// @type String
-/// @link https://github.com/sass-mq/sass-mq#adding-custom-breakpoints Full documentation and examples
-$mq-static-breakpoint: desktop !default;
-
-/// Show breakpoints in the top right corner
-///
-/// If you want to display the currently active breakpoint in the top
-/// right corner of your site during development, add the breakpoints
-/// to this list, ordered by width, e.g. (mobile, tablet, desktop).
-///
-/// @example scss
-///   $mq-show-breakpoints: (mobile, tablet, desktop);
-///   @import 'path/to/mq';
-///
-/// @type map
-$mq-show-breakpoints: () !default;
-
-/// Customize the media type (e.g. `@media screen` or `@media print`)
-/// By default sass-mq uses an "all" media type (`@media all and …`)
-///
-/// @type String
-/// @link https://github.com/sass-mq/sass-mq#changing-media-type Full documentation and examples
-$mq-media-type: all !default;
-
-/// Convert pixels to ems
-///
-/// @param {Number} $px - value to convert
-/// @param {Number} $base-font-size ($mq-base-font-size) - `<body>` font size
-///
-/// @example scss
-///  $font-size-in-ems: mq-px2em(16px);
-///  p { font-size: mq-px2em(16px); }
-///
-/// @requires $mq-base-font-size
-/// @returns {Number}
-@function mq-px2em($px, $base-font-size: $mq-base-font-size) {
-    @if unitless($px) {
-        @warn "Assuming #{$px} to be in pixels, attempting to convert it into pixels.";
-        @return mq-px2em($px * 1px, $base-font-size);
-    } @else if unit($px) == em {
-        @return $px;
-    }
-    @return ($px / $base-font-size) * 1em;
-}
-
-/// Get a breakpoint's width
-///
-/// @param {String} $name - Name of the breakpoint. One of $mq-breakpoints
-///
-/// @example scss
-///  $tablet-width: mq-get-breakpoint-width(tablet);
-///  @media (min-width: mq-get-breakpoint-width(desktop)) {}
-///
-/// @requires {Variable} $mq-breakpoints
-///
-/// @returns {Number} Value in pixels
-@function mq-get-breakpoint-width($name, $breakpoints: $mq-breakpoints) {
-    @if map-has-key($breakpoints, $name) {
-        @return map-get($breakpoints, $name);
-    } @else {
-        @warn "Breakpoint #{$name} wasn't found in $breakpoints.";
-    }
-}
-
-/// Media Query mixin
-///
-/// @param {String | Boolean} $from (false) - One of $mq-breakpoints
-/// @param {String | Boolean} $until (false) - One of $mq-breakpoints
-/// @param {String | Boolean} $and (false) - Additional media query parameters
-/// @param {String} $media-type ($mq-media-type) - Media type: screen, print…
-///
-/// @ignore Undocumented API, for advanced use only:
-/// @ignore @param {Map} $breakpoints ($mq-breakpoints)
-/// @ignore @param {String} $static-breakpoint ($mq-static-breakpoint)
-///
-/// @content styling rules, wrapped into a @media query when $responsive is true
-///
-/// @requires {Variable} $mq-media-type
-/// @requires {Variable} $mq-breakpoints
-/// @requires {Variable} $mq-static-breakpoint
-/// @requires {function} mq-px2em
-/// @requires {function} mq-get-breakpoint-width
-///
-/// @link https://github.com/sass-mq/sass-mq#responsive-mode-on-default Full documentation and examples
-///
-/// @example scss
-///  .element {
-///    @include mq($from: mobile) {
-///      color: red;
-///    }
-///    @include mq($until: tablet) {
-///      color: blue;
-///    }
-///    @include mq(mobile, tablet) {
-///      color: green;
-///    }
-///    @include mq($from: tablet, $and: '(orientation: landscape)') {
-///      color: teal;
-///    }
-///    @include mq(950px) {
-///      color: hotpink;
-///    }
-///    @include mq(tablet, $media-type: screen) {
-///      color: hotpink;
-///    }
-///    // Advanced use:
-///    $my-breakpoints: (L: 900px, XL: 1200px);
-///    @include mq(L, $breakpoints: $my-breakpoints, $static-breakpoint: L) {
-///      color: hotpink;
-///    }
-///  }
-@mixin mq(
-    $from: false,
-    $until: false,
-    $and: false,
-    $media-type: $mq-media-type,
-    $breakpoints: $mq-breakpoints,
-    $responsive: $mq-responsive,
-    $static-breakpoint: $mq-static-breakpoint
-) {
-    $min-width: 0;
-    $max-width: 0;
-    $media-query: '';
-
-    // From: this breakpoint (inclusive)
-    @if $from {
-        @if type-of($from) == number {
-            $min-width: mq-px2em($from);
-        } @else {
-            $min-width: mq-px2em(mq-get-breakpoint-width($from, $breakpoints));
-        }
-    }
-
-    // Until: that breakpoint (exclusive)
-    @if $until {
-        @if type-of($until) == number {
-            $max-width: mq-px2em($until);
-        } @else {
-            $max-width: mq-px2em(mq-get-breakpoint-width($until, $breakpoints)) - .01em;
-        }
-    }
-
-    // Responsive support is disabled, rasterize the output outside @media blocks
-    // The browser will rely on the cascade itself.
-    @if $responsive == false {
-        $static-breakpoint-width: mq-get-breakpoint-width($static-breakpoint, $breakpoints);
-        $target-width: mq-px2em($static-breakpoint-width);
-
-        // Output only rules that start at or span our target width
-        @if (
-            $and == false
-            and $min-width <= $target-width
-            and (
-                $until == false or $max-width >= $target-width
-            )
-            and $media-type != 'print'
-        ) {
-            @content;
-        }
-    }
-
-    // Responsive support is enabled, output rules inside @media queries
-    @else {
-        @if $min-width != 0 { $media-query: '#{$media-query} and (min-width: #{$min-width})'; }
-        @if $max-width != 0 { $media-query: '#{$media-query} and (max-width: #{$max-width})'; }
-        @if $and            { $media-query: '#{$media-query} and #{$and}'; }
-
-        // Remove unnecessary media query prefix 'all and '
-        @if ($media-type == 'all' and $media-query != '') {
-            $media-type: '';
-            $media-query: str-slice(unquote($media-query), 6);
-        }
-
-        @media #{$media-type + $media-query} {
-            @content;
-        }
-    }
-}
-
-/// Quick sort
-///
-/// @author Sam Richards
-/// @access private
-/// @param {List} $list - List to sort
-/// @returns {List} Sorted List
-@function _mq-quick-sort($list) {
-    $less:  ();
-    $equal: ();
-    $large: ();
-
-    @if length($list) > 1 {
-        $seed: nth($list, ceil(length($list) / 2));
-
-        @each $item in $list {
-            @if ($item == $seed) {
-                $equal: append($equal, $item);
-            } @else if ($item < $seed) {
-                $less: append($less, $item);
-            } @else if ($item > $seed) {
-                $large: append($large, $item);
-            }
-        }
-
-        @return join(join(_mq-quick-sort($less), $equal), _mq-quick-sort($large));
-    }
-
-    @return $list;
-}
-
-/// Sort a map by values (works with numbers only)
-///
-/// @access private
-/// @param {Map} $map - Map to sort
-/// @returns {Map} Map sorted by value
-@function _mq-map-sort-by-value($map) {
-    $map-sorted: ();
-    $map-keys: map-keys($map);
-    $map-values: map-values($map);
-    $map-values-sorted: _mq-quick-sort($map-values);
-
-    // Reorder key/value pairs based on key values
-    @each $value in $map-values-sorted {
-        $index: index($map-values, $value);
-        $key: nth($map-keys, $index);
-        $map-sorted: map-merge($map-sorted, ($key: $value));
-
-        // Unset the value in $map-values to prevent the loop
-        // from finding the same index twice
-        $map-values: set-nth($map-values, $index, 0);
-    }
-
-    @return $map-sorted;
-}
-
-/// Add a breakpoint
-///
-/// @param {String} $name - Name of the breakpoint
-/// @param {Number} $width - Width of the breakpoint
-///
-/// @requires {Variable} $mq-breakpoints
-///
-/// @example scss
-///  @include mq-add-breakpoint(tvscreen, 1920px);
-///  @include mq(tvscreen) {}
-@mixin mq-add-breakpoint($name, $width) {
-    $new-breakpoint: ($name: $width);
-    $mq-breakpoints: map-merge($mq-breakpoints, $new-breakpoint) !global;
-    $mq-breakpoints: _mq-map-sort-by-value($mq-breakpoints) !global;
-}
-
-/// Show the active breakpoint in the top right corner of the viewport
-/// @link https://github.com/sass-mq/sass-mq#seeing-the-currently-active-breakpoint
-///
-/// @param {List} $show-breakpoints ($mq-show-breakpoints) - List of breakpoints to show in the top right corner
-/// @param {Map} $breakpoints ($mq-breakpoints) - Breakpoint names and sizes
-///
-/// @requires {Variable} $mq-breakpoints
-/// @requires {Variable} $mq-show-breakpoints
-///
-/// @example scss
-///  // Show breakpoints using global settings
-///  @include mq-show-breakpoints;
-///
-///  // Show breakpoints using custom settings
-///  @include mq-show-breakpoints((L, XL), (S: 300px, L: 800px, XL: 1200px));
-@mixin mq-show-breakpoints($show-breakpoints: $mq-show-breakpoints, $breakpoints: $mq-breakpoints) {
-    body:before {
-        background-color: #FCF8E3;
-        border-bottom: 1px solid #FBEED5;
-        border-left: 1px solid #FBEED5;
-        color: #C09853;
-        font: small-caption;
-        padding: 3px 6px;
-        pointer-events: none;
-        position: fixed;
-        right: 0;
-        top: 0;
-        z-index: 100;
-
-        // Loop through the breakpoints that should be shown
-        @each $show-breakpoint in $show-breakpoints {
-            $width: mq-get-breakpoint-width($show-breakpoint, $breakpoints);
-            @include mq($show-breakpoint, $breakpoints: $breakpoints) {
-                content: "#{$show-breakpoint} ≥ #{$width} (#{mq-px2em($width)})";
-            }
-        }
-    }
-}
-
-@if length($mq-show-breakpoints) > 0 {
-    @include mq-show-breakpoints;
-}
diff --git a/_sass/page/inuitcss/utilities/_utilities.clearfix.scss b/_sass/page/inuitcss/utilities/_utilities.clearfix.scss
deleted file mode 100755
index bc64cab..0000000
--- a/_sass/page/inuitcss/utilities/_utilities.clearfix.scss
+++ /dev/null
@@ -1,11 +0,0 @@
-/* ==========================================================================
-   #CLEARFIX
-   ========================================================================== */
-
-/**
- * Attach our clearfix mixin to a utility class.
- */
-
-.u-clearfix {
-  @include inuit-clearfix();
-}
diff --git a/_sass/page/inuitcss/utilities/_utilities.headings.scss b/_sass/page/inuitcss/utilities/_utilities.headings.scss
deleted file mode 100755
index a4a4a74..0000000
--- a/_sass/page/inuitcss/utilities/_utilities.headings.scss
+++ /dev/null
@@ -1,36 +0,0 @@
-/* ==========================================================================
-   #HEADINGS
-   ========================================================================== */
-
-/**
- * Redefine all of our basic heading styles against utility classes so as to
- * allow for double stranded heading hierarchy, e.g. we semantically need an H2,
- * but we want it to be sized like an H1:
- *
- *   <h2 class="u-h1"></h2>
- *
- */
-
-.u-h1 {
-  @include inuit-font-size($inuit-font-size-h1, $important: true);
-}
-
-.u-h2 {
-  @include inuit-font-size($inuit-font-size-h2, $important: true);
-}
-
-.u-h3 {
-  @include inuit-font-size($inuit-font-size-h3, $important: true);
-}
-
-.u-h4 {
-  @include inuit-font-size($inuit-font-size-h4, $important: true);
-}
-
-.u-h5 {
-  @include inuit-font-size($inuit-font-size-h5, $important: true);
-}
-
-.u-h6 {
-  @include inuit-font-size($inuit-font-size-h6, $important: true);
-}
diff --git a/_sass/page/inuitcss/utilities/_utilities.hide.scss b/_sass/page/inuitcss/utilities/_utilities.hide.scss
deleted file mode 100755
index e0832ed..0000000
--- a/_sass/page/inuitcss/utilities/_utilities.hide.scss
+++ /dev/null
@@ -1,21 +0,0 @@
-/* ==========================================================================
-   #HIDE
-   ========================================================================== */
-
-/**
- * Hide only visually, but have it available for screen readers:
- * http://snook.ca/archives/html_and_css/hiding-content-for-accessibility
- */
-
-.u-hidden-visually {
-  @include inuit-hidden-visually();
-}
-
-
-/**
- * Hide visually and from screen readers.
- */
-
-.u-hidden {
-  display: none !important;
-}
diff --git a/_sass/page/inuitcss/utilities/_utilities.print.scss b/_sass/page/inuitcss/utilities/_utilities.print.scss
deleted file mode 100755
index 7a0b980..0000000
--- a/_sass/page/inuitcss/utilities/_utilities.print.scss
+++ /dev/null
@@ -1,88 +0,0 @@
-/* ==========================================================================
-   #PRINT
-   ========================================================================== */
-
-/**
- * Very crude, reset-like styles taken from the HTML5 Boilerplate:
- * https://github.com/h5bp/html5-boilerplate/blob/5.3.0/dist/doc/css.md#print-styles
- * https://github.com/h5bp/html5-boilerplate/blob/master/dist/css/main.css#L205-L282
- */
-
-@media print {
-
-  /**
-   * 1. Black prints faster: http://www.sanbeiji.com/archives/953
-   */
-
-  *,
-  *:before,
-  *:after {
-    background: transparent !important;
-    color: #000 !important; /* [1] */
-    box-shadow: none !important;
-    text-shadow: none !important;
-  }
-
-
-  a,
-  a:visited {
-    text-decoration: underline;
-  }
-
-  a[href]:after {
-    content: " (" attr(href) ")";
-  }
-
-  abbr[title]:after {
-    content: " (" attr(title) ")";
-  }
-
-
-  /**
-   * Don’t show links that are fragment identifiers, or use the `javascript:`
-   * pseudo protocol.
-   */
-
-  a[href^="#"]:after,
-  a[href^="javascript:"]:after {
-    content: "";
-  }
-
-  pre,
-  blockquote {
-    border: 1px solid #999;
-    page-break-inside: avoid;
-  }
-
-
-  /**
-   * Printing Tables: http://css-discuss.incutio.com/wiki/Printing_Tables
-   */
-
-  thead {
-    display: table-header-group;
-  }
-
-  tr,
-  img {
-    page-break-inside: avoid;
-  }
-
-
-  img {
-    max-width: 100% !important;
-  }
-
-  p,
-  h2,
-  h3 {
-    orphans: 3;
-    widows: 3;
-  }
-
-  h2,
-  h3 {
-    page-break-after: avoid;
-  }
-
-}
diff --git a/_sass/page/inuitcss/utilities/_utilities.responsive-spacings.scss b/_sass/page/inuitcss/utilities/_utilities.responsive-spacings.scss
deleted file mode 100755
index 49683b0..0000000
--- a/_sass/page/inuitcss/utilities/_utilities.responsive-spacings.scss
+++ /dev/null
@@ -1,124 +0,0 @@
-/* ==========================================================================
-   #RESPONSIVE-SPACINGS
-   ========================================================================== */
-
-/**
- * Utility classes enhancing the normal spacing classes by adding responsiveness
- * to them. By default, there are not responsive spacings defined. You can
- * generate responsive spacings by adding entries to the following three Sass
- * maps, e.g.:
- *
- *   $inuit-responsive-spacing-directions: (
- *     null: null,
- *     bottom: bottom,
- *   );
- *
- *   $inuit-responsive-spacing-properties: (
- *     "margin": "margin",
- *   );
- *
- *   $inuit-responsive-spacing-sizes: (
- *     "-small": $inuit-global-spacing-unit-small,
- *   );
- *
- * This would bring us the following classes:
- *
- *   .u-margin-small@mobile {}
- *   .u-margin-small@tablet {}
- *   .u-margin-small@desktop {}
- *   .u-margin-small@wide {}
- *   .u-margin-bottom-small@mobile {}
- *   .u-margin-bottom-small@tablet {}
- *   .u-margin-bottom-small@desktop {}
- *   .u-margin-bottom-small@wide {}
- *
- * You can change the generated CSS classes by further extending the Sass maps.
- * If you want every ‘normal’ spacing (those from `utilities.spacings`) also as
- * a responsive version, you can just mirror the ‘normal’ spacings:
- *
- *   $inuit-responsive-spacing-directions: $inuit-spacing-directions !default;
- *
- *   $inuit-responsive-spacing-properties: $inuit-spacing-properties !default;
- *
- *   $inuit-responsive-spacing-sizes: $inuit-spacing-sizes !default;
- *
- * BUT BE AWARE: This can generate a huge chunk of extra CSS, depending on the
- * amount of breakpoints you defined. So please check your CSS’ output and
- * filesize!
- */
-
-
-
-// The responsive spacings just make sense and work properly when the ‘normal’
-// spacings are included, too. In case they're not, we set `_utilities.spacings`
-// to `null`.
-$inuit-spacing-directions: null !default;
-
-// If the ‘normal’ spacings partial is not included, we provide an error message
-// to indicate this.
-@if $inuit-spacing-directions == null {
-  @error "In order to use responsive spacings, you also need to include `_utilities.spacings.scss`!";
-}
-@else {
-
-
-
-// When using Sass-MQ, this defines the separator for the breakpoints suffix
-// in the class name. By default, we are generating the responsive suffixes
-// for the classes with a `@` symbol so you get classes like:
-//
-//   <div class="u-margin-bottom@mobile">
-//
-// Be aware that since the `@` symbol is a reserved symbol in CSS, it has to be
-// escaped with a `\`. In the markup though, you write your classes without the
-// backslash (e.g. `u-margin-bottom@mobile`).
-$inuit-widths-breakpoint-separator: \@ !default;
-
-
-
-$inuit-responsive-spacing-directions: null !default;
-
-$inuit-responsive-spacing-properties: null !default;
-
-$inuit-responsive-spacing-sizes: null !default;
-
-
-
-/* stylelint-disable max-nesting-depth */
-
-// Don't output anything if no responsive spacings are defined.
-@if ($inuit-responsive-spacing-properties != null) {
-
-  @each $property-namespace, $property in $inuit-responsive-spacing-properties {
-
-    @each $direction-namespace, $direction-rules in $inuit-responsive-spacing-directions {
-
-      @each $size-namespace, $size in $inuit-responsive-spacing-sizes {
-
-        @each $inuit-bp-name, $inuit-bp-value in $mq-breakpoints {
-
-          @include mq($from: $inuit-bp-name) {
-
-            .u-#{$property-namespace}#{$direction-namespace}#{$size-namespace}#{$inuit-widths-breakpoint-separator}#{$inuit-bp-name} {
-
-              @each $direction in $direction-rules {
-                #{$property}#{$direction}: $size !important;
-              }
-
-            }
-
-          }
-
-        }
-
-      }
-
-    }
-
-  }
-
-}
-
-/* stylelint-enable max-nesting-depth */
-
-}
diff --git a/_sass/page/inuitcss/utilities/_utilities.spacings.scss b/_sass/page/inuitcss/utilities/_utilities.spacings.scss
deleted file mode 100755
index dee519b..0000000
--- a/_sass/page/inuitcss/utilities/_utilities.spacings.scss
+++ /dev/null
@@ -1,60 +0,0 @@
-/* ==========================================================================
-   #SPACINGS
-   ========================================================================== */
-
-/**
- * Utility classes to put specific spacing values onto elements. The below loop
- * will generate us a suite of classes like:
- *
- *   .u-margin-top {}
- *   .u-padding-left-large {}
- *   .u-margin-right-small {}
- *   .u-padding {}
- *   .u-padding-right-none {}
- *   .u-padding-horizontal {}
- *   .u-padding-vertical-small {}
- */
-
-$inuit-spacing-directions: (
-  null: null,
-  "-top": "-top",
-  "-right": "-right",
-  "-bottom": "-bottom",
-  "-left": "-left",
-  "-horizontal": "-left" "-right",
-  "-vertical": "-top" "-bottom",
-) !default;
-
-$inuit-spacing-properties: (
-  "padding": "padding",
-  "margin": "margin",
-) !default;
-
-$inuit-spacing-sizes: (
-  null: $inuit-global-spacing-unit,
-  "-tiny": $inuit-global-spacing-unit-tiny,
-  "-small": $inuit-global-spacing-unit-small,
-  "-large": $inuit-global-spacing-unit-large,
-  "-huge": $inuit-global-spacing-unit-huge,
-  "-none": 0
-) !default;
-
-@each $property-namespace, $property in $inuit-spacing-properties {
-
-  @each $direction-namespace, $direction-rules in $inuit-spacing-directions {
-
-    @each $size-namespace, $size in $inuit-spacing-sizes {
-
-      .u-#{$property-namespace}#{$direction-namespace}#{$size-namespace} {
-
-        @each $direction in $direction-rules {
-          #{$property}#{$direction}: $size !important;
-        }
-
-      }
-
-    }
-
-  }
-
-}
diff --git a/_sass/page/inuitcss/utilities/_utilities.widths.scss b/_sass/page/inuitcss/utilities/_utilities.widths.scss
deleted file mode 100755
index 50c8cdb..0000000
--- a/_sass/page/inuitcss/utilities/_utilities.widths.scss
+++ /dev/null
@@ -1,184 +0,0 @@
-/* ==========================================================================
-   #WIDTHS
-   ========================================================================== */
-
-/**
- * inuitcss generates a series of utility classes that give a fluid width to
- * whichever element they’re applied, e.g.:
- *
- *   <img src="" alt="" class="u-1/2" />
- *
- * These classes are most commonly used in conjunction with our layout system,
- * e.g.:
- *
- *   <div class="o-layout__item  u-1/2">
- *
- * By default, inuitcss will also generate responsive variants of each of these
- * classes by using your Sass MQ configuration, e.g.:
- *
- *   <div class="o-layout__item  u-1/1  u-1/2@tablet  u-1/3@desktop">
- *
- * Optionally, inuitcss can generate offset classes which can push and pull
- * elements left and right by a specified amount, e.g.:
- *
- *   <div class="o-layout__item  u-2/3  u-pull-1/3">
- *
- * This is useful for making very granular changes to the rendered order of
- * items in a layout.
- *
- * N.B. This option is turned off by default.
- */
-
-
-
-// Which fractions would you like in your grid system(s)? By default, inuitcss
-// provides you fractions of one whole, halves, thirds, quarters and fifths,
-// e.g.:
-//
-//   .u-1/2
-//   .u-2/5
-//   .u-3/4
-//   .u-2/3
-
-$inuit-fractions: 1 2 3 4 5 !default;
-
-
-
-
-
-// Optionally, inuitcss can generate classes to offset items by a certain width.
-// Would you like to generate these types of class as well? E.g.:
-//
-//   .u-push-1/3
-//   .u-pull-2/4
-//   .u-pull-1/5
-//   .u-push-2/3
-
-$inuit-offsets: false !default;
-
-
-
-
-
-// By default, inuitcss uses fractions-like classes like `<div class="u-1/4">`.
-// You can change the `/` to whatever you fancy with this variable.
-$inuit-widths-delimiter: \/ !default;
-
-
-
-
-
-// When using Sass-MQ, this defines the separator for the breakpoints suffix
-// in the class name. By default, we are generating the responsive suffixes
-// for the classes with a `@` symbol so you get classes like:
-// <div class="u-3/12@mobile">
-$inuit-widths-breakpoint-separator: \@ !default;
-
-
-
-
-
-// A mixin to spit out our width classes. Pass in the columns we want the widths
-// to have, and an optional suffix for responsive widths. E.g. to create thirds
-// and quarters for a small breakpoint:
-//
-// @include widths(3 4, -sm);
-
-@mixin inuit-widths($columns, $breakpoint: null) {
-
-  // Loop through the number of columns for each denominator of our fractions.
-  @each $denominator in $columns {
-
-    // Begin creating a numerator for our fraction up until we hit the
-    // denominator.
-    @for $numerator from 1 through $denominator {
-
-      // Build a class in the format `.u-3/4[@<breakpoint>]`.
-
-      .u-#{$numerator}#{$inuit-widths-delimiter}#{$denominator}#{$breakpoint} {
-        width: ($numerator / $denominator) * 100% !important;
-      }
-
-      @if ($inuit-offsets == true) {
-
-        /**
-         * 1. Reset any leftover or conflicting `left`/`right` values.
-         */
-
-        // Build a class in the format `.u-push-1/2[@<breakpoint>]`.
-
-        .u-push-#{$numerator}#{$inuit-widths-delimiter}#{$denominator}#{$breakpoint} {
-          position: relative !important;
-          right: auto !important; /* [1] */
-          left: ($numerator / $denominator) * 100% !important;
-        }
-
-        // Build a class in the format `.u-pull-5/6[@<breakpoint>]`.
-
-        .u-pull-#{$numerator}#{$inuit-widths-delimiter}#{$denominator}#{$breakpoint} {
-          position: relative !important;
-          right: ($numerator / $denominator) * 100% !important;
-          left: auto !important; /* [1] */
-        }
-
-      }
-
-    }
-
-  }
-
-  @if ($inuit-offsets == true and $breakpoint != null) {
-
-    // Create auto push and pull classes.
-
-    .u-push-none#{$breakpoint} {
-      left: auto !important;
-    }
-
-    .u-pull-none#{$breakpoint} {
-      right: auto !important;
-    }
-
-  }
-
-}
-
-
-
-
-
-/**
- * A series of width helper classes that you can use to size things like grid
- * systems. Classes take a fraction-like format (e.g. `.u-2/3`). Use these in
- * your markup:
- *
- * <div class="u-7/12">
- *
- * The following will generate widths helper classes based on the fractions
- * defined in the `$inuit-fractions` list.
- */
-
-@include inuit-widths($inuit-fractions);
-
-
-
-
-
-/**
- * If we’re using Sass-MQ, automatically generate grid system(s) for each of our
- * defined breakpoints, and give them a Responsive Suffix, e.g.:
- *
- * <div class="u-3/12@mobile">
- */
-
-@if (variable-exists(mq-breakpoints)) {
-
-  @each $inuit-bp-name, $inuit-bp-value in $mq-breakpoints {
-
-    @include mq($from: $inuit-bp-name) {
-      @include inuit-widths($inuit-fractions, #{$inuit-widths-breakpoint-separator}#{$inuit-bp-name});
-    }
-
-  }
-
-}
diff --git a/_sass/page/main.scss b/_sass/page/main.scss
deleted file mode 100755
index a0fd0c4..0000000
--- a/_sass/page/main.scss
+++ /dev/null
@@ -1,102 +0,0 @@
-/**
- * This file is the main entry point for a single page's stylings. It should
- * be self-contained CSS for *one* page. There is another SCSS file for
- * styling a *book* that depends on this one.
- *
- * It uses the Inuit CSS framework: https://github.com/inuitcss/inuitcss
- *
- * InuitCSS only provides basic typographic stylings and helper functions. All
- * framework files live in ./inuitcss/ . All of our custom CSS lives in one of
- * these 7 folders:
- *
- * `/settings`: Global variables, site-wide settings, config switches, etc.
- * `/tools`: Site-wide mixins and functions.
- * `/generic`: Low-specificity, far-reaching rulesets (e.g. resets).
- * `/elements`: Unclassed HTML elements (e.g. `a {}`, `blockquote {}`, `address {}`).
- * `/objects`: Objects, abstractions, and design patterns (e.g. `.o-layout {}`).
- * `/components`: Discrete, complete chunks of UI (e.g. `.c-carousel {}`).
- * `/utilities`: High-specificity, very explicit selectors. Overrides and
-   helper classes (e.g. `.u-hidden {}`).
- *
- * CONTENTS
- *
- * SETTINGS
- * Core.................inuitcss’ core and setup settings.
- * Global...............Project-wide variables and settings.
- *
- * TOOLS
- * Font-size............A mixin which guarantees baseline-friendly line-heights.
- * Clearfix.............Micro clearfix mixin.
- * Hidden...............Mixin for hiding elements.
- * Sass MQ..............inuitcss’ default media query manager.
- *
- * GENERIC
- * Box-sizing...........Better default `box-sizing`.
- * Normalize.css........A level playing field using @necolas’ Normalize.css.
- * Reset................A tiny reset to complement Normalize.css.
- * Shared...............Sensibly and tersely share some global commonalities
- *                      (particularly useful when managing vertical rhythm).
- *
- * ELEMENTS
- * Page.................Set up our document’s default `font-size` and
- *                      `line-height`.
- * Headings.............Very minimal (i.e. only font-size information) for
- *                      headings 1 through 6.
- * Images...............Base image styles.
- * Tables...............Simple table styles.
- *
- * OBJECTS
- * Wrapper..............Page constraint object.
- *
- * COMPONENTS
- * Buttons..............An example button component, and how it fits into the
- *                      inuitcss framework.
- *
- * UTILITIES
- * Clearfix.............Bind our clearfix onto a utility class.
- */
-
-// SETTINGS
-@import 'inuitcss/settings/settings.core';
-@import 'settings/settings.global';
-
-// // TOOLS
-@import 'inuitcss/tools/tools.font-size';
-@import 'inuitcss/tools/tools.clearfix';
-@import 'inuitcss/tools/tools.hidden';
-@import 'inuitcss/tools/tools.mq';
-
-// // GENERIC
-@import 'inuitcss/generic/generic.box-sizing';
-@import 'inuitcss/generic/generic.normalize';
-@import 'inuitcss/generic/generic.reset';
-@import 'inuitcss/generic/generic.shared';
-@import 'generic/generic.phone-scrolling';
-
-// // ELEMENTS
-@import 'inuitcss/elements/elements.page';
-@import 'inuitcss/elements/elements.headings';
-@import 'inuitcss/elements/elements.images';
-@import 'inuitcss/elements/elements.tables';
-@import 'elements/elements.variables';
-@import 'elements/elements.typography';
-@import 'elements/elements.syntax-highlighting';
-@import 'elements/elements.tables';
-@import 'elements/elements.links';
-
-// // OBJECTS
-@import 'inuitcss/objects/objects.wrapper';
-@import 'inuitcss/objects/objects.layout';
-
-@import 'objects/objects.tooltip';
-@import 'objects/objects.copy-button';
-
-// // COMPONENTS
-@import 'components/components.page';
-@import 'components/components.sidebar-right';
-@import 'components/components.hidecells';
-
-// // UTILITIES
-@import 'inuitcss/utilities/utilities.clearfix';
-@import 'inuitcss/utilities/utilities.hide';
-@import 'inuitcss/utilities/utilities.spacings';
diff --git a/_sass/page/objects/_objects.copy-button.scss b/_sass/page/objects/_objects.copy-button.scss
deleted file mode 100755
index ceff439..0000000
--- a/_sass/page/objects/_objects.copy-button.scss
+++ /dev/null
@@ -1,20 +0,0 @@
-/* Copy buttons */
-.copybtn {
-  position: absolute;
-  top: 0;
-  right: 0;
-
-  margin: $spacing-unit-tiny;
-  width: $spacing-unit;
-  height: $spacing-unit;
-  padding: 0 4px 2px;
-
-  i {
-    font-style: normal;
-    color: #777777
-  }
-}
-
-.input_area .highlight, .text_cell_render .highlight {
-  position: relative;
-}
diff --git a/_sass/page/objects/_objects.tooltip.scss b/_sass/page/objects/_objects.tooltip.scss
deleted file mode 100755
index c808b9b..0000000
--- a/_sass/page/objects/_objects.tooltip.scss
+++ /dev/null
@@ -1,54 +0,0 @@
-/**
- * A minimal CSS-only tooltip copied from:
- *   https://codepen.io/mildrenben/pen/rVBrpK
- *
- * To use, write HTML like the following:
- *
- * <p class="o-tooltip--left" data-tooltip="Hey">Short</p>
- */
-
-$cubic: cubic-bezier(0.64, 0.09, 0.08, 1);
-
-.o-tooltip {
-  position: relative;
-  &:after {
-    opacity: 0;
-    visibility: hidden;
-    position: absolute;
-    content: attr(data-tooltip);
-    padding: 6px;
-    top: 1.4em;
-    left: 50%;
-    transform: translateX(-50%) translateY(-2px);
-    background: grey;
-    font-size: 0.7rem;
-    color: white;
-    white-space: nowrap;
-    z-index: 2;
-    border-radius: 2px;
-    transition: opacity 0.2s $cubic, transform 0.2s $cubic;
-  }
-  &:hover {
-    &:after {
-      display: block;
-      opacity: 1;
-      visibility: visible;
-      transform: translateX(-50%) translateY(0);
-    }
-  }
-}
-
-.o-tooltip--left {
-  @extend .o-tooltip;
-
-  &:after {
-    top: 0;
-    left: 0;
-    transform: translateX(-102%) translateY(0);
-  }
-  &:hover {
-    &:after {
-      transform: translateX(-100%) translateY(0);
-    }
-  }
-}
diff --git a/_sass/page/settings/settings.global.scss b/_sass/page/settings/settings.global.scss
deleted file mode 100755
index 4b6b5cb..0000000
--- a/_sass/page/settings/settings.global.scss
+++ /dev/null
@@ -1,25 +0,0 @@
-///* ========================================================================
-//   #GLOBAL
-//   ======================================================================== */
-
-// The global settings file contains any project-wide variables; things that
-// need to be made available to the entire codebase.
-
-$spacing-unit-tiny: $inuit-global-spacing-unit-tiny;
-$spacing-unit-small: $inuit-global-spacing-unit-small;
-$spacing-unit-med: 0.75 * $inuit-global-spacing-unit;
-$spacing-unit: $inuit-global-spacing-unit;
-$spacing-unit-large: $inuit-global-spacing-unit-large;
-$spacing-unit-huge: $inuit-global-spacing-unit-huge;
-
-$color-light-gray: #f7f7f7;
-$color-dark-gray: #ccc;
-
-$color-text: #222;
-$color-links: #0077d8;
-
-// Text book variables
-$textbook-page-width: 100%;
-$textbook-page-with-sidebar-width: 70%;
-$book-background-color: white;
-
diff --git a/assets/css/styles.scss b/assets/css/styles.scss
deleted file mode 100755
index da50df3..0000000
--- a/assets/css/styles.scss
+++ /dev/null
@@ -1,4 +0,0 @@
----
----
-
-@import 'main';
diff --git a/assets/custom/custom.css b/assets/custom/custom.css
deleted file mode 100755
index be0bcd1..0000000
--- a/assets/custom/custom.css
+++ /dev/null
@@ -1,5 +0,0 @@
-/* Put your custom CSS here */
-.left {
-    margin: 5px 2px 2px 0px !important;
-    float: left !important;
-}
diff --git a/assets/custom/custom.js b/assets/custom/custom.js
deleted file mode 100755
index 792c873..0000000
--- a/assets/custom/custom.js
+++ /dev/null
@@ -1 +0,0 @@
-// Put your custom javascript here
\ No newline at end of file
diff --git a/assets/html/index.html b/assets/html/index.html
deleted file mode 100755
index ce3046f..0000000
--- a/assets/html/index.html
+++ /dev/null
@@ -1,27 +0,0 @@
----
-permalink: /index.html
-title: "Index"
-layout: none
----
-
-<!-- The index page should simply re-direct to the first chapter -->
-{% for chapter in site.data.toc %}
-{% unless chapter.external %}
-  {% if chapter.url %}
-    {% comment %}This ensures that the first link we re-direct to isn't an external site {% endcomment %}
-    {% assign redirectURL = chapter.url | relative_url %}
-    {% break %}
-  {% endif %}
-{% endunless %}
-{% endfor %}
-<!DOCTYPE html>
-<html lang="en-US">
-  <meta charset="utf-8">
-  <title>Redirecting&hellip;</title>
-  <link rel="canonical" href="{{ redirectURL }}">
-  <script>location="{{ redirectURL }}"</script>
-  <meta http-equiv="refresh" content="0; url={{ redirectURL }}">
-  <meta name="robots" content="noindex">
-  <h1>Redirecting&hellip;</h1>
-  <a href="{{ redirectURL }}">Click here if you are not redirected.</a>
-</html>
diff --git a/assets/html/search_form.html b/assets/html/search_form.html
deleted file mode 100755
index b25a8de..0000000
--- a/assets/html/search_form.html
+++ /dev/null
@@ -1,15 +0,0 @@
----
-permalink: /search
-title: "Search the site"
-search_page: true
----
-
-<div class="search-content__inner-wrap">
-    <input type="text" id="lunr_search" class="search-input" tabindex="-1" placeholder="'Enter your search term...''" />
-    <div id="results" class="results"></div>
-</div>
-
-<script>
-    // Add the lunr store since we will now search it
-    {% include search/lunr/lunr-store.js %}
-</script>
\ No newline at end of file
diff --git a/assets/images/download-solid.svg b/assets/images/download-solid.svg
deleted file mode 100755
index 781f993..0000000
--- a/assets/images/download-solid.svg
+++ /dev/null
@@ -1,13 +0,0 @@
-<?xml version="1.0" encoding="utf-8"?>
-<!-- Generator: Adobe Illustrator 23.0.3, SVG Export Plug-In . SVG Version: 6.00 Build 0)  -->
-<svg version="1.1" id="Layer_1" focusable="false" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"
-	 x="0px" y="0px" viewBox="0 0 512 512" style="enable-background:new 0 0 512 512;" xml:space="preserve">
-<style type="text/css">
-	.st0{fill:#FFFFFF;}
-</style>
-<path class="st0" d="M216,0h80c13.3,0,24,10.7,24,24v168h87.7c17.8,0,26.7,21.5,14.1,34.1L269.7,378.3c-7.5,7.5-19.8,7.5-27.3,0
-	L90.1,226.1c-12.6-12.6-3.7-34.1,14.1-34.1H192V24C192,10.7,202.7,0,216,0z M512,376v112c0,13.3-10.7,24-24,24H24
-	c-13.3,0-24-10.7-24-24V376c0-13.3,10.7-24,24-24h146.7l49,49c20.1,20.1,52.5,20.1,72.6,0l49-49H488C501.3,352,512,362.7,512,376z
-	 M388,464c0-11-9-20-20-20s-20,9-20,20s9,20,20,20S388,475,388,464z M452,464c0-11-9-20-20-20s-20,9-20,20s9,20,20,20
-	S452,475,452,464z"/>
-</svg>
diff --git a/assets/images/edit-button.svg b/assets/images/edit-button.svg
deleted file mode 100755
index 48ca987..0000000
--- a/assets/images/edit-button.svg
+++ /dev/null
@@ -1,13 +0,0 @@
-<?xml version="1.0" encoding="utf-8"?>
-<!-- Generator: Adobe Illustrator 23.0.3, SVG Export Plug-In . SVG Version: 6.00 Build 0)  -->
-<svg version="1.1" id="Layer_1" focusable="false" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"
-	 x="0px" y="0px" viewBox="0 0 512 512" style="enable-background:new 0 0 512 512;" xml:space="preserve">
-<style type="text/css">
-	.st0{fill:#808080;}
-</style>
-<path class="st0" d="M497.9,142.1l-46.1,46.1c-4.7,4.7-12.3,4.7-17,0l-111-111c-4.7-4.7-4.7-12.3,0-17l46.1-46.1
-	c18.7-18.7,49.1-18.7,67.9,0l60.1,60.1C516.7,92.9,516.7,123.3,497.9,142.1z M284.2,99.8L21.6,362.4L0.4,483.9
-	c-2.9,16.4,11.4,30.6,27.8,27.8l121.5-21.3l262.6-262.6c4.7-4.7,4.7-12.3,0-17l-111-111C296.5,95.1,288.9,95.1,284.2,99.8
-	L284.2,99.8z M124.1,339.9c-5.5-5.5-5.5-14.3,0-19.8l154-154c5.5-5.5,14.3-5.5,19.8,0s5.5,14.3,0,19.8l-154,154
-	C138.4,345.4,129.6,345.4,124.1,339.9L124.1,339.9z M88,424h48v36.3l-64.5,11.3l-31.1-31.1L51.7,376H88V424z"/>
-</svg>
diff --git a/assets/images/list-solid.svg b/assets/images/list-solid.svg
deleted file mode 100755
index 541f861..0000000
--- a/assets/images/list-solid.svg
+++ /dev/null
@@ -1,14 +0,0 @@
-<?xml version="1.0" encoding="utf-8"?>
-<!-- Generator: Adobe Illustrator 23.0.3, SVG Export Plug-In . SVG Version: 6.00 Build 0)  -->
-<svg version="1.1" id="Layer_1" focusable="false" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"
-	 x="0px" y="0px" viewBox="0 0 512 512" style="enable-background:new 0 0 512 512;" xml:space="preserve">
-<style type="text/css">
-	.st0{fill:#5A5A5A;}
-</style>
-<path class="st0" d="M80,368H16c-8.8,0-16,7.2-16,16v64c0,8.8,7.2,16,16,16h64c8.8,0,16-7.2,16-16v-64C96,375.2,88.8,368,80,368z
-	 M80,48H16C7.2,48,0,55.2,0,64v64c0,8.8,7.2,16,16,16h64c8.8,0,16-7.2,16-16V64C96,55.2,88.8,48,80,48z M80,208H16
-	c-8.8,0-16,7.2-16,16v64c0,8.8,7.2,16,16,16h64c8.8,0,16-7.2,16-16v-64C96,215.2,88.8,208,80,208z M496,384H176c-8.8,0-16,7.2-16,16
-	v32c0,8.8,7.2,16,16,16h320c8.8,0,16-7.2,16-16v-32C512,391.2,504.8,384,496,384z M496,64H176c-8.8,0-16,7.2-16,16v32
-	c0,8.8,7.2,16,16,16h320c8.8,0,16-7.2,16-16V80C512,71.2,504.8,64,496,64z M496,224H176c-8.8,0-16,7.2-16,16v32c0,8.8,7.2,16,16,16
-	h320c8.8,0,16-7.2,16-16v-32C512,231.2,504.8,224,496,224z"/>
-</svg>
diff --git a/assets/images/logo_binder.svg b/assets/images/logo_binder.svg
deleted file mode 100755
index 16390ee..0000000
--- a/assets/images/logo_binder.svg
+++ /dev/null
@@ -1,19 +0,0 @@
-<?xml version="1.0" encoding="utf-8"?>
-<!-- Generator: Adobe Illustrator 23.0.1, SVG Export Plug-In . SVG Version: 6.00 Build 0)  -->
-<svg version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px"
-	 viewBox="0 0 44.4 44.4" style="enable-background:new 0 0 44.4 44.4;" xml:space="preserve">
-<style type="text/css">
-	.st0{fill:none;stroke:#F5A252;stroke-width:5;stroke-miterlimit:10;}
-	.st1{fill:none;stroke:#579ACA;stroke-width:5;stroke-miterlimit:10;}
-	.st2{fill:none;stroke:#E66581;stroke-width:5;stroke-miterlimit:10;}
-</style>
-<title>logo</title>
-<g>
-	<path class="st0" d="M33.9,6.4c3.6,3.9,3.4,9.9-0.5,13.5s-9.9,3.4-13.5-0.5s-3.4-9.9,0.5-13.5l0,0C24.2,2.4,30.2,2.6,33.9,6.4z"/>
-	<path class="st1" d="M35.1,27.3c2.6,4.6,1.1,10.4-3.5,13c-4.6,2.6-10.4,1.1-13-3.5s-1.1-10.4,3.5-13l0,0
-		C26.6,21.2,32.4,22.7,35.1,27.3z"/>
-	<path class="st2" d="M25.9,17.8c2.6,4.6,1.1,10.4-3.5,13s-10.4,1.1-13-3.5s-1.1-10.4,3.5-13l0,0C17.5,11.7,23.3,13.2,25.9,17.8z"/>
-	<path class="st1" d="M19.2,26.4c3.1-4.3,9.1-5.2,13.3-2.1c1.1,0.8,2,1.8,2.7,3"/>
-	<path class="st0" d="M19.9,19.4c-3.6-3.9-3.4-9.9,0.5-13.5s9.9-3.4,13.5,0.5"/>
-</g>
-</svg>
diff --git a/assets/images/logo_jupyterhub.svg b/assets/images/logo_jupyterhub.svg
deleted file mode 100755
index c584bad..0000000
--- a/assets/images/logo_jupyterhub.svg
+++ /dev/null
@@ -1 +0,0 @@
-<svg id="Layer_1" data-name="Layer 1" xmlns="http://www.w3.org/2000/svg" width="38.73" height="50" viewBox="0 0 38.73 50"><defs><style>.cls-1{fill:#767677;}.cls-2{fill:#f37726;}.cls-3{fill:#9e9e9e;}.cls-4{fill:#616262;}.cls-5{font-size:17.07px;fill:#fff;font-family:Roboto-Regular, Roboto;}</style></defs><title>logo_jupyterhub</title><g id="Canvas"><path id="path7_fill" data-name="path7 fill" class="cls-1" d="M39.51,3.53a3,3,0,0,1-1.7,2.9A3,3,0,0,1,34.48,6a3,3,0,0,1-.82-3.26,3,3,0,0,1,1.05-1.41A3,3,0,0,1,37.52.86a2.88,2.88,0,0,1,1,.6,3,3,0,0,1,.7.93,3.18,3.18,0,0,1,.28,1.14Z" transform="translate(-1.87 -0.69)"/><path id="path8_fill" data-name="path8 fill" class="cls-2" d="M21.91,38.39c-8,0-15.06-2.87-18.7-7.12a19.93,19.93,0,0,0,37.39,0C37,35.52,30,38.39,21.91,38.39Z" transform="translate(-1.87 -0.69)"/><path id="path9_fill" data-name="path9 fill" class="cls-2" d="M21.91,10.78c8,0,15.05,2.87,18.69,7.12a19.93,19.93,0,0,0-37.39,0C6.85,13.64,13.86,10.78,21.91,10.78Z" transform="translate(-1.87 -0.69)"/><path id="path10_fill" data-name="path10 fill" class="cls-3" d="M10.88,46.66a3.86,3.86,0,0,1-.52,2.15,3.81,3.81,0,0,1-1.62,1.51,3.93,3.93,0,0,1-2.19.34,3.79,3.79,0,0,1-2-.94,3.73,3.73,0,0,1-1.14-1.9,3.79,3.79,0,0,1,.1-2.21,3.86,3.86,0,0,1,1.33-1.78,3.92,3.92,0,0,1,3.54-.53,3.85,3.85,0,0,1,2.14,1.93,3.74,3.74,0,0,1,.37,1.43Z" transform="translate(-1.87 -0.69)"/><path id="path11_fill" data-name="path11 fill" class="cls-4" d="M4.12,9.81A2.18,2.18,0,0,1,2.9,9.48a2.23,2.23,0,0,1-.84-1A2.26,2.26,0,0,1,1.9,7.26a2.13,2.13,0,0,1,.56-1.13,2.18,2.18,0,0,1,2.36-.56,2.13,2.13,0,0,1,1,.76,2.18,2.18,0,0,1,.42,1.2A2.22,2.22,0,0,1,4.12,9.81Z" transform="translate(-1.87 -0.69)"/></g><text class="cls-5" transform="translate(5.24 30.01)">Hub</text></svg>
\ No newline at end of file
diff --git a/assets/images/search-solid.svg b/assets/images/search-solid.svg
deleted file mode 100755
index 94764a2..0000000
--- a/assets/images/search-solid.svg
+++ /dev/null
@@ -1,12 +0,0 @@
-<?xml version="1.0" encoding="utf-8"?>
-<!-- Generator: Adobe Illustrator 23.0.3, SVG Export Plug-In . SVG Version: 6.00 Build 0)  -->
-<svg version="1.1" id="Layer_1" focusable="false" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"
-	 x="0px" y="0px" viewBox="0 0 512 512" style="enable-background:new 0 0 512 512;" xml:space="preserve">
-<style type="text/css">
-	.st0{fill:#5A5A5A;}
-</style>
-<path class="st0" d="M505,442.7L405.3,343c-4.5-4.5-10.6-7-17-7H372c27.6-35.3,44-79.7,44-128C416,93.1,322.9,0,208,0S0,93.1,0,208
-	s93.1,208,208,208c48.3,0,92.7-16.4,128-44v16.3c0,6.4,2.5,12.5,7,17l99.7,99.7c9.4,9.4,24.6,9.4,33.9,0l28.3-28.3
-	C514.3,467.3,514.3,452.1,505,442.7z M208,336c-70.7,0-128-57.2-128-128c0-70.7,57.2-128,128-128c70.7,0,128,57.2,128,128
-	C336,278.7,278.8,336,208,336z"/>
-</svg>
diff --git a/assets/js/page/anchors.js b/assets/js/page/anchors.js
deleted file mode 100755
index ad922cf..0000000
--- a/assets/js/page/anchors.js
+++ /dev/null
@@ -1,12 +0,0 @@
-const initAnchors = () => {
-  if (window.anchors === undefined) {
-    setTimeout(initAnchors, 250);
-    return;
-  }
-  anchors.add("main h1, main h2, main h3, main h4");
-
-  // Disable Turbolinks for anchors
-  document.querySelectorAll('.anchorjs-link')
-    .forEach(it => it.dataset['turbolinks'] = false);
-}
-initFunction(initAnchors);
diff --git a/assets/js/page/copy-button.js b/assets/js/page/copy-button.js
deleted file mode 100755
index eb09ad1..0000000
--- a/assets/js/page/copy-button.js
+++ /dev/null
@@ -1,61 +0,0 @@
-/**
- * Set up copy/paste for code blocks
- */
-const copySVG = `<svg aria-labelledby="title" aria-hidden="true" data-prefix="far" data-icon="copy" class="svg-inline--fa fa-copy fa-w-14" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512">
-  <title id="title" lang="en">Copy code</title>
-  <path fill="#777" d="M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z"></path>
-  </svg>`
-
-const clipboardButton = id =>
-  `<a id="copy-button-${id}" class="btn copybtn o-tooltip--left" data-tooltip="Copy" data-clipboard-target="#${id}">
-    ${copySVG}
-  </a>`
-
-// Clears selected text since ClipboardJS will select the text when copying
-const clearSelection = () => {
-  if (window.getSelection) {
-    window.getSelection().removeAllRanges()
-  } else if (document.selection) {
-    document.selection.empty()
-  }
-}
-
-// Changes tooltip text for two seconds, then changes it back
-const temporarilyChangeTooltip = (el, newText) => {
-  const oldText = el.getAttribute('data-tooltip')
-  el.setAttribute('data-tooltip', newText)
-  setTimeout(() => el.setAttribute('data-tooltip', oldText), 2000)
-}
-
-const addCopyButtonToCodeCells = () => {
-  // If ClipboardJS hasn't loaded, wait a bit and try again. This
-  // happens because we load ClipboardJS asynchronously.
-  if (window.ClipboardJS === undefined) {
-    setTimeout(addCopyButtonToCodeCells, 250)
-    return
-  }
-
-  pageElements['codeCells'].forEach((codeCell) => {
-    const id = codeCell.getAttribute('id')
-    if (document.getElementById("copy-button" + id) == null) {
-      codeCell.insertAdjacentHTML('afterend', clipboardButton(id));
-    }
-  })
-
-  const clipboard = new ClipboardJS('.copybtn')
-  clipboard.on('success', event => {
-    clearSelection()
-    temporarilyChangeTooltip(event.trigger, 'Copied!')
-  })
-
-  clipboard.on('error', event => {
-    temporarilyChangeTooltip(event.trigger, 'Failed to copy')
-  })
-
-  // Get rid of clipboard before the next page visit to avoid memory leak
-  document.addEventListener('turbolinks:before-visit', () =>
-    clipboard.destroy()
-  )
-}
-
-initFunction(addCopyButtonToCodeCells);
\ No newline at end of file
diff --git a/assets/js/page/documentSelectors.js b/assets/js/page/documentSelectors.js
deleted file mode 100755
index 794d165..0000000
--- a/assets/js/page/documentSelectors.js
+++ /dev/null
@@ -1,35 +0,0 @@
-/**
- * Select various elements on the page for later use
- */
-
-// IDs we'll attach to cells
-const codeCellId = index => `codecell${index}`
-const inputCellId = index => `inputcell${index}`
-
-pageElements = {}
-
-// All code cells
-findCodeCells = function() {
-    var codeCells = document.querySelectorAll('div.c-textbook__content > div.highlighter-rouge > div.highlight > pre, div.input_area pre, div.text_cell_render div.highlight pre')
-    pageElements['codeCells'] = codeCells;
-
-    codeCells.forEach((codeCell, index) => {
-      const id = codeCellId(index)
-      codeCell.setAttribute('id', id)
-    })
-};
-
-initFunction(findCodeCells);
-
-// All cells in general
-findInputCells = function() {
-    var inputCells = document.querySelectorAll('div.jb_cell')
-    pageElements['inputCells'] = inputCells;
-
-    inputCells.forEach((inputCell, index) => {
-        const id = inputCellId(index)
-        inputCell.setAttribute('id', id)
-    })
-};
-
-initFunction(findInputCells);
\ No newline at end of file
diff --git a/assets/js/page/dom-update.js b/assets/js/page/dom-update.js
deleted file mode 100755
index 916e27f..0000000
--- a/assets/js/page/dom-update.js
+++ /dev/null
@@ -1,17 +0,0 @@
-const runWhenDOMLoaded = cb => {
-  if (document.readyState != 'loading') {
-    cb()
-  } else if (document.addEventListener) {
-    document.addEventListener('DOMContentLoaded', cb)
-  } else {
-    document.attachEvent('onreadystatechange', function() {
-      if (document.readyState == 'complete') cb()
-    })
-  }
-}
-
-// Helper function to init things quickly
-initFunction = function(myfunc) {
-  runWhenDOMLoaded(myfunc);
-  document.addEventListener('turbolinks:load', myfunc);
-};
\ No newline at end of file
diff --git a/assets/js/page/hide-cell.js b/assets/js/page/hide-cell.js
deleted file mode 100755
index c7cd04c..0000000
--- a/assets/js/page/hide-cell.js
+++ /dev/null
@@ -1,86 +0,0 @@
-/**
-  Add buttons to hide code cells
-*/
-var setCellVisibility = function (inputField, kind) {
-    // Update the image and class for hidden
-    var id = inputField.getAttribute('data-id');
-    var element = document.querySelector(`#${id}`);
-
-    if (kind === "visible") {
-        element.classList.remove('hidden');
-        inputField.checked = true;
-    } else {
-        element.classList.add('hidden');
-        inputField.checked = false;
-    }
-}
-
-var toggleCellVisibility = function (event) {
-    // The label is clicked, and now we decide what to do based on the input field's clicked status
-    if (event.target.tagName === "LABEL") {
-        var inputField = event.target.previousElementSibling;
-    } else {
-        // It is the span inside the target
-        var inputField = event.target.parentElement.previousElementSibling;
-    }
-
-    if (inputField.checked === true) {
-        setCellVisibility(inputField, "visible");
-    } else {
-        setCellVisibility(inputField, "hidden");
-    }
-}
-
-
-// Button constructor
-const hideCodeButton = id => `<input class="hidebtn" type="checkbox" id="hidebtn${id}" data-id="${id}"><label title="Toggle cell" for="hidebtn${id}" class="plusminus"><span class="pm_h"></span><span class="pm_v"></span></label>`
-
-var addHideButton = (element, id) => {
-    // Add a hide button to an HTML element.
-    element.setAttribute("id", id)
-    // Insert the button just inside the end of the next div
-    element.insertAdjacentHTML('afterend', hideCodeButton(id))
-
-    // Set up the visibility toggle
-    // The label will be two-sibings deep from the element to-be hidden
-    hideLink = element.nextElementSibling.nextElementSibling;
-    hideLink.addEventListener('click', toggleCellVisibility)
-}
-
-var addAllHideButtons = function () {
-    // If a hide button is already added, don't add another
-    if (document.querySelector('input.hidebtn') !== null) {
-        return;
-    }
-
-    // Find the input cells and add a hide button
-    hideIdNum = 0;
-    pageElements['inputCells'].forEach((cell) => {
-        const id = cell.getAttribute('id')
-
-        if (cell.classList.contains("tag_hide_input")) {
-            addHideButton(cell.querySelector('div.inner_cell'), `hide-${hideIdNum}`);
-            hideIdNum ++;
-        }
-
-        if (cell.classList.contains("tag_hide_output")) {
-            addHideButton(cell.querySelector('div.output'), `hide-${hideIdNum}`);
-            hideIdNum ++;
-        }
-    });
-}
-
-
-// Initialize the hide buttos
-var initHiddenCells = function () {
-    // Add hide buttons to the cells
-    addAllHideButtons();
-
-    // Toggle the code cells that should be hidden
-    document.querySelectorAll('div.tag_hide_input input, div.tag_hide_output input').forEach(function (item) {
-        setCellVisibility(item, 'hidden');
-        item.checked = true;
-    })
-}
-
-initFunction(initHiddenCells);
\ No newline at end of file
diff --git a/assets/js/page/tocbot.js b/assets/js/page/tocbot.js
deleted file mode 100755
index ff4e25f..0000000
--- a/assets/js/page/tocbot.js
+++ /dev/null
@@ -1,38 +0,0 @@
-const initToc = () => {
-  if (window.tocbot === undefined) {
-    setTimeout(initToc, 250);
-    return;
-  }
-
-  // Check whether we have any sidebar content. If not, then show the sidebar earlier.
-  var SIDEBAR_CONTENT_TAGS = ['.tag_full_width', '.tag_popout'];
-  var sidebar_content_query = SIDEBAR_CONTENT_TAGS.join(', ')
-  if (document.querySelectorAll(sidebar_content_query).length === 0) {
-    document.querySelector('nav.onthispage').classList.add('no_sidebar_content')
-  }
-
-  // Initialize the TOC bot if we have TOC headers
-  const tocContent = '.c-textbook__content';
-  const tocHeaders = 'h1, h2, h3';
-  var headers = document.querySelector(tocContent).querySelectorAll(tocHeaders);
-  if (headers.length > 0) {
-    document.querySelector('aside.sidebar__right').classList.remove('hidden');
-    tocbot.init({
-      tocSelector: 'nav.onthispage',
-      contentSelector: tocContent,
-      headingSelector: tocHeaders,
-      orderedList: false,
-      collapseDepth: 6,
-      listClass: 'toc__menu',
-      activeListItemClass: " ",  // Not using, can't be empty
-      activeLinkClass: " ", // Not using, can't be empty
-    });
-  } else {
-    document.querySelector('aside.sidebar__right').classList.add('hidden');
-  }
-
-  // Disable Turbolinks for TOC links
-  document.querySelectorAll('.toc-list-item a')
-    .forEach(it => it.dataset['turbolinks'] = false);
-}
-initFunction(initToc);
diff --git a/assets/js/scripts.js b/assets/js/scripts.js
deleted file mode 100755
index b624667..0000000
--- a/assets/js/scripts.js
+++ /dev/null
@@ -1,257 +0,0 @@
-/**
- * Site-wide JS that sets up:
- *
- * [1] MathJax rendering on navigation
- * [2] Sidebar toggling
- * [3] Sidebar scroll preserving
- * [4] Keyboard navigation
- * [5] Right sidebar scroll highlighting / navbar show
- */
-
-const togglerId = 'js-sidebar-toggle'
-const textbookId = 'js-textbook'
-const togglerActiveClass = 'is-active'
-const textbookActiveClass = 'js-show-sidebar'
-const mathRenderedClass = 'js-mathjax-rendered'
-const icon_path = document.location.origin + `${site_basename}assets`;
-
-const getToggler = () => document.getElementById(togglerId)
-const getTextbook = () => document.getElementById(textbookId)
-
-// [1] Run MathJax when Turbolinks navigates to a page.
-// When Turbolinks caches a page, it also saves the MathJax rendering. We mark
-// each page with a CSS class after rendering to prevent double renders when
-// navigating back to a cached page.
-document.addEventListener('turbolinks:load', () => {
-  const textbook = getTextbook()
-  if (window.MathJax && !textbook.classList.contains(mathRenderedClass)) {
-    MathJax.Hub.Queue(
-      ["resetEquationNumbers", MathJax.InputJax.TeX],
-      ['Typeset', MathJax.Hub]
-    )
-    textbook.classList.add(mathRenderedClass)
-  }
-})
-
-var initMathAnchors = () => {
-  // Disable Turbolinks for MathJax links
-  if (typeof MathJax === 'undefined') {
-    setTimeout(initMathAnchors, 250);
-    return;
-  }
-  MathJax.Hub.Queue(function () {
-    document.querySelectorAll('.MathJax a')
-      .forEach(it => it.dataset['turbolinks'] = false);
-  });
-}
-
-initFunction(initMathAnchors);
-
-/**
- * [2] Toggles sidebar and menu icon
- */
-const toggleSidebar = () => {
-  const toggler = getToggler()
-  const textbook = getTextbook()
-
-  if (textbook.classList.contains(textbookActiveClass)) {
-    textbook.classList.remove(textbookActiveClass)
-    toggler.classList.remove(togglerActiveClass)
-  } else {
-    textbook.classList.add(textbookActiveClass)
-    toggler.classList.add(togglerActiveClass)
-  }
-}
-
-/**
- * Keep the variable below in sync with the tablet breakpoint value in
- * _sass/inuitcss/tools/_tools.mq.scss
- *
- */
-const autoCloseSidebarBreakpoint = 769
-
-// Set up event listener for sidebar toggle button
-const sidebarButtonHandler = () => {
-  getToggler().addEventListener('click', toggleSidebar)
-
-  /**
-   * Auto-close sidebar on smaller screens after page load.
-   *
-   * Having the sidebar be open by default then closing it on page load for
-   * small screens gives the illusion that the sidebar closes in response
-   * to selecting a page in the sidebar. However, it does cause a bit of jank
-   * on the first page load.
-   *
-   * Since we don't want to persist state in between page navigation, this is
-   * the best we can do while optimizing for larger screens where most
-   * viewers will read the textbook.
-   *
-   * The code below assumes that the sidebar is open by default.
-   */
-  if (window.innerWidth < autoCloseSidebarBreakpoint) toggleSidebar()
-}
-
-initFunction(sidebarButtonHandler);
-
-/**
- * [3] Preserve sidebar scroll when navigating between pages
- */
-let sidebarScrollTop = 0
-const getSidebar = () => document.getElementById('js-sidebar')
-
-document.addEventListener('turbolinks:before-visit', () => {
-  sidebarScrollTop = getSidebar().scrollTop
-})
-
-document.addEventListener('turbolinks:load', () => {
-  getSidebar().scrollTop = sidebarScrollTop
-})
-
-/**
- * Focus textbook page by default so that user can scroll with spacebar
- */
-const focusPage = () => {
-  document.querySelector('.c-textbook__page').focus()
-}
-
-initFunction(focusPage);
-
-/**
- * [4] Use left and right arrow keys to navigate forward and backwards.
- */
-const LEFT_ARROW_KEYCODE = 37
-const RIGHT_ARROW_KEYCODE = 39
-
-const getPrevUrl = () => document.getElementById('js-page__nav__prev').href
-const getNextUrl = () => document.getElementById('js-page__nav__next').href
-const initPageNav = (event) => {
-  const keycode = event.which
-
-  if (keycode === LEFT_ARROW_KEYCODE) {
-    Turbolinks.visit(getPrevUrl())
-  } else if (keycode === RIGHT_ARROW_KEYCODE) {
-    Turbolinks.visit(getNextUrl())
-  }
-};
-
-var keyboardListener = false;
-const initListener = () => {
-  if (keyboardListener === false) {
-    document.addEventListener('keydown', initPageNav)
-    keyboardListener = true;
-  }
-}
-initFunction(initListener);
-
-/**
- * [5] Scrolling functions:
- *   * Right sidebar scroll highlighting
- *   * Top navbar hiding for scrolling
- */
-
-var didScroll;
-
-initScrollFunc = function() {
-  var content = document.querySelector('.c-textbook__page');
-  var topbar = document.getElementById("top-navbar");
-  var prevScrollpos = content.scrollTop; // Initializing
-
-  scrollFunc = function() {
-    // This is the function that does all the stuff when scrolling happens
-
-    var position = content.scrollTop; // Because we use this differently for sidebar
-
-    // Decide to show the navbar
-    var currentScrollPos = content.scrollTop;
-    var delta = 10;
-    var scrollDiff = prevScrollpos - currentScrollPos;
-    if (scrollDiff >= delta) {
-      // If we scrolled down, consider showing the menu
-      topbar.classList.remove("hidetop")
-    } else if (Math.abs(scrollDiff) >= delta) {
-      // If we scrolled up, consider hiding the menu
-      topbar.classList.add("hidetop")
-    } else {
-      // Do nothing because we didn't scroll enough
-    }
-    prevScrollpos = currentScrollPos;
-
-    // Highlight the right sidebar section
-    position = position + (window.innerHeight / 4);  // + Manual offset
-
-    content.querySelectorAll('h2, h3').forEach((header, index) => {
-      // Highlight based on location from the top of the screen
-      var target = header.getBoundingClientRect().top
-      var pixelOffset = 300;  // Number of pixels from top to be highlighted
-      var id = header.id;
-      if (target < pixelOffset) {
-        var query = 'ul.toc__menu a[href="#' + id + '"]';
-        document.querySelectorAll('ul.toc__menu li').forEach((item) => {item.classList.remove('active')});
-        document.querySelectorAll(query).forEach((item) => {item.parentElement.classList.add('active')});
-      }
-    });
-  }
-
-  // Our event listener just sets "yep, I scrolled" to true.
-  // The interval function will set it to false after it runs.
-  content.addEventListener('scroll', () => {didScroll = true;});
-  scrollWait = 250;
-  setInterval(() => {
-    if (didScroll) {
-      scrollFunc();
-      didScroll = false;
-    }
-  }, scrollWait)
-}
-
-initFunction(initScrollFunc);
-
-
-/**
- * [6] Left sidebar highlight
- *   Loop through the left sidebar links and show / highlight the relevant ones
- */
-
-var updateSidebar = () => {
-  var currentUrl = window.location.href;
-  var chapters = document.querySelector('ul.c-sidebar__chapters')
-  chapters.querySelectorAll('li.c-sidebar__chapter').forEach((chapter, index) => {
-    var sections = chapter.nextElementSibling;
-    if (currentUrl.endsWith(chapter.dataset.url + '.html')) {
-      chapter.querySelector('a').classList.add('c-sidebar__entry--active')
-      if (sections.classList.contains('c-sidebar__sections')) {
-        sections.classList.remove('u-hidden-visually');
-      }
-    }
-
-    // Loop through sections to highlight as needed
-    if (sections) {
-      sections.querySelectorAll('li.c-sidebar__section').forEach((section, ix_section) => {
-        var subsections = section.nextElementSibling;
-
-        // If we're in a top-level section page, show the section
-        if (currentUrl.endsWith(section.dataset.url + '.html')) {
-          section.querySelector('a').classList.add('c-sidebar__entry--active');
-          sections.classList.remove('u-hidden-visually');
-
-          // If we have subsections, show them if we've clicked the parent section
-          if (subsections.classList.contains('c-sidebar__subsections')) {
-            subsections.classList.remove('u-hidden-visually');
-          }
-        }
-
-        // Loop through subections to highlight if needed
-        if (subsections) {
-          subsections.querySelectorAll('li.c-sidebar__subsection').forEach((subsection, ix_subsection) => {
-            if (currentUrl.endsWith(subsection.dataset.url + '.html')) {
-              subsection.querySelector('a').classList.add('c-sidebar__entry--active');
-              sections.classList.remove('u-hidden-visually');
-              subsections.classList.remove('u-hidden-visually');
-            }
-          })
-        }
-      })
-    }
-  });
-}
-initFunction(updateSidebar);
diff --git a/content/LICENSE.md b/content/LICENSE.md
deleted file mode 100755
index 6236026..0000000
--- a/content/LICENSE.md
+++ /dev/null
@@ -1,5 +0,0 @@
-# License for this book
-
-All content in this book (ie, any files and content in the `content/` folder)
-is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/)
-(CC BY-SA 4.0) license.
\ No newline at end of file
diff --git a/content/features/Intro_clases.ipynb b/content/features/Intro_clases.ipynb
deleted file mode 120000
index 4bd6770..0000000
--- a/content/features/Intro_clases.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/Intro_clases.ipynb
\ No newline at end of file
diff --git a/content/features/LagrangePoly.ipynb b/content/features/LagrangePoly.ipynb
deleted file mode 120000
index 1eda3db..0000000
--- a/content/features/LagrangePoly.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/LagrangePoly.ipynb
\ No newline at end of file
diff --git a/content/features/LagrangePolynomial.py b/content/features/LagrangePolynomial.py
deleted file mode 120000
index 9d50d73..0000000
--- a/content/features/LagrangePolynomial.py
+++ /dev/null
@@ -1 +0,0 @@
-../../material/LagrangePolynomial.py
\ No newline at end of file
diff --git a/content/features/Minimization.ipynb b/content/features/Minimization.ipynb
deleted file mode 120000
index 6beb69c..0000000
--- a/content/features/Minimization.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/Minimization.ipynb
\ No newline at end of file
diff --git a/content/features/Pandas.ipynb b/content/features/Pandas.ipynb
deleted file mode 120000
index de1723c..0000000
--- a/content/features/Pandas.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/Pandas.ipynb
\ No newline at end of file
diff --git a/content/features/differential-equations.ipynb b/content/features/differential-equations.ipynb
deleted file mode 120000
index 21f6aaa..0000000
--- a/content/features/differential-equations.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/differential-equations.ipynb
\ No newline at end of file
diff --git a/content/features/figures b/content/features/figures
deleted file mode 120000
index 8555e60..0000000
--- a/content/features/figures
+++ /dev/null
@@ -1 +0,0 @@
-../../material/figures/
\ No newline at end of file
diff --git a/content/features/interpolation.ipynb b/content/features/interpolation.ipynb
deleted file mode 120000
index f189606..0000000
--- a/content/features/interpolation.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/interpolation.ipynb
\ No newline at end of file
diff --git a/content/features/interpolation_details.ipynb b/content/features/interpolation_details.ipynb
deleted file mode 120000
index 7de8eb4..0000000
--- a/content/features/interpolation_details.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/interpolation_details.ipynb
\ No newline at end of file
diff --git a/content/features/ipython-notebooks.ipynb b/content/features/ipython-notebooks.ipynb
deleted file mode 120000
index 28d4dea..0000000
--- a/content/features/ipython-notebooks.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/ipython-notebooks.ipynb
\ No newline at end of file
diff --git a/content/features/least_action_minimization.ipynb b/content/features/least_action_minimization.ipynb
deleted file mode 120000
index 7871de6..0000000
--- a/content/features/least_action_minimization.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/least_action_minimization.ipynb
\ No newline at end of file
diff --git a/content/features/linear-algebra-diagonalization.ipynb b/content/features/linear-algebra-diagonalization.ipynb
deleted file mode 120000
index f15a829..0000000
--- a/content/features/linear-algebra-diagonalization.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/linear-algebra-diagonalization.ipynb
\ No newline at end of file
diff --git a/content/features/linear-algebra-extra.ipynb b/content/features/linear-algebra-extra.ipynb
deleted file mode 120000
index 5451c79..0000000
--- a/content/features/linear-algebra-extra.ipynb
+++ /dev/null
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-../../material/linear-algebra-extra.ipynb
\ No newline at end of file
diff --git a/content/features/linear-algebra.ipynb b/content/features/linear-algebra.ipynb
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\ No newline at end of file
diff --git a/content/features/matplotlib.ipynb b/content/features/matplotlib.ipynb
deleted file mode 120000
index fee8094..0000000
--- a/content/features/matplotlib.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/matplotlib.ipynb
\ No newline at end of file
diff --git a/content/features/numerical-calculus-integration.ipynb b/content/features/numerical-calculus-integration.ipynb
deleted file mode 120000
index 83a293a..0000000
--- a/content/features/numerical-calculus-integration.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/numerical-calculus-integration.ipynb
\ No newline at end of file
diff --git a/content/features/numerical-calculus.ipynb b/content/features/numerical-calculus.ipynb
deleted file mode 120000
index 6ef087d..0000000
--- a/content/features/numerical-calculus.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/numerical-calculus.ipynb
\ No newline at end of file
diff --git a/content/features/one-variable-equations-fixed-point.ipynb b/content/features/one-variable-equations-fixed-point.ipynb
deleted file mode 120000
index f87383b..0000000
--- a/content/features/one-variable-equations-fixed-point.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/one-variable-equations-fixed-point.ipynb
\ No newline at end of file
diff --git a/content/features/one-variable-equations.ipynb b/content/features/one-variable-equations.ipynb
deleted file mode 120000
index 9dc3a89..0000000
--- a/content/features/one-variable-equations.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/one-variable-equations.ipynb
\ No newline at end of file
diff --git a/content/features/overview-python.ipynb b/content/features/overview-python.ipynb
deleted file mode 120000
index b372565..0000000
--- a/content/features/overview-python.ipynb
+++ /dev/null
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-../../material/overview-python.ipynb
\ No newline at end of file
diff --git a/content/features/scientific-libraries.ipynb b/content/features/scientific-libraries.ipynb
deleted file mode 120000
index c91073a..0000000
--- a/content/features/scientific-libraries.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/scientific-libraries.ipynb
\ No newline at end of file
diff --git a/content/features/statistics.ipynb b/content/features/statistics.ipynb
deleted file mode 120000
index e7f104b..0000000
--- a/content/features/statistics.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../../material/statistics.ipynb
\ No newline at end of file
diff --git a/content/images/C-3PO_droid.png b/content/images/C-3PO_droid.png
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z``j;Zw-4Fqq*4I%C|k|r9wm~>t4jdTA$GsO9b)8qZ+2jEg>0?n+~ThO8tilpq=0OX
z&bhS|WG%>55cHh?x`wkLp92&FAcf<(g;O}|@!=a$1cDd0=ay9<$WP8hesTx$UA9hd
zR;>X-7Vv0n0nlkoPx4t@wGM2PZrdM=hR+UUJe?kh+E~945NWOb(fMF_qL^(0`&PGA
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diff --git a/content/index.ipynb b/content/index.ipynb
deleted file mode 120000
index 9199f99..0000000
--- a/content/index.ipynb
+++ /dev/null
@@ -1 +0,0 @@
-../index.ipynb
\ No newline at end of file

From e92602f926984db11964d515b3f2ae4bc34e1dae Mon Sep 17 00:00:00 2001
From: Diego Restrepo <restrepo@udea.edu.co>
Date: Mon, 6 Feb 2023 19:57:52 -0500
Subject: [PATCH 3/9] clean

---
 Gemfile       |  21 ----
 Gemfile.lock  | 309 --------------------------------------------------
 Guardfile     |   8 --
 Makefile      |  34 ------
 book_build.sh |  12 --
 runtime.txt   |   1 -
 6 files changed, 385 deletions(-)
 delete mode 100755 Gemfile
 delete mode 100644 Gemfile.lock
 delete mode 100755 Guardfile
 delete mode 100755 Makefile
 delete mode 100755 book_build.sh
 delete mode 100755 runtime.txt

diff --git a/Gemfile b/Gemfile
deleted file mode 100755
index 5230ce5..0000000
--- a/Gemfile
+++ /dev/null
@@ -1,21 +0,0 @@
-source 'https://rubygems.org'
-
-group :jekyll_plugins do
-  gem 'github-pages'
-  gem 'jekyll-feed', '~> 0.6'
-
-  # Textbook plugins
-  gem 'jekyll-redirect-from'
-  gem 'jekyll-scholar'
-end
-
-# Windows does not include zoneinfo files, so bundle the tzinfo-data gem
-gem 'tzinfo-data', platforms: [:mingw, :mswin, :x64_mingw, :jruby]
-
-# Performance-booster for watching directories on Windows
-gem 'wdm', '~> 0.1.0' if Gem.win_platform?
-
-# Development tools
-gem 'guard', '~> 2.14.2'
-gem 'guard-jekyll-plus', '~> 2.0.2'
-gem 'guard-livereload', '~> 2.5.2'
diff --git a/Gemfile.lock b/Gemfile.lock
deleted file mode 100644
index 3cfafaf..0000000
--- a/Gemfile.lock
+++ /dev/null
@@ -1,309 +0,0 @@
-GEM
-  remote: https://rubygems.org/
-  specs:
-    activesupport (4.2.11.1)
-      i18n (~> 0.7)
-      minitest (~> 5.1)
-      thread_safe (~> 0.3, >= 0.3.4)
-      tzinfo (~> 1.1)
-    addressable (2.8.0)
-      public_suffix (>= 2.0.2, < 5.0)
-    bibtex-ruby (4.4.7)
-      latex-decode (~> 0.0)
-    citeproc (1.0.9)
-      namae (~> 1.0)
-    citeproc-ruby (1.1.10)
-      citeproc (~> 1.0, >= 1.0.9)
-      csl (~> 1.5)
-    coderay (1.1.2)
-    coffee-script (2.4.1)
-      coffee-script-source
-      execjs
-    coffee-script-source (1.11.1)
-    colorator (1.1.0)
-    commonmarker (0.17.13)
-      ruby-enum (~> 0.5)
-    concurrent-ruby (1.1.5)
-    csl (1.5.0)
-      namae (~> 1.0)
-    csl-styles (1.0.1.9)
-      csl (~> 1.0)
-    dnsruby (1.61.3)
-      addressable (~> 2.5)
-    em-websocket (0.5.1)
-      eventmachine (>= 0.12.9)
-      http_parser.rb (~> 0.6.0)
-    ethon (0.12.0)
-      ffi (>= 1.3.0)
-    eventmachine (1.2.7)
-    execjs (2.7.0)
-    faraday (0.17.0)
-      multipart-post (>= 1.2, < 3)
-    ffi (1.11.1)
-    formatador (0.2.5)
-    forwardable-extended (2.6.0)
-    gemoji (3.0.1)
-    github-pages (202)
-      activesupport (= 4.2.11.1)
-      github-pages-health-check (= 1.16.1)
-      jekyll (= 3.8.5)
-      jekyll-avatar (= 0.6.0)
-      jekyll-coffeescript (= 1.1.1)
-      jekyll-commonmark-ghpages (= 0.1.6)
-      jekyll-default-layout (= 0.1.4)
-      jekyll-feed (= 0.11.0)
-      jekyll-gist (= 1.5.0)
-      jekyll-github-metadata (= 2.12.1)
-      jekyll-mentions (= 1.4.1)
-      jekyll-optional-front-matter (= 0.3.0)
-      jekyll-paginate (= 1.1.0)
-      jekyll-readme-index (= 0.2.0)
-      jekyll-redirect-from (= 0.14.0)
-      jekyll-relative-links (= 0.6.0)
-      jekyll-remote-theme (= 0.4.0)
-      jekyll-sass-converter (= 1.5.2)
-      jekyll-seo-tag (= 2.5.0)
-      jekyll-sitemap (= 1.2.0)
-      jekyll-swiss (= 0.4.0)
-      jekyll-theme-architect (= 0.1.1)
-      jekyll-theme-cayman (= 0.1.1)
-      jekyll-theme-dinky (= 0.1.1)
-      jekyll-theme-hacker (= 0.1.1)
-      jekyll-theme-leap-day (= 0.1.1)
-      jekyll-theme-merlot (= 0.1.1)
-      jekyll-theme-midnight (= 0.1.1)
-      jekyll-theme-minimal (= 0.1.1)
-      jekyll-theme-modernist (= 0.1.1)
-      jekyll-theme-primer (= 0.5.3)
-      jekyll-theme-slate (= 0.1.1)
-      jekyll-theme-tactile (= 0.1.1)
-      jekyll-theme-time-machine (= 0.1.1)
-      jekyll-titles-from-headings (= 0.5.1)
-      jemoji (= 0.10.2)
-      kramdown (= 1.17.0)
-      liquid (= 4.0.0)
-      listen (= 3.1.5)
-      mercenary (~> 0.3)
-      minima (= 2.5.0)
-      nokogiri (>= 1.10.4, < 2.0)
-      rouge (= 3.11.0)
-      terminal-table (~> 1.4)
-    github-pages-health-check (1.16.1)
-      addressable (~> 2.3)
-      dnsruby (~> 1.60)
-      octokit (~> 4.0)
-      public_suffix (~> 3.0)
-      typhoeus (~> 1.3)
-    guard (2.14.2)
-      formatador (>= 0.2.4)
-      listen (>= 2.7, < 4.0)
-      lumberjack (>= 1.0.12, < 2.0)
-      nenv (~> 0.1)
-      notiffany (~> 0.0)
-      pry (>= 0.9.12)
-      shellany (~> 0.0)
-      thor (>= 0.18.1)
-    guard-compat (1.2.1)
-    guard-jekyll-plus (2.0.2)
-      guard (~> 2.10, >= 2.10.3)
-      guard-compat (~> 1.1)
-      jekyll (>= 1.0.0)
-    guard-livereload (2.5.2)
-      em-websocket (~> 0.5)
-      guard (~> 2.8)
-      guard-compat (~> 1.0)
-      multi_json (~> 1.8)
-    html-pipeline (2.12.0)
-      activesupport (>= 2)
-      nokogiri (>= 1.4)
-    http_parser.rb (0.6.0)
-    i18n (0.9.5)
-      concurrent-ruby (~> 1.0)
-    jekyll (3.8.5)
-      addressable (~> 2.4)
-      colorator (~> 1.0)
-      em-websocket (~> 0.5)
-      i18n (~> 0.7)
-      jekyll-sass-converter (~> 1.0)
-      jekyll-watch (~> 2.0)
-      kramdown (~> 1.14)
-      liquid (~> 4.0)
-      mercenary (~> 0.3.3)
-      pathutil (~> 0.9)
-      rouge (>= 1.7, < 4)
-      safe_yaml (~> 1.0)
-    jekyll-avatar (0.6.0)
-      jekyll (~> 3.0)
-    jekyll-coffeescript (1.1.1)
-      coffee-script (~> 2.2)
-      coffee-script-source (~> 1.11.1)
-    jekyll-commonmark (1.3.1)
-      commonmarker (~> 0.14)
-      jekyll (>= 3.7, < 5.0)
-    jekyll-commonmark-ghpages (0.1.6)
-      commonmarker (~> 0.17.6)
-      jekyll-commonmark (~> 1.2)
-      rouge (>= 2.0, < 4.0)
-    jekyll-default-layout (0.1.4)
-      jekyll (~> 3.0)
-    jekyll-feed (0.11.0)
-      jekyll (~> 3.3)
-    jekyll-gist (1.5.0)
-      octokit (~> 4.2)
-    jekyll-github-metadata (2.12.1)
-      jekyll (~> 3.4)
-      octokit (~> 4.0, != 4.4.0)
-    jekyll-mentions (1.4.1)
-      html-pipeline (~> 2.3)
-      jekyll (~> 3.0)
-    jekyll-optional-front-matter (0.3.0)
-      jekyll (~> 3.0)
-    jekyll-paginate (1.1.0)
-    jekyll-readme-index (0.2.0)
-      jekyll (~> 3.0)
-    jekyll-redirect-from (0.14.0)
-      jekyll (~> 3.3)
-    jekyll-relative-links (0.6.0)
-      jekyll (~> 3.3)
-    jekyll-remote-theme (0.4.0)
-      addressable (~> 2.0)
-      jekyll (~> 3.5)
-      rubyzip (>= 1.2.1, < 3.0)
-    jekyll-sass-converter (1.5.2)
-      sass (~> 3.4)
-    jekyll-scholar (5.16.0)
-      bibtex-ruby (~> 4.0, >= 4.0.13)
-      citeproc-ruby (~> 1.0)
-      csl-styles (~> 1.0)
-      jekyll (~> 3.0)
-    jekyll-seo-tag (2.5.0)
-      jekyll (~> 3.3)
-    jekyll-sitemap (1.2.0)
-      jekyll (~> 3.3)
-    jekyll-swiss (0.4.0)
-    jekyll-theme-architect (0.1.1)
-      jekyll (~> 3.5)
-      jekyll-seo-tag (~> 2.0)
-    jekyll-theme-cayman (0.1.1)
-      jekyll (~> 3.5)
-      jekyll-seo-tag (~> 2.0)
-    jekyll-theme-dinky (0.1.1)
-      jekyll (~> 3.5)
-      jekyll-seo-tag (~> 2.0)
-    jekyll-theme-hacker (0.1.1)
-      jekyll (~> 3.5)
-      jekyll-seo-tag (~> 2.0)
-    jekyll-theme-leap-day (0.1.1)
-      jekyll (~> 3.5)
-      jekyll-seo-tag (~> 2.0)
-    jekyll-theme-merlot (0.1.1)
-      jekyll (~> 3.5)
-      jekyll-seo-tag (~> 2.0)
-    jekyll-theme-midnight (0.1.1)
-      jekyll (~> 3.5)
-      jekyll-seo-tag (~> 2.0)
-    jekyll-theme-minimal (0.1.1)
-      jekyll (~> 3.5)
-      jekyll-seo-tag (~> 2.0)
-    jekyll-theme-modernist (0.1.1)
-      jekyll (~> 3.5)
-      jekyll-seo-tag (~> 2.0)
-    jekyll-theme-primer (0.5.3)
-      jekyll (~> 3.5)
-      jekyll-github-metadata (~> 2.9)
-      jekyll-seo-tag (~> 2.0)
-    jekyll-theme-slate (0.1.1)
-      jekyll (~> 3.5)
-      jekyll-seo-tag (~> 2.0)
-    jekyll-theme-tactile (0.1.1)
-      jekyll (~> 3.5)
-      jekyll-seo-tag (~> 2.0)
-    jekyll-theme-time-machine (0.1.1)
-      jekyll (~> 3.5)
-      jekyll-seo-tag (~> 2.0)
-    jekyll-titles-from-headings (0.5.1)
-      jekyll (~> 3.3)
-    jekyll-watch (2.2.1)
-      listen (~> 3.0)
-    jemoji (0.10.2)
-      gemoji (~> 3.0)
-      html-pipeline (~> 2.2)
-      jekyll (~> 3.0)
-    kramdown (1.17.0)
-    latex-decode (0.3.1)
-    liquid (4.0.0)
-    listen (3.1.5)
-      rb-fsevent (~> 0.9, >= 0.9.4)
-      rb-inotify (~> 0.9, >= 0.9.7)
-      ruby_dep (~> 1.2)
-    lumberjack (1.0.13)
-    mercenary (0.3.6)
-    method_source (0.9.2)
-    mini_portile2 (2.6.1)
-    minima (2.5.0)
-      jekyll (~> 3.5)
-      jekyll-feed (~> 0.9)
-      jekyll-seo-tag (~> 2.1)
-    minitest (5.13.0)
-    multi_json (1.14.1)
-    multipart-post (2.1.1)
-    namae (1.0.1)
-    nenv (0.3.0)
-    nokogiri (1.12.5)
-      mini_portile2 (~> 2.6.1)
-      racc (~> 1.4)
-    notiffany (0.1.3)
-      nenv (~> 0.1)
-      shellany (~> 0.0)
-    octokit (4.14.0)
-      sawyer (~> 0.8.0, >= 0.5.3)
-    pathutil (0.16.2)
-      forwardable-extended (~> 2.6)
-    pry (0.12.2)
-      coderay (~> 1.1.0)
-      method_source (~> 0.9.0)
-    public_suffix (3.1.1)
-    racc (1.5.2)
-    rb-fsevent (0.10.3)
-    rb-inotify (0.10.0)
-      ffi (~> 1.0)
-    rouge (3.11.0)
-    ruby-enum (0.7.2)
-      i18n
-    ruby_dep (1.5.0)
-    rubyzip (2.0.0)
-    safe_yaml (1.0.5)
-    sass (3.7.4)
-      sass-listen (~> 4.0.0)
-    sass-listen (4.0.0)
-      rb-fsevent (~> 0.9, >= 0.9.4)
-      rb-inotify (~> 0.9, >= 0.9.7)
-    sawyer (0.8.2)
-      addressable (>= 2.3.5)
-      faraday (> 0.8, < 2.0)
-    shellany (0.0.1)
-    terminal-table (1.8.0)
-      unicode-display_width (~> 1.1, >= 1.1.1)
-    thor (0.20.3)
-    thread_safe (0.3.6)
-    typhoeus (1.3.1)
-      ethon (>= 0.9.0)
-    tzinfo (1.2.5)
-      thread_safe (~> 0.1)
-    unicode-display_width (1.6.0)
-
-PLATFORMS
-  ruby
-
-DEPENDENCIES
-  github-pages
-  guard (~> 2.14.2)
-  guard-jekyll-plus (~> 2.0.2)
-  guard-livereload (~> 2.5.2)
-  jekyll-feed (~> 0.6)
-  jekyll-redirect-from
-  jekyll-scholar
-  tzinfo-data
-
-BUNDLED WITH
-   1.17.2
diff --git a/Guardfile b/Guardfile
deleted file mode 100755
index fbf9911..0000000
--- a/Guardfile
+++ /dev/null
@@ -1,8 +0,0 @@
-guard 'jekyll-plus', serve: true do
-  watch /.*/
-  ignore /^_site/
-end
-
-guard 'livereload' do
-  watch /.*/
-end
diff --git a/Makefile b/Makefile
deleted file mode 100755
index cc37ba9..0000000
--- a/Makefile
+++ /dev/null
@@ -1,34 +0,0 @@
-.PHONY: help book clean serve
-
-help:
-	@echo "Please use 'make <target>' where <target> is one of:"
-	@echo "  install     to install the necessary dependencies for jupyter-book to build"
-	@echo "  book        to convert the content/ folder into Jekyll markdown in _build/"
-	@echo "  clean       to clean out site build files"
-	@echo "  runall      to run all notebooks in-place, capturing outputs with the notebook"
-	@echo "  serve       to serve the repository locally with Jekyll"
-	@echo "  build       to build the site HTML and store in _site/"
-	@echo "  site 		 to build the site HTML, store in _site/, and serve with Jekyll"
-
-
-install:
-	jupyter-book install ./
-
-book:
-	jupyter-book build ./
-
-runall:
-	jupyter-book run ./content
-
-clean:
-	python scripts/clean.py
-
-serve:
-	bundle exec guard
-
-build:
-	jupyter-book build ./ --overwrite
-
-site: build
-	bundle exec jekyll build
-	touch _site/.nojekyll
diff --git a/book_build.sh b/book_build.sh
deleted file mode 100755
index 84fe018..0000000
--- a/book_build.sh
+++ /dev/null
@@ -1,12 +0,0 @@
-#!/bin/bash
-if [ ! "$(which python| grep anaconda)" ]; then
-    echo ERROR: conda activate base
-    exit
-fi
-
-
-jupyter-book build --overwrite .
-jupyter-book build .
-git add _build/*
-
-git submodule update --remote Evaluacion_2020-2
diff --git a/runtime.txt b/runtime.txt
deleted file mode 100755
index 548d713..0000000
--- a/runtime.txt
+++ /dev/null
@@ -1 +0,0 @@
-3.7
\ No newline at end of file

From 70e9f1e25b91086a3e6ca00de0c0dc8fbccc035f Mon Sep 17 00:00:00 2001
From: Diego Restrepo <restrepo@udea.edu.co>
Date: Tue, 7 Feb 2023 11:43:44 -0500
Subject: [PATCH 4/9] lc

---
 _config.yml   | 32 ++++++++++++++++++++++++++++++++
 _toc.yml      | 13 +++++++++++++
 build_book.sh |  1 +
 push_build.sh |  1 +
 4 files changed, 47 insertions(+)
 create mode 100644 _config.yml
 create mode 100644 _toc.yml
 create mode 100755 build_book.sh
 create mode 100755 push_build.sh

diff --git a/_config.yml b/_config.yml
new file mode 100644
index 0000000..98c1f67
--- /dev/null
+++ b/_config.yml
@@ -0,0 +1,32 @@
+# Book settings
+# Learn more at https://jupyterbook.org/customize/config.html
+
+title: Advanced Computation
+author: Diego Restrepo (UdeA)
+logo: logo.png
+
+# Force re-execution of notebooks on each build.
+# See https://jupyterbook.org/content/execute.html
+execute:
+  execute_notebooks: 'off'
+
+# Define the name of the latex output file for PDF builds
+latex:
+  latex_documents:
+    targetname: book.tex
+
+# Add a bibtex file so that we can create citations
+bibtex_bibfiles:
+  - references.bib
+
+# Information about where the book exists on the web
+repository:
+  url: https://github.com/restrepo/Advanced_Computation # Online location of your book
+  path_to_book: docs  # Optional path to your book, relative to the repository root
+  branch: master  # Which branch of the repository should be used when creating links (optional)
+
+# Add GitHub buttons to your book
+# See https://jupyterbook.org/customize/config.html#add-a-link-to-your-repository
+html:
+  use_issues_button: true
+  use_repository_button: true
diff --git a/_toc.yml b/_toc.yml
new file mode 100644
index 0000000..ee86fc5
--- /dev/null
+++ b/_toc.yml
@@ -0,0 +1,13 @@
+# Table of contents
+# Learn more at https://jupyterbook.org/customize/toc.html
+
+format: jb-book
+root: intro
+chapters:
+- file: Multiprocessing
+- file: DevOps
+- file: Data_Analysis
+- file: Intro_clases
+- file: fullstack
+- file: markdown-notebooks
+
diff --git a/build_book.sh b/build_book.sh
new file mode 100755
index 0000000..38ede32
--- /dev/null
+++ b/build_book.sh
@@ -0,0 +1 @@
+jupyter-book build --all $(pwd)
diff --git a/push_build.sh b/push_build.sh
new file mode 100755
index 0000000..1533304
--- /dev/null
+++ b/push_build.sh
@@ -0,0 +1 @@
+ghp-import -n -p -f _build/html

From d457b5e5b46f273adc9e0cf111e7fe857f42497a Mon Sep 17 00:00:00 2001
From: Diego Restrepo <restrepo@udea.edu.co>
Date: Tue, 7 Feb 2023 11:45:23 -0500
Subject: [PATCH 5/9] lc

---
 logo.png | Bin 0 -> 17659 bytes
 1 file changed, 0 insertions(+), 0 deletions(-)
 create mode 100644 logo.png

diff --git a/logo.png b/logo.png
new file mode 100644
index 0000000000000000000000000000000000000000..b43e201d931c2d3f21518ba711d9b987750756e5
GIT binary patch
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From 76fb5e020be530f36fff5791ef908700513ba5bb Mon Sep 17 00:00:00 2001
From: Diego Restrepo <restrepo@udea.edu.co>
Date: Tue, 7 Feb 2023 11:56:42 -0500
Subject: [PATCH 6/9] lc

---
 _config.yml | 4 ++--
 _toc.yml    | 9 ++-------
 2 files changed, 4 insertions(+), 9 deletions(-)

diff --git a/_config.yml b/_config.yml
index 98c1f67..a4c8612 100644
--- a/_config.yml
+++ b/_config.yml
@@ -1,7 +1,7 @@
 # Book settings
 # Learn more at https://jupyterbook.org/customize/config.html
 
-title: Advanced Computation
+title: Computational Methods
 author: Diego Restrepo (UdeA)
 logo: logo.png
 
@@ -21,7 +21,7 @@ bibtex_bibfiles:
 
 # Information about where the book exists on the web
 repository:
-  url: https://github.com/restrepo/Advanced_Computation # Online location of your book
+  url: https://github.com/restrepo/ComputationalMethods # Online location of your book
   path_to_book: docs  # Optional path to your book, relative to the repository root
   branch: master  # Which branch of the repository should be used when creating links (optional)
 
diff --git a/_toc.yml b/_toc.yml
index ee86fc5..2a8fef5 100644
--- a/_toc.yml
+++ b/_toc.yml
@@ -2,12 +2,7 @@
 # Learn more at https://jupyterbook.org/customize/toc.html
 
 format: jb-book
-root: intro
+root: index
 chapters:
-- file: Multiprocessing
-- file: DevOps
-- file: Data_Analysis
-- file: Intro_clases
-- file: fullstack
-- file: markdown-notebooks
+- file: material/overview-python
 

From ba05b055d5cb0a294909951566b7fc344fb33352 Mon Sep 17 00:00:00 2001
From: Diego Restrepo <restrepo@udea.edu.co>
Date: Tue, 7 Feb 2023 15:05:09 -0500
Subject: [PATCH 7/9] new book version

---
 _config.yml | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/_config.yml b/_config.yml
index a4c8612..014563f 100644
--- a/_config.yml
+++ b/_config.yml
@@ -30,3 +30,5 @@ repository:
 html:
   use_issues_button: true
   use_repository_button: true
+
+only_build_toc_files: true
\ No newline at end of file

From 2a97347cef183aefb74a8e010080a850bd1fe274 Mon Sep 17 00:00:00 2001
From: Diego Restrepo <restrepo@udea.edu.co>
Date: Tue, 7 Feb 2023 15:35:49 -0500
Subject: [PATCH 8/9] bad subsections

---
 _toc.yml | 11 +++++++++++
 1 file changed, 11 insertions(+)

diff --git a/_toc.yml b/_toc.yml
index 2a8fef5..c25ea82 100644
--- a/_toc.yml
+++ b/_toc.yml
@@ -3,6 +3,17 @@
 
 format: jb-book
 root: index
+options:  # The options key will be applied to all chapters, but not sub-sections
+  numbered: True
+
 chapters:
 - file: material/overview-python
+- file: material/computer-arithmetics
+- file: material/one-variable-equations
+- file: material/interpolation
+- file: material/numerical-calculus
+- file: material/linear-algebra
+- file: material/differential-equations
+- file: material/statistics
+- file: material/Intro_clases
 

From f144bce76cf01b149b379cbee93d6918d4c385a7 Mon Sep 17 00:00:00 2001
From: Diego Restrepo <restrepo@udea.edu.co>
Date: Wed, 8 Feb 2023 15:09:44 -0500
Subject: [PATCH 9/9] alpha version of new book

---
 logo.png                              | Bin 17659 -> 7718 bytes
 material/Intro_clases.ipynb           |  44 ++---
 material/computer-arithmetics.ipynb   |  20 +--
 material/differential-equations.ipynb | 130 +++++++++------
 material/interpolation.ipynb          |  91 +++++++----
 material/linear-algebra.ipynb         | 135 +++++++++-------
 material/numerical-calculus.ipynb     |  77 ++++-----
 material/one-variable-equations.ipynb |  80 ++++++----
 material/overview-python.ipynb        | 222 +++++++++++++++++---------
 material/statistics.ipynb             |  59 +++----
 10 files changed, 511 insertions(+), 347 deletions(-)

diff --git a/logo.png b/logo.png
index b43e201d931c2d3f21518ba711d9b987750756e5..946f0b7f86a82f966bfc01f6a4b39ff352e2c6ba 100644
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diff --git a/material/Intro_clases.ipynb b/material/Intro_clases.ipynb
index a702481..6049030 100644
--- a/material/Intro_clases.ipynb
+++ b/material/Intro_clases.ipynb
@@ -10,13 +10,19 @@
     "<a href=\"https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/Intro_clases.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Clases"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {
     "id": "Y6JhrcAwKDy2"
    },
    "source": [
-    "# Clases\n",
     "En Python, todo es un objeto, es decir, una instancia o realización de un clase. Cada objeto tiene unos _métodos_:\n",
     "```python\n",
     "objeto.metodo(...)\n",
@@ -605,7 +611,7 @@
     }
    ],
    "source": [
-    "n=int() # ⬄ to n=0\n",
+    "n=int() ## ⬄ to n=0\n",
     "n"
    ]
   },
@@ -640,7 +646,7 @@
     }
    ],
    "source": [
-    "x=float(6) # ⬄ x=3.\n",
+    "x=float(6) ## ⬄ x=3.\n",
     "x"
    ]
   },
@@ -664,7 +670,7 @@
     }
    ],
    "source": [
-    "l=list() # ⬄ l=[]\n",
+    "l=list() ## ⬄ l=[]\n",
     "l"
    ]
   },
@@ -688,7 +694,7 @@
     }
    ],
    "source": [
-    "d=dict() # ⬄ d={}\n",
+    "d=dict() ## ⬄ d={}\n",
     "d"
    ]
   },
@@ -744,8 +750,8 @@
     "...\n",
     "class DataFrame(NDFrame):\n",
     "    ...\n",
-    "    # ----------------------------------------------------------------------\n",
-    "    # Constructors\n",
+    "    ## ----------------------------------------------------------------------\n",
+    "    ## Constructors\n",
     "\n",
     "    def __init__(self, data=None, index=None, columns=None, dtype=None,\n",
     "                 copy=False):\n",
@@ -946,7 +952,7 @@
     "id": "yMv7DnahY-Iz"
    },
    "source": [
-    "### Programación por clases"
+    "#### Programación por clases"
    ]
   },
   {
@@ -1218,7 +1224,7 @@
     "id": "udotZC4Ich2u"
    },
    "source": [
-    "### Herencia\n",
+    "#### Herencia\n",
     "Los métodos especiales se pueden heredar automáticamente desde la clase inicial, que llamaremos `superclass`. Para ello inicializamos la clase con la `superclass` como argumento, es decir, en forma genérica como:\n",
     "```python\n",
     "class subclass(superclass):\n",
@@ -1388,7 +1394,7 @@
     "id": "IM9gxUyXQrg4"
    },
    "source": [
-    "### Reescribir método de `superclass`"
+    "#### Reescribir método de `superclass`"
    ]
   },
   {
@@ -1469,7 +1475,7 @@
     "id": "6MnnPiCsQHYi"
    },
    "source": [
-    "### Use un método de la `superclass` con `super`\n",
+    "#### Use un método de la `superclass` con `super`\n",
     "Si lo que queremos es _modificar_ el comportomiento de un método de una superclase entonces debemos usar la función `super`.\n",
     "\n",
     "Para mantener la herencia el comportamiento de los métodos de una clase inicial, ls _super_ clase, se usa la función `super` que mantiene  la __herencia__. Ver:\n",
@@ -2697,11 +2703,11 @@
     "id": "-hmU3is6ch3A"
    },
    "source": [
-    "# Implementation of [arXiv:1905.13729](https://arxiv.org/abs/1905.13729) \n",
+    "## Implementation of [arXiv:1905.13729](https://arxiv.org/abs/1905.13729) \n",
     "See also: DOI: [10.1103/PhysRevD.101.095032](http://doi.org/10.1103/PhysRevD.101.095032)\n",
     "\n",
     "Repo: https://github.com/restrepo/anomalies\n",
-    "## General solution to the U(1) anomaly equations\n",
+    "### General solution to the U(1) anomaly equations\n",
     "Let a vector $\\boldsymbol{z}$ with $N$ integer entries such that\n",
     "$$ \\sum_{i=1}^N z_i=0\\,,\\qquad \\sum_{i=1}^N z_i^3=0\\,.$$\n",
     "We like to build this set of of $N$ from two subsets $\\boldsymbol{\\ell}$ and $\\boldsymbol{k}$ with sizes\n",
@@ -2785,7 +2791,7 @@
     "        y=free( l+[k[0]]+[0]+[-i for i in l ]+[-k[0]])\n",
     "    xfac=0\n",
     "    yfac=0\n",
-    "    # Build not trivial solution\n",
+    "    ## Build not trivial solution\n",
     "    zz=x+y\n",
     "    if sort:\n",
     "        zz=sorted( zz ,key=abs, reverse=reverse ) \n",
@@ -3112,7 +3118,7 @@
     "id": "UxNY3JWNch3D"
    },
    "source": [
-    "## Python implmentation\n",
+    "### Python implmentation\n",
     "Obtain a numpy array `z` of `N` integers which satisfy the Diophantine equations\n",
     "```python\n",
     ">>> z.sum()\n",
@@ -3262,7 +3268,7 @@
     "id": "Ch88lXOsch3F"
    },
    "source": [
-    "## Analysis\n",
+    "### Analysis\n",
     "`solutions` class → Initialize the object to obtain anomaly free solutions for any set of `N` integers"
    ]
   },
@@ -3285,7 +3291,7 @@
     "    q=(q/GCD).astype(int)\n",
     "    if ( #not 0 in z and \n",
     "          0 not in [ sum(p) for p in itertools.permutations(q, 2) ] and #avoid vector-like and multiple 0's\n",
-    "          #q.size > np.unique(q).size and # check for at least a duplicated entry\n",
+    "          #q.size > np.unique(q).size and ## check for at least a duplicated entry\n",
     "          np.abs(q).max()<=q_max\n",
     "           ):\n",
     "        return q,GCD\n",
@@ -3298,7 +3304,7 @@
     "    Call the initialize object with N and get the solutions:\n",
     "    Example:\n",
     "    >>> s=solutions()\n",
-    "    >>> s(6) # N = 6\n",
+    "    >>> s(6) ## N = 6\n",
     "    \n",
     "    Redefine the self.chiral function to implement further restrictions:\n",
     "    inherit from this class and define the new chiral function\n",
@@ -3729,7 +3735,7 @@
     "id": "o0M5XpiEch3J"
    },
    "source": [
-    "## Example\n",
+    "### Example\n",
     "Simple csv to json converter"
    ]
   },
diff --git a/material/computer-arithmetics.ipynb b/material/computer-arithmetics.ipynb
index d14e8bd..5f18f95 100644
--- a/material/computer-arithmetics.ipynb
+++ b/material/computer-arithmetics.ipynb
@@ -199,7 +199,7 @@
     }
    },
    "source": [
-    "### Integers"
+    "#### Integers"
    ]
   },
   {
@@ -906,7 +906,7 @@
     "id": "KL8J_d0y7jNI"
    },
    "source": [
-    "#### Full implementation"
+    "##### Full implementation"
    ]
   },
   {
@@ -1041,7 +1041,7 @@
     "id": "siRJmwCAekWg"
    },
    "source": [
-    "## Binary machine numbers    "
+    "### Binary machine numbers    "
    ]
   },
   {
@@ -1065,7 +1065,7 @@
     "id": "mTaL_rTLekWh"
    },
    "source": [
-    "### Single-precision numbers"
+    "#### Single-precision numbers"
    ]
   },
   {
@@ -1218,7 +1218,7 @@
    },
    "outputs": [],
    "source": [
-    "# %load numtobin.py\n",
+    "## %load numtobin.py\n",
     "def num32( binary ):\n",
     "    #Inverting binary string\n",
     "    binary = binary[::-1]\n",
@@ -1327,7 +1327,7 @@
     "id": "8qJB0mI5ekW0"
    },
    "source": [
-    "### Double-precision numbers"
+    "#### Double-precision numbers"
    ]
   },
   {
@@ -1443,7 +1443,7 @@
     "id": "Frp255HEekW1"
    },
    "source": [
-    "### **ACTIVITY**\n",
+    "### ACTIVITY\n",
     "\n",
     "**1.** Write a python script that calculates the double precision number represented by a 64-bits binary.\n",
     "\n",
@@ -1483,7 +1483,7 @@
     "id": "EMlWF2h_ekW3"
    },
    "source": [
-    "### Addition"
+    "#### Addition"
    ]
   },
   {
@@ -1557,7 +1557,7 @@
     "id": "vdb_-iBaekW_"
    },
    "source": [
-    "### Multiplication"
+    "#### Multiplication"
    ]
   },
   {
@@ -1774,7 +1774,7 @@
     "id": "uvZ8RXy6ekXE"
    },
    "source": [
-    "**ACTIVITY**\n",
+    "### ACTIVITY\n",
     "\n",
     "Find the error associated to the finite representation in the next operations \n",
     "\n",
diff --git a/material/differential-equations.ipynb b/material/differential-equations.ipynb
index 8a2db7e..0dd4665 100644
--- a/material/differential-equations.ipynb
+++ b/material/differential-equations.ipynb
@@ -27,8 +27,6 @@
     "id": "65f7lpUoAujd"
    },
    "source": [
-    "<a href=\"https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/differential-equations.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>\n",
-    "\n",
     "Differential equations is without doubt one of the more useful branch of mathematics in science. They are used to model problems involving the change of some variable with respect to another. Differential equations cover a wide range of different applications, ranging from ordinary differential equations (ODE) until boundary-value problems involving many variables. For the sake of simplicity, throughout this section we shall cover only ODE systems as they are more elemental and equally useful. First, we shall cover first order methods, then second order methods and finally, system of differential equations.\n",
     "\n",
     "See: \n",
@@ -96,7 +94,7 @@
    "outputs": [],
    "source": [
     "import numpy as np\n",
-    "# JSAnimation import available at https://github.com/jakevdp/JSAnimation\n",
+    "## JSAnimation import available at https://github.com/jakevdp/JSAnimation\n",
     "#from JSAnimation import IPython_display\n",
     "from matplotlib import animation"
    ]
@@ -115,7 +113,8 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "ng_1qE8bAujv"
+    "id": "ng_1qE8bAujv",
+    "tags": []
    },
    "source": [
     "## Physical motivation\n",
@@ -157,7 +156,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
+   },
    "source": [
     "### Example: Drag force\n",
     "For a body falling in the air, in addition to the gravitiy force $\\boldsymbol{W}$, there is a dragging force in the opposite direction given by [[see here for details]](https://en.wikipedia.org/wiki/Drag_(physics))![img](https://raw.githubusercontent.com/restrepo/ComputationalMethods/master/material/figures/friction.svg)\n",
@@ -213,10 +215,19 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "iF1Wi1W6Aujz"
+    "id": "iF1Wi1W6Aujz",
+    "tags": []
+   },
+   "source": [
+    "## Euler's method"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": []
    },
    "source": [
-    "## Euler's method\n",
     "### First order differential equations"
    ]
   },
@@ -263,7 +274,7 @@
     "id": "9FEdijqEAuj1"
    },
    "source": [
-    "### Second order differential equations"
+    "#### Second order differential equations"
    ]
   },
   {
@@ -302,7 +313,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Euler method for second order differential equations\n",
+    "#### Euler method for second order differential equations\n",
     "We can define the column vectors\n",
     "\\begin{align}\n",
     "\\boldsymbol{U}=\\begin{bmatrix}\n",
@@ -327,14 +338,24 @@
     "$$"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "### Newton's second law of motion"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "2Gru5svcAuj4"
+    "id": "2Gru5svcAuj4",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## Newton's second law of motion\n",
     "In fact, the equation of motion can be seen as the system of equations\n",
     "\\begin{align}\n",
     "\\frac{\\boldsymbol{p}}{m}=&\\frac{\\operatorname{d}\\boldsymbol{x}}{\\operatorname{d}t}\\\\\n",
@@ -409,14 +430,14 @@
     "import numpy as np\n",
     "import pandas as pd\n",
     "df=pd.DataFrame()\n",
-    "m=.5 # Kg\n",
-    "g=9.8 # m/s^2\n",
-    "# intial conditions\n",
+    "m=.5 ## Kg\n",
+    "g=9.8 ## m/s^2\n",
+    "## intial conditions\n",
     "x=np.array([0,50,0]) #m\n",
-    "v=np.array([5,0,0]) # m/s\n",
+    "v=np.array([5,0,0]) ## m/s\n",
     "p=m*v\n",
     "Δt=0.01 #s\n",
-    "# Analysis for the first 3 s\n",
+    "## Analysis for the first 3 s\n",
     "df=df.append({'x':x,'p':p},ignore_index=True)\n",
     "Fg=np.array([0,-m*g,0]) #N (Weight)\n",
     "\n",
@@ -669,7 +690,7 @@
     }
    ],
    "source": [
-    "#               x       y\n",
+    "##               x       y\n",
     "plt.plot(df.x.str[0],df.x.str[1])\n",
     "plt.xlabel('$x$ [m]',size=15)\n",
     "plt.ylabel('$y$ [m]',size=15)\n",
@@ -1085,7 +1106,7 @@
     "id": "T5kY_8ymAukz"
    },
    "source": [
-    "### Example. Spring\n",
+    "#### Example. Spring\n",
     "In order to apply this, let's assume a simple mass-spring.\n",
     "\n",
     "![](https://raw.githubusercontent.com/sbustamante/ComputationalMethods/master/material/figures/mass_spring.png)\n",
@@ -1110,10 +1131,11 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "LFHDsXTcAukz"
+    "id": "LFHDsXTcAukz",
+    "tags": []
    },
    "source": [
-    "## <font color='red'>     **Activity** </font>"
+    "### <font color='red'>     **Activity** </font>"
    ]
   },
   {
@@ -1335,9 +1357,12 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
+   },
    "source": [
-    "## Implementation in Scipy"
+    "### Implementation in Scipy"
    ]
   },
   {
@@ -1381,7 +1406,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### First order ordinary differential equations\n",
+    "#### First order ordinary differential equations\n",
     "`integrate.odeint`:\n",
     "Integrate a system of ordinary differential equations\n",
     "\n",
@@ -1410,7 +1435,7 @@
     "id": "Nh-pSqV1LcVn"
    },
    "source": [
-    "### Example\n",
+    "#### Example\n",
     "Consider the following differential equation\n",
     "$$ \\frac{dy(t)}{dt}=-k y(t),$$\n",
     "where $k$ is constant and with initial condition $y(0)=y_0=5$"
@@ -1467,7 +1492,7 @@
     }
    ],
    "source": [
-    "# initial condition\n",
+    "## initial condition\n",
     "y0 = 5\n",
     "t=[0,0.1]\n",
     "integrate.odeint(f,y0,t)"
@@ -1526,13 +1551,13 @@
     }
    ],
    "source": [
-    "# time points\n",
+    "## time points\n",
     "t = np.linspace(0,20)\n",
     "\n",
-    "# solve ODE\n",
+    "## solve ODE\n",
     "y = integrate.odeint(f,y0,t)\n",
     "\n",
-    "# plot results\n",
+    "## plot results\n",
     "plt.plot(t,y)\n",
     "plt.xlabel('$t$',size=15)\n",
     "plt.ylabel('$y(t)$',size=15)\n",
@@ -1569,7 +1594,7 @@
     }
    ],
    "source": [
-    "# time points\n",
+    "## time points\n",
     "t = np.linspace(0,20)\n",
     "\n",
     "#k = 0.3 solve ODE\n",
@@ -1577,7 +1602,7 @@
     "y = integrate.odeint(f,y0,t)\n",
     "\n",
     "k=0.3\n",
-    "# plot results\n",
+    "## plot results\n",
     "plt.plot(t,y,'c-',label='ODE solution')\n",
     "plt.plot(t,y0*np.exp(-k*t),'k:',label='Analytical solution')\n",
     "plt.xlabel('$t$',size=15)\n",
@@ -1594,7 +1619,7 @@
     "id": "ocorkhCzAuk3"
    },
    "source": [
-    "### Second order differential equations"
+    "#### Second order differential equations"
    ]
   },
   {
@@ -1627,7 +1652,7 @@
     "id": "hvmkwHOvAuk6"
    },
    "source": [
-    "### Some examples\n",
+    "#### Some examples\n",
     "1. http://www.southampton.ac.uk/~fangohr/teaching/python/book/Python-for-Computational-Science-and-Engineering.pdf\n",
     "   * http://www.southampton.ac.uk/~fangohr/teaching/python/book/ipynb/\n",
     "1. http://sam-dolan.staff.shef.ac.uk/mas212/\n",
@@ -1757,7 +1782,7 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "U0=[0,0] # → x=0"
+    "U0=[0,0] ## → x=0"
    ]
   },
   {
@@ -1987,7 +2012,7 @@
     "    Here U is a vector such that y=U[0] and z=U[1]. \n",
     "    This function should return [y', z']\n",
     "    '''    \n",
-    "    y,z=U # y → U[0]; z → U[1] \n",
+    "    y,z=U ## y → U[0]; z → U[1] \n",
     "    return [        z, \n",
     "            -2*z - 2*y + np.cos(2*x)]\n",
     "\n",
@@ -2117,7 +2142,7 @@
     "            -m*g-c*U[3]/m]\n",
     "\n",
     "m=0.5\n",
-    "# p_x(0)=5*m → v(0)=5 m/s\n",
+    "## p_x(0)=5*m → v(0)=5 m/s\n",
     "U0 = [0, 50,m*5,0]\n",
     "t = np.linspace(0, 3, 200)"
    ]
@@ -2284,20 +2309,20 @@
     "t = np.linspace(0, 3, 200)\n",
     "\n",
     "c=0\n",
-    "Us = integrate.odeint(dU_dt, U0, t,args=(c,) ) # c = 0.5 \n",
+    "Us = integrate.odeint(dU_dt, U0, t,args=(c,) ) ## c = 0.5 \n",
     "xs = Us[:,0]\n",
     "ys = Us[:,1]\n",
     "plt.plot(xs,ys,label=f'c={c}')\n",
     "\n",
     "\n",
     "c=0.1\n",
-    "Us = integrate.odeint(dU_dt, U0, t,args=(c,) ) # c = 0.5 \n",
+    "Us = integrate.odeint(dU_dt, U0, t,args=(c,) ) ## c = 0.5 \n",
     "xs = Us[:,0]\n",
     "ys = Us[:,1]\n",
     "plt.plot(xs,ys,label=f'c={c}')\n",
     "\n",
     "c=0.5\n",
-    "Us = integrate.odeint(dU_dt, U0, t,args=(c,) ) # c = 0.5 \n",
+    "Us = integrate.odeint(dU_dt, U0, t,args=(c,) ) ## c = 0.5 \n",
     "xs = Us[:,0]\n",
     "ys = Us[:,1]\n",
     "plt.plot(xs,ys,label=f'c={c}')\n",
@@ -2329,10 +2354,12 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "sMt9pUzLAulK"
+    "id": "sMt9pUzLAulK",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## Example 2"
+    "### Example 2"
    ]
   },
   {
@@ -2816,10 +2843,13 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
+   },
    "source": [
     "- - -\n",
-    "# Appendix"
+    "## Appendix"
    ]
   },
   {
@@ -2829,7 +2859,7 @@
     "id": "iTnx0JtaAuli"
    },
    "source": [
-    "## <font color='red'>     **Activity** </font>"
+    "### <font color='red'>     **Activity** </font>"
    ]
   },
   {
@@ -2851,7 +2881,7 @@
     "id": "nZwszgJLAulj"
    },
    "source": [
-    "## Fourth-order Runge-Kutta method"
+    "### Fourth-order Runge-Kutta method"
    ]
   },
   {
@@ -2886,7 +2916,7 @@
     "id": "EhfNFEysAulk"
    },
    "source": [
-    "## <font color='red'>     **Activity** </font>"
+    "### <font color='red'>     **Activity** </font>"
    ]
   },
   {
@@ -2923,7 +2953,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Second-order Runge-Kutta methods\n",
+    "### Second-order Runge-Kutta methods\n",
     "For this method, let's assume a problem of the form:\n",
     "\n",
     "$$ \\frac{dy}{dt}=y'=f(t,y),\\ \\ \\ a\\leq t\\leq b, \\ \\ \\ \\ y(a) = \\alpha $$\n",
@@ -2981,7 +3011,7 @@
     "id": "lcFSKLTdAuln"
    },
    "source": [
-    "# Two-Point Boundary Value Problems"
+    "## Two-Point Boundary Value Problems"
    ]
   },
   {
@@ -3033,7 +3063,7 @@
     "id": "MQ5hmF5kAulo"
    },
    "source": [
-    "## Example 3"
+    "### Example 3"
    ]
   },
   {
@@ -3094,7 +3124,7 @@
     "id": "vXfXbWkSAulq"
    },
    "source": [
-    "## Implementation of the pendulum phase space"
+    "### Implementation of the pendulum phase space"
    ]
   },
   {
@@ -3230,7 +3260,7 @@
     "        omega.append( omegai )\n",
     "        time.append( t )\n",
     "        #if i==10:\n",
-    "        #    break\n",
+    "        ##    break\n",
     "    #Plotting solution\n",
     "    plt.plot( theta, omega, lw=0.1, color = \"blue\" )\n",
     "    if jj==50:\n",
diff --git a/material/interpolation.ipynb b/material/interpolation.ipynb
index 5c40100..eff2d0d 100644
--- a/material/interpolation.ipynb
+++ b/material/interpolation.ipynb
@@ -14,11 +14,11 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "XGh1ScX5EZVp"
+    "id": "XGh1ScX5EZVp",
+    "tags": []
    },
    "source": [
-    "# Interpolation Methods\n",
-    "<a href=\"https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/interpolation.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
+    "# Interpolation Methods"
    ]
   },
   {
@@ -235,10 +235,12 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "Z-txzMoJEZV7"
+    "id": "Z-txzMoJEZV7",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "# NumPy Polynomials"
+    "## NumPy Polynomials"
    ]
   },
   {
@@ -1291,10 +1293,11 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "vWJZfq_YEZWe"
+    "id": "vWJZfq_YEZWe",
+    "tags": []
    },
    "source": [
-    "# Linear Interpolation"
+    "## Linear Interpolation"
    ]
   },
   {
@@ -1335,10 +1338,12 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "WsnL9PoDEZWi"
+    "id": "WsnL9PoDEZWi",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## Steps LI"
+    "### Steps LI"
    ]
   },
   {
@@ -1401,10 +1406,12 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "OZXQtU4WEZWj"
+    "id": "OZXQtU4WEZWj",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## Example 1"
+    "### Example 1"
    ]
   },
   {
@@ -1781,10 +1788,12 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "3Zgj3wv8EZW6"
+    "id": "3Zgj3wv8EZW6",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## Example: \n",
+    "### Example: \n",
     "Generate three points that do not lie upon a stright line, and try to make a manual interpolation with a polynomial of degree two."
    ]
   },
@@ -1981,7 +1990,9 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "BP0sX_6fEZXA"
+    "id": "BP0sX_6fEZXA",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
     "### Polynomial object in `numpy`\n",
@@ -2187,10 +2198,11 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "oKqUP0SaEZXZ"
+    "id": "oKqUP0SaEZXZ",
+    "tags": []
    },
    "source": [
-    "# Lagrange Polynomial"
+    "## Lagrange Polynomial"
    ]
   },
   {
@@ -2217,10 +2229,11 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "jN_ZqgxNEZXs"
+    "id": "jN_ZqgxNEZXs",
+    "tags": []
    },
    "source": [
-    "## Derivation"
+    "### Derivation"
    ]
   },
   {
@@ -2264,14 +2277,25 @@
     "Although all this procedure may seem unnecessary for a polynomial of degree 1, a generalization to polynomials of larger degrees is now possible."
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
+   },
+   "source": [
+    "### General case"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "t5benGvIEZXt"
+    "id": "t5benGvIEZXt",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## General case\n",
     "Let's assume again a well-behaved and unknown function $f$ sampled by using a set of $n+1$ data $(x_m,y_m)$ ($0\\leq m \\leq n$).\n",
     "We call the set of $[x_0,x_1,\\ldots,x_n]$ as the _node_ points of the _interpolation polynomial in the Lagrange form_, $P_n(x)$, where:\n",
     "$$f(x)\\approx P_n(x)\\,,$$\n",
@@ -2312,7 +2336,9 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "HAld-e8YEZXu"
+    "id": "HAld-e8YEZXu",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
     "### Example:\n",
@@ -2334,10 +2360,13 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
+   },
    "source": [
     "Until now we can only guarantee that $P_n(x_i)=f(x_i)$. To calculate the function from any $x$ in the interpolation interval we have the following Theorem (See [here](https://www3.nd.edu/~zxu2/acms40390F12/Lec-3.1.pdf))\n",
-    "## Theorem\n",
+    "### Theorem\n",
     "Suppose $X_0,\\ldots,x_n$ are distinct numbers in the interval $[a,b]$ and $f\\in[a,b]$.Then for each $x$ in $[a,b]$, a number $\\xi(x)$ between $x_0,\\ldots,x_n$, and hence in $[a,b]$, exists with\n",
     "$$\n",
     "f(x)=P_n(x)+E_n(x)\\,,\\qquad \\text{such that } E_n(x_i)=0\\,,\n",
@@ -3106,7 +3135,9 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "WSrNnaKcEZX2"
+    "id": "WSrNnaKcEZX2",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
     "### Implementation in `sympy`\n",
@@ -3374,7 +3405,7 @@
     "id": "FEj64uDmEZX8"
    },
    "source": [
-    "## Steps LP"
+    "### Steps LP"
    ]
   },
   {
@@ -3399,7 +3430,7 @@
     "id": "FbuCBnS4EZX9"
    },
    "source": [
-    "**Activity**\n",
+    "### Activity ###\n",
     "\n",
     "Along with the professor, write an implementation of the previous algorithm during classtime."
    ]
@@ -3411,7 +3442,7 @@
     "id": "-1r8xBk8EZX9"
    },
    "source": [
-    "## Activity LP"
+    "### Activity LP"
    ]
   },
   {
@@ -3534,7 +3565,9 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "IAYkvn-_EZYD"
+    "id": "IAYkvn-_EZYD",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
     "### Lets us check! "
@@ -3951,7 +3984,7 @@
     "id": "VSYozPLFEZYl"
    },
    "source": [
-    "# Appendix\n",
+    "## Appendix\n",
     "Thecnical details of interpolation functions: [interpolation_details.ipynb](./interpolation_details.ipynb)"
    ]
   },
diff --git a/material/linear-algebra.ipynb b/material/linear-algebra.ipynb
index deb5dd0..d2e1ffa 100644
--- a/material/linear-algebra.ipynb
+++ b/material/linear-algebra.ipynb
@@ -17,8 +17,7 @@
     "id": "wNc9QP1rZeic"
    },
    "source": [
-    "# Linear Algebra\n",
-    "<a href=\"https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/linear-algebra.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
+    "# Linear Algebra"
    ]
   },
   {
@@ -95,7 +94,7 @@
     "\n",
     "import numpy as np\n",
     "import matplotlib.pyplot as plt\n",
-    "# JSAnimation import available at https://github.com/jakevdp/JSAnimation\n",
+    "## JSAnimation import available at https://github.com/jakevdp/JSAnimation\n",
     "#from JSAnimation import IPython_display\n",
     "from matplotlib import animation\n",
     "#Interpolation add-on\n",
@@ -153,10 +152,12 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "6jt5J-u-ZejE"
+    "id": "6jt5J-u-ZejE",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "# Matrices in Python"
+    "## Matrices in Python"
    ]
   },
   {
@@ -170,7 +171,7 @@
     "\n",
     "NumPy, besides the extreme useful NumPy array objects, which are like tensors of any rank, also provides the Matrix objects that are overloaded with proper matrix operations.\n",
     "\n",
-    "### Numpy arrays\n",
+    "#### Numpy arrays\n",
     "* A matrix can be represented by an array of two indices"
    ]
   },
@@ -323,7 +324,7 @@
     "id": "tfkDRtrKZejO"
    },
    "source": [
-    "#### Special arrays:\n",
+    "##### Special arrays:\n",
     "Let $n$ the range of the matrix\n",
     "* zero matrix"
    ]
@@ -650,7 +651,7 @@
     "id": "5rSf6mZlZekT"
    },
    "source": [
-    "### Analytical matrices\n",
+    "#### Analytical matrices\n",
     "Some times it is necessary to work with analytical Matrices. In such a case we can use the symbolic module `Sympy`"
    ]
   },
@@ -665,8 +666,8 @@
    "outputs": [],
    "source": [
     "import sympy\n",
-    "x =sympy.Symbol('x') # declare analytical varibles\n",
-    "sympy.init_printing() # Use LaTeX to print sympy obejects"
+    "x =sympy.Symbol('x') ## declare analytical varibles\n",
+    "sympy.init_printing() ## Use LaTeX to print sympy obejects"
    ]
   },
   {
@@ -734,7 +735,7 @@
    },
    "source": [
     "In the following we will focus only in numerical matrices as `numpy` arrays, but also use `sympy` to print the matrices in a more readable way\n",
-    "### Matrix operations (numpy)\n",
+    "#### Matrix operations (numpy)\n",
     "Both as functions or array's methods \n",
     "* Transpose"
    ]
@@ -982,7 +983,7 @@
    ],
    "source": [
     "display ( sympy.Matrix(  Mc.conjugate() ) )\n",
-    "display ( sympy.Matrix(  Mc.conjugate().transpose() ) ) # hermitian-conjugate"
+    "display ( sympy.Matrix(  Mc.conjugate().transpose() ) ) ## hermitian-conjugate"
    ]
   },
   {
@@ -1549,7 +1550,7 @@
     "id": "F9oa0NBZZemN"
    },
    "source": [
-    "### Element by element operations"
+    "#### Element by element operations"
    ]
   },
   {
@@ -1667,7 +1668,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Submatrix from matrix: "
+    "#### Submatrix from matrix: "
    ]
   },
   {
@@ -1712,7 +1713,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Extract rows"
+    "#### Extract rows"
    ]
   },
   {
@@ -1826,7 +1827,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Extract columns"
+    "#### Extract columns"
    ]
   },
   {
@@ -1882,7 +1883,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Reordering"
+    "#### Reordering"
    ]
   },
   {
@@ -1987,7 +1988,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### It is possible to reasign a full set of rows to a matrix"
+    "#### It is possible to reasign a full set of rows to a matrix"
    ]
   },
   {
@@ -2182,7 +2183,7 @@
     }
    ],
    "source": [
-    "B=A.copy() # two different memory spaces\n",
+    "B=A.copy() ## two different memory spaces\n",
     "B[0]=4\n",
     "print('A =',A)\n",
     "print('B =',B)"
@@ -2192,7 +2193,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Matrix object"
+    "#### Matrix object"
    ]
   },
   {
@@ -2320,7 +2321,7 @@
     "id": "1bTZjPRGZeml"
    },
    "source": [
-    "## Basic operations with matrices"
+    "### Basic operations with matrices"
    ]
   },
   {
@@ -2344,7 +2345,7 @@
     "id": "0RGhUwogZemm"
    },
    "source": [
-    "### Extract rows from an array"
+    "#### Extract rows from an array"
    ]
   },
   {
@@ -2354,7 +2355,7 @@
     "id": "4X53NZ-oZemn"
    },
    "source": [
-    "## <span style='color:red'> Activity </span>"
+    "### <span style='color:red'> Activity </span>"
    ]
   },
   {
@@ -2640,10 +2641,12 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "Q8nfRt1uZenB"
+    "id": "Q8nfRt1uZenB",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "# Gaussian Elimination"
+    "## Gaussian Elimination"
    ]
   },
   {
@@ -2653,7 +2656,7 @@
     "id": "IO5SMLhJZejA"
    },
    "source": [
-    "## Example 1"
+    "### Example 1"
    ]
   },
   {
@@ -2724,7 +2727,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### <span style=\"color:red\"> Activity </span>\n",
+    "#### <span style=\"color:red\"> Activity </span>\n",
     "Use numpy functions to build the augmented matrix M1"
    ]
   },
@@ -3048,10 +3051,12 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "6Jp7iXYHZei2"
+    "id": "6Jp7iXYHZei2",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "# Linear Systems of Equations"
+    "## Linear Systems of Equations"
    ]
   },
   {
@@ -3083,7 +3088,7 @@
     "id": "uahDSxiBZei7"
    },
    "source": [
-    "## Matrices and vectors"
+    "### Matrices and vectors"
    ]
   },
   {
@@ -3169,7 +3174,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Implementation in Scipy"
+    "#### Implementation in Scipy"
    ]
   },
   {
@@ -3350,7 +3355,7 @@
     "id": "NGgCWPrTZeoD"
    },
    "source": [
-    "## General Gaussian elimination"
+    "### General Gaussian elimination"
    ]
   },
   {
@@ -3559,7 +3564,7 @@
     "id": "adCyEV1hZepz"
    },
    "source": [
-    "## Pivoting Strategies"
+    "### Pivoting Strategies"
    ]
   },
   {
@@ -3625,7 +3630,7 @@
     "id": "n8B6eI0HZepn"
    },
    "source": [
-    "## Computing time"
+    "### Computing time"
    ]
   },
   {
@@ -3653,7 +3658,7 @@
     "id": "xQgyFrSWZepo"
    },
    "source": [
-    "## Example 2"
+    "### Example 2"
    ]
   },
   {
@@ -3760,7 +3765,7 @@
     "id": "0WYsnMsGZepu"
    },
    "source": [
-    "## <span style='color:red'> Activity </span>"
+    "### <span style='color:red'> Activity </span>"
    ]
   },
   {
@@ -3790,7 +3795,7 @@
     "id": "M8_l9AohZepz"
    },
    "source": [
-    "## Partial pivoting"
+    "### Partial pivoting"
    ]
   },
   {
@@ -3814,7 +3819,7 @@
     "id": "ZMjWc8XAZep1"
    },
    "source": [
-    "## <span style='color:red'> Activity </span>"
+    "### <span style='color:red'> Activity </span>"
    ]
   },
   {
@@ -3841,10 +3846,12 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "FqTRdCLBZep2"
+    "id": "FqTRdCLBZep2",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "# Matrix Inversion"
+    "## Matrix Inversion"
    ]
   },
   {
@@ -3964,7 +3971,7 @@
     "id": "PHRKvZk2Zep5"
    },
    "source": [
-    "## <span style='color:red'> Activity </span>"
+    "### <span style='color:red'> Activity </span>"
    ]
   },
   {
@@ -3991,10 +3998,12 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "LH99CU2rZep7"
+    "id": "LH99CU2rZep7",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "# Determinant of a Matrix"
+    "## Determinant of a Matrix"
    ]
   },
   {
@@ -4014,7 +4023,7 @@
     "id": "vaeTK6sjZep7"
    },
    "source": [
-    "## Calculating determinants"
+    "### Calculating determinants"
    ]
   },
   {
@@ -4050,7 +4059,7 @@
     "id": "JQX4y_lAZeqB"
    },
    "source": [
-    "## Computing time of determinants"
+    "### Computing time of determinants"
    ]
   },
   {
@@ -4094,7 +4103,7 @@
     "id": "Ar5GDOlQZeqC"
    },
    "source": [
-    "## <span style='color:red'> Activity </span>"
+    "### <span style='color:red'> Activity </span>"
    ]
   },
   {
@@ -4114,7 +4123,7 @@
     "id": "0_eLCjf2ZeqC"
    },
    "source": [
-    "## Properties of determinants"
+    "### Properties of determinants"
    ]
   },
   {
@@ -4164,7 +4173,7 @@
     "id": "q8dFyFkRZeqD"
    },
    "source": [
-    "## <span style='color:red'> Activity </span>"
+    "### <span style='color:red'> Activity </span>"
    ]
   },
   {
@@ -4181,7 +4190,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Existence of inverse\n",
+    "### Existence of inverse\n",
     "A matrix $A$ has an inverse if $\\det{A}\\ne 0$. See for example [here](http://www.sosmath.com/matrix/inverse/inverse.html)\n",
     "\n",
     "If the matrix $A$ has an inverse, then\n",
@@ -4387,7 +4396,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Numpy implementation"
+    "#### Numpy implementation"
    ]
   },
   {
@@ -4437,10 +4446,12 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "sbOkNC-hZep7"
+    "id": "sbOkNC-hZep7",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "# Matrix diagonalization\n",
+    "## Matrix diagonalization\n",
     "See [linear-algebra-diagonalization](./linear-algebra-diagonalization.ipynb)"
    ]
   },
@@ -4458,10 +4469,12 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "DqQum4S8ZeqF"
+    "id": "DqQum4S8ZeqF",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "# LU Factorization"
+    "## LU Factorization"
    ]
   },
   {
@@ -4514,7 +4527,7 @@
     "id": "h-xWNkyPZeqG"
    },
    "source": [
-    "## <span style='color:red'> Activity </span>"
+    "### <span style='color:red'> Activity </span>"
    ]
   },
   {
@@ -4534,7 +4547,7 @@
     "id": "ODqeEny5ZeqI"
    },
    "source": [
-    "## Derivation of LU factorization"
+    "### Derivation of LU factorization"
    ]
   },
   {
@@ -4626,7 +4639,7 @@
     "id": "WMhljipEZeqI"
    },
    "source": [
-    "## Algorithm for LU factorization"
+    "### Algorithm for LU factorization"
    ]
   },
   {
@@ -4664,7 +4677,7 @@
     "id": "exUdG00wZeqI"
    },
    "source": [
-    "## <span style='color:red'> Activity </span>"
+    "### <span style='color:red'> Activity </span>"
    ]
   },
   {
@@ -4800,7 +4813,7 @@
     "id": "ZMBQ61z8ZeqU"
    },
    "source": [
-    "## <span style='color:red'> Eigenvalues and Eigenvectors activity  </span>"
+    "### <span style='color:red'> Eigenvalues and Eigenvectors activity  </span>"
    ]
   },
   {
@@ -4810,7 +4823,7 @@
     "id": "o2NzCnm8ZeqV"
    },
    "source": [
-    "### Electron interacting with a magnetic field\n",
+    "#### Electron interacting with a magnetic field\n",
     "\n",
     "An electron is placed to interact with an uniform magnetic field. To give account of the possible allowed levels of the electron in the presence of the magnetic field it is necessary to solve the next equation\n",
     "\n",
@@ -4877,7 +4890,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.12"
+   "version": "3.9.2"
   }
  },
  "nbformat": 4,
diff --git a/material/numerical-calculus.ipynb b/material/numerical-calculus.ipynb
index 6956037..608e572 100644
--- a/material/numerical-calculus.ipynb
+++ b/material/numerical-calculus.ipynb
@@ -12,13 +12,9 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {
-    "colab_type": "text",
-    "id": "UKJxA2XF1BYh"
-   },
+   "metadata": {},
    "source": [
-    "# Numerical Calculus\n",
-    "<a href=\"https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/numerical-calculus.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
+    "# Numerical Calculus"
    ]
   },
   {
@@ -33,7 +29,7 @@
     "---\n",
     "* [Class activities with CoCalc](https://cocalc.com/projects/a8330cfb-9dfb-442e-9b2b-ba664e31a685/files/numerical-calculus.ipynb?fullscreen=default&session=default)\n",
     "---\n",
-    "### Bibliography\n",
+    "#### Bibliography\n",
     "* https://github.com/restrepo/Calculus\n",
     "* Thomas J. Sargent John Stachurski, [Python Programming for Quantitative Economics GitHub](https://python-programming.quantecon.org/scipy.html)"
    ]
@@ -97,10 +93,12 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "t4sf-c691BYu"
+    "id": "t4sf-c691BYu",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "# Numerical Differentiation"
+    "## Numerical Differentiation"
    ]
   },
   {
@@ -128,7 +126,7 @@
     "id": "wRMWf0971BYw"
    },
    "source": [
-    "## Example 1"
+    "### Example 1"
    ]
   },
   {
@@ -476,7 +474,7 @@
     "id": "7N5o9BA31BZW"
    },
    "source": [
-    "##  Implementation of the derivate of the function inside a full range\n",
+    "###  Implementation of the derivate of the function inside a full range\n",
     "We now generalize the `derivative` function to allow the evaluation of the derivate in a full range of values. It will be designed such that the evaluation in just one point can be still possible. In this way, the new function  can be used as a full replacement of `derivative` function.\n",
     "\n",
     "To implement this function we need three keys ingredients of `python`:\n",
@@ -484,7 +482,7 @@
     "1. Function definition with generic mandatory and optional arguments\n",
     "1. Conversion of a float function into a vectorized fuction\n",
     "\n",
-    "### `try` and `except` python progamming sctructure\n",
+    "#### `try` and `except` python progamming sctructure\n",
     "First we introduce the `try` and `except` python progamming sctructure, wich is used to bypass one python error. For example a zero dimension array has not a shape attribute, so that the following error, of type `IndexError`, is generated:"
    ]
   },
@@ -890,7 +888,7 @@
     "        else:\n",
     "            return False\n",
     "\n",
-    "# Test the functions for typical cases        \n",
+    "## Test the functions for typical cases        \n",
     "assert is_integer(3.)\n",
     "assert is_integer(3.5)==False\n",
     "assert is_integer(3)\n",
@@ -908,7 +906,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Testing"
+    "### Testing"
    ]
   },
   {
@@ -992,7 +990,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Function parameters\n",
+    "#### Function parameters\n",
     "In python the function can be defined with mandatory and optional arguments"
    ]
   },
@@ -1025,7 +1023,7 @@
     "id": "NMDk53KY1BZl"
    },
    "source": [
-    "### Function definition with generic mandatory and optional arguments\n",
+    "#### Function definition with generic mandatory and optional arguments\n",
     "We need to introduce another important concept about functions. There is a powerfull way to define a function with generic arguments:\n",
     "1. Instead of the mandatory arguments we can use the generic list pointer: `*args`\n",
     "1. Instead of the optional arguments we can use the generic dictionary pointer: `**kwargs`"
@@ -1235,7 +1233,7 @@
     "id": "K372aP-a1BZ5"
    },
    "source": [
-    "### Conversion of a float function into a vectorized function\n",
+    "#### Conversion of a float function into a vectorized function\n",
     "An important `numpy` function is:"
    ]
   },
@@ -1352,7 +1350,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Vectorization of Scipy method `derivative`\n",
+    "#### Vectorization of Scipy method `derivative`\n",
     "\n",
     "The problem is that `derivative` only works for one a point"
    ]
@@ -1506,7 +1504,7 @@
     "id": "GgksedR51BaG"
    },
    "source": [
-    "## Code for the derivate of the function inside a full range\n",
+    "### Code for the derivate of the function inside a full range\n",
     "\n",
     "To recover the evaluation of a float into a float we can force the `IndexError` in order to use of the pure `scipy` `derivative` function when a float is given as input. The full implemention combine the three previos ingredients into a very compact pythonic-way function definition as shwon below"
    ]
@@ -1542,7 +1540,7 @@
     "    '''\n",
     "    try:\n",
     "        #x0: can be an array or a list  \n",
-    "        nn=np.asarray(x0).shape[0] # force error if float is used \n",
+    "        nn=np.asarray(x0).shape[0] ## force error if float is used \n",
     "        fp=np.vectorize(misc.derivative)\n",
     "    except IndexError:\n",
     "        fp=misc.derivative\n",
@@ -1634,7 +1632,7 @@
     "    '''\n",
     "    try:\n",
     "        #x0: can be an array or a list  \n",
-    "        nn=np.asarray(x0).shape[0] # force error if float is used \n",
+    "        nn=np.asarray(x0).shape[0] ## force error if float is used \n",
     "        fp=np.vectorize(misc.derivative)\n",
     "    except IndexError:\n",
     "        fp=misc.derivative\n",
@@ -1813,7 +1811,7 @@
     "id": "KsNQouL31Baa"
    },
    "source": [
-    "### Test\n",
+    "#### Test\n",
     "Let us check the implementation with the previous function"
    ]
   },
@@ -1982,7 +1980,7 @@
     "id": "fJsq3pgq1Bay"
    },
    "source": [
-    "### Example\n",
+    "#### Example\n",
     "Finally we check the function $f(x)=\\cos x$ in the range $[0,2\\pi]$"
    ]
   },
@@ -2087,7 +2085,7 @@
     "        from scipy.misc import derivative\n",
     "        derivative?\n",
     "    '''\n",
-    "    if isinstance(x0,list): # or ... or ...\n",
+    "    if isinstance(x0,list): ## or ... or ...\n",
     "        print(x0)\n",
     "        fp=np.vectorize(misc.derivative)\n",
     "    else:\n",
@@ -2142,7 +2140,7 @@
     "id": "ZyOV6qnu1BbC"
    },
    "source": [
-    "## Example: Heat transfer in a 1D bar "
+    "### Example: Heat transfer in a 1D bar "
    ]
   },
   {
@@ -2224,7 +2222,7 @@
     "def Temp(x):\n",
     "    return x**3 + 3*x-1\n",
     "\n",
-    "# Points where function is known\n",
+    "## Points where function is known\n",
     "Xn = np.linspace(0,10,100)\n",
     "\n",
     "plt.figure( figsize=(8,7) )\n",
@@ -2243,7 +2241,7 @@
     "id": "HXUbjyh81BbP"
    },
    "source": [
-    "## Activity"
+    "### Activity"
    ]
   },
   {
@@ -2264,7 +2262,7 @@
     "id": "QtPqZYqG1BbU"
    },
    "source": [
-    "## Activity\n",
+    "### Activity\n",
     "\n",
     "    \n",
     "The Poisson's equation relates the matter content of a body with the gravitational potential through the next equation\n",
@@ -2286,7 +2284,7 @@
     "id": "uIC7_pDp1BbU"
    },
    "source": [
-    "## Activity"
+    "### Activity"
    ]
   },
   {
@@ -2333,15 +2331,18 @@
    "cell_type": "markdown",
    "metadata": {
     "colab_type": "text",
-    "id": "MR7WLv5v1BbX"
+    "id": "MR7WLv5v1BbX",
+    "tags": []
    },
    "source": [
-    "# [Numerical Integration](./numerical-calculus-integration.ipynb)  "
+    "## [Numerical Integration](./numerical-calculus-integration.ipynb)  "
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "tags": []
+   },
    "source": [
     "## Appendix"
    ]
@@ -2353,7 +2354,7 @@
     "id": "tZz5pIMR1Ba3"
    },
    "source": [
-    "### One implementation of `derivative` algorithm"
+    "#### One implementation of `derivative` algorithm"
    ]
   },
   {
@@ -2443,7 +2444,7 @@
     "id": "lGExPWl81Ba-"
    },
    "source": [
-    "## n+1-point formula"
+    "### n+1-point formula"
    ]
   },
   {
@@ -2510,7 +2511,7 @@
     "id": "73Bfd3Bb1BbA"
    },
    "source": [
-    "## Endpoint formulas"
+    "### Endpoint formulas"
    ]
   },
   {
@@ -2545,7 +2546,7 @@
     "id": "xPu3StrM1BbB"
    },
    "source": [
-    "## Midpoint formulas"
+    "### Midpoint formulas"
    ]
   },
   {
@@ -2633,7 +2634,7 @@
     "def TMP( Ynh,Ynmh, h = 0.01 ):    \n",
     "    return (Ynh-Ynmh)/(2*h)\n",
     "\n",
-    "# Points where function is known\n",
+    "## Points where function is known\n",
     "Xn = np.linspace(0,10,100)\n",
     "Tn = Temp(Xn)\n",
     "#Magnitude of heat flux array\n",
@@ -2689,7 +2690,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.12"
+   "version": "3.9.2"
   }
  },
  "nbformat": 4,
diff --git a/material/one-variable-equations.ipynb b/material/one-variable-equations.ipynb
index 92777ca..80e1c16 100644
--- a/material/one-variable-equations.ipynb
+++ b/material/one-variable-equations.ipynb
@@ -1,5 +1,19 @@
 {
  "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<a href=\"https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/one-variable-equations.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# One Variable Equations"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {
@@ -9,14 +23,10 @@
     "id": "cx_2LAjUw7VH",
     "slideshow": {
      "slide_type": "slide"
-    }
+    },
+    "tags": []
    },
    "source": [
-    "<a href=\"https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/one-variable-equations.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>\n",
-    "\n",
-    "# One Variable Equations\n",
-    "<a href=\"https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/one-variable-equations.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>\n",
-    "\n",
     "Throughout this section and the next ones we shall cover the topic of solutions to one variable equations. Many different problems in physics and astronomy require the use of complex expressions, even with implicit dependence of variables. When it is necessary to solve for one of those variable, an analytical approach is not usually the best solution, because of its complexity or even because it does not exist at all. Different approaches for dealing with this comprehend series expansions and numerical solutions. Among the most widely used numerical approaches are the Bisection or Binary-search method, fixed-point iteration, Newton's methods.\n",
     "\n",
     "For further details see for example Chap. 5 of Ref. ([1](../index.ipynb#Bibliography)) or Chap. 4 of Ref. ([3](../index.ipynb#Bibliography))\n"
@@ -28,7 +38,8 @@
     "cell_id": "00001-9b5c5089-cfa0-452e-9212-f426a9b2eab6",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "sExxbMPEw7VM"
+    "id": "sExxbMPEw7VM",
+    "tags": []
    },
    "source": [
     "- - -\n",
@@ -55,7 +66,7 @@
     "    - [Steps](./one-variable-equations-fixed-point.ipynb#Steps-SM)\n",
     "\n",
     "- - -\n",
-    "## Bibliography\n",
+    "##### Bibliography\n",
     "[1] [Kiusalaas, Numerical Methods in Engineering with Python](https://drive.google.com/file/d/0BxoOXsn2EUNIQUdFVkctR2xWRUk/view?usp=sharing)<br/>\n",
     "[2] [Jensen, Computational_Physics](https://drive.google.com/file/d/0BxoOXsn2EUNIekRUMFVPYXVsSTg/view?usp=sharing).  Companion repos: https://github.com/mhjensen [Web page](http://compphysics.github.io/ComputationalPhysics/doc/web/course)  <br/>\n",
     "[3] E. Ayres, [Computational Physics With Python](http://bit.ly/CPwPython)<br/>\n"
@@ -164,10 +175,11 @@
     "id": "RNEo8n--w7Vc",
     "slideshow": {
      "slide_type": "slide"
-    }
+    },
+    "tags": []
    },
    "source": [
-    "# Bisection Method"
+    "## Bisection Method"
    ]
   },
   {
@@ -194,7 +206,7 @@
     }
    },
    "source": [
-    "## Steps BM"
+    "#### Steps BM"
    ]
   },
   {
@@ -241,10 +253,11 @@
     "id": "7uoUfZn9w7Vk",
     "slideshow": {
      "slide_type": "slide"
-    }
+    },
+    "tags": []
    },
    "source": [
-    "## Stop condition BM"
+    "### Stop condition BM"
    ]
   },
   {
@@ -286,10 +299,11 @@
     "cell_id": "00012-ff797fa2-7ac6-4758-babd-e6465b4c2cf0",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "nBQBHO23w7Vm"
+    "id": "nBQBHO23w7Vm",
+    "tags": []
    },
    "source": [
-    "## Error analysis BM"
+    "### Error analysis BM"
    ]
   },
   {
@@ -387,7 +401,7 @@
    },
    "outputs": [],
    "source": [
-    "# plt.plot( x,y  ) # points joined with straight lines "
+    "#### plt.plot( x,y  ) # points joined with straight lines "
    ]
   },
   {
@@ -404,7 +418,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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\n",
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",
       "text/plain": [
        "<Figure size 432x288 with 1 Axes>"
       ]
@@ -416,7 +430,7 @@
     }
    ],
    "source": [
-    "# 1/2**i → exp2(-i)\n",
+    "#### 1/2**i → exp2(-i)\n",
     "plt.semilogy(i,np.exp2(-i))\n",
     "plt.grid()"
    ]
@@ -435,7 +449,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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\n",
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",
       "text/plain": [
        "<Figure size 432x432 with 1 Axes>"
       ]
@@ -470,10 +484,12 @@
     "cell_id": "00019-b2770622-2ad4-402d-a96e-536b3ad56f04",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "pd9PwTe1w7V-"
+    "id": "pd9PwTe1w7V-",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## Example 1"
+    "### Example 1"
    ]
   },
   {
@@ -1111,7 +1127,8 @@
     "cell_id": "00036-c0648f60-b13b-4d60-b251-69523c1d13fb",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "yC9nyRCwGJaw"
+    "id": "yC9nyRCwGJaw",
+    "tags": []
    },
    "source": [
     "## General arguments and options of a SciPy function"
@@ -1589,7 +1606,8 @@
     "cell_id": "00060-7cfe2241-3f28-42a9-bddf-0e04f4537090",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "IJIfqraOG_fz"
+    "id": "IJIfqraOG_fz",
+    "tags": []
    },
    "source": [
     "### Example\n",
@@ -2151,7 +2169,7 @@
     "id": "Mf8r3RfJw7XG"
    },
    "source": [
-    "### <font color='red'> ACTIVITY </font>"
+    "###### <font color='red'> ACTIVITY </font>"
    ]
   },
   {
@@ -2271,7 +2289,7 @@
     }
    ],
    "source": [
-    "# Solo usar para constante\n",
+    "#### Solo usar para constante\n",
     "np.e"
    ]
   },
@@ -2311,10 +2329,12 @@
     "cell_id": "00092-16a277e5-2710-41fa-8654-b258dc4899c3",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "hBSyBUNrw7XJ"
+    "id": "hBSyBUNrw7XJ",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## Example 2"
+    "### Example 2"
    ]
   },
   {
@@ -2478,7 +2498,7 @@
     },
     {
      "data": {
-      "image/png": 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\n",
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C0+P6o8AxZtbCzHoCvYGXgalA79hDrjmhQ8OjMUNPIlwxAQwHHsllnUQkT3r1gmefheuuC9NO/O1voaxTJ43mXQSa1r9L1uxNaEZ7y8xej2UXE3q97Qo4MAc4DcDdZ5jZvcDbhB53I9x9NYCZnQk8SWjSG+PuM+L5fguMN7MrgNcIiU9EikGTJnDWWWEUhh/9CKqr126rGc0bNGNyI5S3ROTuLwKWZtOEDRwzinDfqHb5hHTHufts1jbtiUgx+o//gG+/Xb+8ZjRvJaJGRyMriEjjU9e4dBrNu1FSIhKRxmdDo3lPnJi/OCQrlIhEpPHZ0Gjehx0GY8YkE5dsFCUiEWl80o3mfccdcMEFsGoVnHwy/Od/aqy6RiKfveZERLKnoiJ9x4RevWDEiHDVNHs23H47tGiR//ikwXRFJCLF5bTT4O9/h9at4e674eCD4fPPk45KNkCJSESKz6GHwosvhkFUX3gBfvhDeP/9pKOSOigRiUhx2mUXmDIFdt4Z3nsPBgyAf/0r6agkDSUiESleXbqEK6IhQ+Czz2DQILj//vqPk7xSIhKR4tamTbhndNppsHIlHH00XHWVetQVECUiESl+TZvCjTfClVeG9xdcAGecEbp6S+Lq7b5tZu0bcJ417v7FpocjIpIjZiEB9egRppG46aYwWOo994TZYCUxDXmO6MO4pBuwtEYZsIExN0RECsTPfx7uHf3kJ/D447DvvmF68s6dk46sZDWkaW6mu/dy9551LYRJ6UREGocf/jD0qOvdG954A/bcE37/+3C1pPmN8q4hiWivLO0jIlI4ttsudOfeZx9YuBAuvjg01bmvnd9IySgv6k1E7r4iG/uIiBScDh3CaN21B1CFtfMbSc41eKw5M+sPjAS6x+MMcHffOUexiYjk3mabwTffpN+m+Y3yIpNBT6uA84G3gDW5CUdEJAHduoXmuHTlknOZPEf0qbs/6u4fuPvcmiVnkYmI5Eu6+Y0ATj89/7GUoEwS0aVmdquZHWtmR9YsOYtMRCRfas9v1LJlKL/hhtCRQXIqk0R0IrArMAQ4PC4/zkFMIiL5V1EBc+bAmjVh2oi994YFC8JI3l9+mXR0RS2Te0S7u/v2OYtERKRQtGwJjzwSktFbb8FPfxoeftUEezmRyRXR/5lZn5xFIiJSSDp0gCeegK23hkmT4MQTw9WSZF0miWgA8LqZvWtmb5rZW2b2ZkMPNrOuZjbJzN42sxlmdnYsb29mE81sVnxtF8vNzK43s+r4eX1TzjU87j/LzIanlPeLcVXHYzc0LJGIyIb16AETJqyd7fXCC5OOqChlkoiGAL2Bway9P3R4BsevAs5z9z6EpDYiXmFdCDzj7r2BZ+J7gEPj5/UGKoEb4d+DsF4K7AnsQehE0S4ecyNwaspxQzKIT0RkfbvtBg8+GEbwvuoq+N//TTqiopNJIuqY2m07dt3+QUMPdveP3P3VuP4VMBPoDAwFxsbdxgJHxPWhwJ0eTAHamtk2wCHARHdf7O5LgInAkLitjbtPcXcH7kw5l4jIxjv4YLjttrB+9tnwwAPJxlNkMumscIuZDXP36QBmdixwDvBYph9qZj2A3YCXgHJ3/yhu+hgoj+udgfkphy2IZRsqX5CmPN3nVxKusigvL2fy5MmZViHnli1bVpBx5YLqWnyKsp7dutHtlFPodeutrDn2WN7405/4cuedi7OuaeSynpkkoqOA+83sOGBfYBihmS4jZtYaeAA4x92Xpt7GcXc3s5xPm+juo4HRAP379/eBAwfm+iMzNnnyZAoxrlxQXYtP0dZz//2hWTOa3Hgju112Gfzzn0yG4qxrLbn8mTa4ac7dZwPHAA8CPwMGu3tGnevNrBkhCVW5+4OxeFFsViO+fhLLFwJdUw7vEss2VN4lTbmISHaYhXtEQ4fCkiUwZAjNP/ss6agavXoTUU3vuNhD7n6gPdATeCnDXnMG3EaY3+ialE2PAjU934YDj6SUD4u95wYAX8YmvCeBwWbWLnZSGAw8GbctNbMB8bOGpZxLRCQ7ysrgrrtgwACYN4+df/tbWLo06agatYY0zWVr9IS9geOBt8zs9Vh2MfAH4F4zOxmYC/w8bpsAHAZUA8sJIzvg7ovN7HfA1Ljff7v74rh+BnAH0BJ4PC4iItnVqhX8/e+w9960fu89OPLI0M27efOkI2uUGpKIHnL3vhvawcxerW8fd3+RuqcbPzDN/g6MqONcY4AxacqnATttKA4Rkazo2BGeeIJv+/Wj+TPPwEknwZ13hhleJSMNSUQ71NMEZ8CWWYpHRKTx6NmTN3//e/qfd16YzbVLF/jDH5KOqtFpSCL6fgP2Wb2pgYiINEbLtt8e7r8fDj8crrwSunaFEWkbc6QODZkqfG4DlgX1nUdEpGgNGQK33BLWzzwTOnUKTXQ9eoQrJdkgNWaKiGTDCSfAUUeF9U8/Bfcw62tlpZJRPZSIRESy5eWX1y9bvhxGjsx/LI2IEpGISLbMn5++fN68/MbRyDQ4EcUHS39pZpfE993MbI/chSYi0sh065ZZuQCZXRH9FdgLODa+/wr4S9YjEhFprEaNCg+71vab3+Q/lkYkk0S0p7uPAFYAxCkY9BixiEiNigoYPRq6dw/j0m22WSh/6qnQeUHSyiQRfWdmZYADmNlWgObNFRFJVVEBc+aEacVnzYI2bcJwQHfdlXRkBSuTRHQ98BDQycxGAS8Cv89JVCIixaBLF7gmjvF81lnw8cfJxlOgMpkGogq4gJB8PgKOcPd7cxWYiEhROOkkGDwYFi8OIy6oiW49mfSau9Ld33H3v7j7De4+08yuzGVwIiKNnlkYdaF1a3jwQbjvvqQjKjiZNM0dnKbs0GwFIiJStLp1g6uuCusjRoSRF+TfGjIx3ulm9hawfc0EeXH5AHgr9yGKiBSBykoYNAg++wx+/eukoykoDbkiugs4nDBj6uEpSz93r8hhbCIixaNJE7j11vCc0T33wEMPJR1RwWjI6Ntfuvscdz8WWAqUA92Bncxsv1wHKCJSNHr1Wjtf0emnw+efJxtPgciks8IpwPPAk8Dl8fWy3IQlIlKkRoyAffeFRYvgnHOSjqYgZNJZ4Wxgd2Cuuw8CdgO+yEVQIiJFq0kTuO22MOrCuHHw2GNJR5S4TBLRCndfAWBmLdz9HWD73IQlIlLEevcO49IBnHYafPFFouEkLZNEtMDM2gIPAxPN7BFgbi6CEhEpemefDXvtBR9+COeem3Q0icpkZIWfuvsX7n4Z8F/AbcDQXAUmIlLUyspgzBho0QJuvx2eeCLpiBKTSWeFFmZ2nJldDOwP7ApclMHxY8zsEzObnlJ2mZktNLPX43JYyraLzKzazN41s0NSyofEsmozuzClvKeZvRTL7zEzjQwuIoXt+9+Hyy8P66eeCkuXJhtPQjJpmnuEcAW0Cvg6ZWmoO4Ahacqvdfdd4zIBwMz6AMcAO8Zj/mpmZXH0778QRnToAxwb9wW4Mp5rO2AJcHIGsYmIJOO882D33WHBAjj//KSjSUTTDPbt4u7pEkmDuPvzZtajgbsPBca7+0rgAzOrBmpmg61299kAZjYeGGpmM4EDgOPiPmMJXctv3Nh4RUTyomnT0ETXt2+Yy+jnP4cDD0w6qrzK5Iro/8zsBzmI4cw4ZNAYM2sXyzoDqZO/L4hldZV3AL5w91W1ykVECt9OO8Ell4T1U06BZcuSjSfPMrki2gc4IY4xtxIwwN195034/BuB3xEm2/sdcDVw0iacr0HMrBKoBCgvL2fy5Mm5/siMLVu2rCDjygXVtfiUSj0he3W1AQPo27s3W8yaxapOnShbsYKVnTox+5RT+OSggzY90E2Uy59pJoko6yNtu/uimnUzuwWoebJrIdA1ZdcusYw6yj8H2ppZ03hVlLp/us8dDYwG6N+/vw8cOHDTKpIDkydPphDjygXVtfiUSj0hy3U98US4+GKafvMNAJstWkSfa6+lzw47hJlfE5TLn2km3bfnpls25cPNbJuUtz8FanrUPQocE3vq9QR6Ay8DU4HesYdcc0KHhkfd3YFJwFHx+OGEzhUiIo3HzTevX7Z8OYwcmf9Y8qjeKyIze9Hd9zGzrwhNaJb66u5tGvJBZnY3MBDoaGYLgEuBgWa2azzfHOA0wklnmNm9wNuEXnoj3H11PM+ZhHHuyoAx7j4jfsRvgfFmdgXwGuE5JxGRxmPevMzKi0S9icjd94mvW2zKB8XRu2urM1m4+yhgVJryCcCENOWzWduzTkSk8enWDeamaWjq1i3/seRRQ66INjj2hLtfk71wRERK2KhRYQK95cvXljVvvnZcuiLVkM4KNVdC2xNG3340vj+ccN9GRESyoaZDwsiRa6+MOneG446r+5gi0JCJ8S5398sJPdH6uvt57n4e0A8o7utFEZF8q6iAOXPCVdFWW8EHH8BzzyUdVU5l8kBrOfBtyvtvY5mIiGRby5Zw5plh/aqrko0lxzJJRHcCL8eBSi8DXiKMHyciIrlwxhkhIU2YADNm1L9/I9WgRGRmRkhEJxIGFF0CnOjuv89hbCIipa1jx/CQK8Cf/pRsLDnUoEQUHxid4O6vuvt1cXktx7GJiMi554bpxauqYGGdA8Y0apk0zb1qZrvnLBIREVnf974HRx4J330H11+fdDQ5kUki2hP4l5m9H0fLfsvM3sxVYCIiEtXMU3TTTUU5eV4mg54eUv8uIiKSdXvsAfvtB88/D7fcEibTKyKJDnoqIiINVHNV9Oc/h2a6IpJJ0xxmtouZnRmXXXIVlIiI1HLYYbDDDmFK8XvuSTqarGpwIjKzs4EqoFNcxpnZr3MVmIiIpGjSBH7zm7B+1VXgnmw8WZTJFdHJwJ7ufom7XwIMAE7NTVgiIrKeigrYemt4802YODHpaLImk0RkwOqU96tjmYiI5EOLFnDWWWG9iIb9ySQR3Q68FIf4uRyYgiafExHJr1/9CjbfHJ5+Gl4rjnEFMuk1dw1hiJ/FwGeEIX7+nKO4REQknXbt4NR4V6RIhv3JqNccoTnO47Im++GIiEi9zjkHyspC77l0M7o2MhvTa64j6jUnIpKc7t3hF7+A1avDc0WN3Mb0mrtUveZERBJW84DrLbfAkiXJxrKJ1GtORKQx2nVXOOgg+PpruPnmpKPZJBvba+4yQq+5MTmJSkRE6lfzgOt118HKlcnGsgk2ttfcYkKvuWtzFZiIiNRj8GDYeWf4+OMwX1EjlUlnhbHAbHe/3t2vB+aYWYOviMxsjJl9YmbTU8ram9lEM5sVX9vFcjOz682sOk450TflmOFx/1lmNjylvF+cmqI6HqtmQxEpbmZrr4pOOy0MA9SjR6NLSpk0ze3s7l/UvHH3JcBuGRx/BzCkVtmFwDPu3ht4Jr4HOBToHZdK4EYIiQu4lDA30h7ApTXJK+5zaspxtT9LRKT4uIeEtGpVWJ87FyorG1UyyiQRNUn5pV+TFBo8n5G7P09o0ks1FBgb18cCR6SU3+nBFKCtmW1DmBNporsvjolwIjAkbmvj7lPitOZ3ppxLRKR4XXLJ+gOgLl8OI0cmE89GyGRivKsJM7TeF98fDYzaxM8vd/eP4vrHQHlc7wzMT9lvQSzbUPmCNOUiIsVt3rzMygtQJlc0d5rZNOCAWHSku7+drUDc3c0sL+Oam1klocmP8vJyJk+enI+PzciyZcsKMq5cUF2LT6nUE5Kv64BOndhs0aL1yld06sSULMaV03q6e94WoAcwPeX9u8A2cX0b4N24fjNwbO39gGOBm1PKb45l2wDvpJSvs9+Gln79+nkhmjRpUtIh5I3qWnxKpZ7uBVDXcePcW7VyDw10YWnVKpRnUe16AtM8S7kh07Hmsu1RoKbn23DgkZTyYbH33ADgSw9NeE8Cg82sXbxfNRh4Mm5bamYDYm+5YSnnEhEpXhUVMHo0bLtteN+kSXjAtaIi2bgykLdEZGZ3A/8CtjezBWZ2MvAH4GAzmwUcFN8DTABmA9XALcAZAO6+GPgdMDUu/x3LiPvcGo95H3g8H/USEUlcRUWYQrxbN1izBvr0STqijDT4HpGZTQR+4+5vbMwHufuxdWw6MM2+Doyo4zxjSDOig7tPA3bamNhERBo9Mzj4YLjttjB7a9++9R9TIDK5Ivot8Gczuz12lxYRkUJy0EHh9emnk40jQ5kM8fOquw8CHgOeMLNLzaxl7kITEZGMHBgbmF54Ab75JtlYMpDRPaLYEeBdwigGvwZmmdnxuQhMREQytNVWsNtuYQDUF19MOpoGy2SsuX8CC4FrCQ+LngAMBPYws9G5CE5ERDJ08MHhdeLEZOPIQCYjK1QCb8eOBKl+bWYzsxiTiIhsrIMOgj/+sVElokzuEc1Ik4Rq/ChL8YiIyKbYZx9o0QJefx0+/TTpaBokK88RufvsbJxHREQ2UcuWsO++Yf2ZZ5KNpYGSHllBRESyrZHdJ1IiEhEpNjXPE02cuP4UEQVIiUhEpNjsuit07Ajz58OsWUlHUy8lIhGRYtOkydqHWxtB85wSkYhIMWpE94mUiEREilFNIpo0CVatSjaWeigRiYgUo27doHdvWLoUXn456Wg2SIlIRKRY1VwVFfho3EpEIiLFqpHcJ1IiEhEpVoMGhR50U6bAV18lHU2dlIhERIrVllvCHnuEzgqTJycdTZ2UiEREilkjuE+kRCQiUswawX0iJSIRkWI2YAC0bg0zZ8KCBUlHk5YSkYhIMWvWDPbfP6wXaPOcEpGISLFr1y68nngi9OgBVVWJhlNbQSQiM5tjZm+Z2etmNi2WtTeziWY2K762i+VmZtebWbWZvWlmfVPOMzzuP8vMhidVHxGRglFVBfffv/b93LlQWVlQyaggElE0yN13dff+8f2FwDPu3ht4Jr4HOBToHZdK4EYIiQu4FNgT2AO4tCZ5iYiUrJEjYcWKdcuWLw/lBaKQElFtQ4GxcX0scERK+Z0eTAHamtk2wCHARHdf7O5LgInAkDzHLCJSWObNy6w8AU2TDiBy4Ckzc+Bmdx8NlLv7R3H7x0B5XO8MzE85dkEsq6t8PWZWSbiaory8nMkF+KDXsmXLCjKuXFBdi0+p1BMKv64DOnVis0WL1itf0akTUzKIO5f1LJREtI+7LzSzTsBEM3sndaO7e0xSWRET3WiA/v37+8CBA7N16qyZPHkyhRhXLqiuxadU6gmNoK5XXx3uCS1fvrasVSs2u/rqjOLOZT0LomnO3RfG10+Ahwj3eBbFJjfi6ydx94VA15TDu8SyuspFREpXRQWMHh2mDgdo1Sq8r6hINq4UiSciM9vczLaoWQcGA9OBR4Ganm/DgUfi+qPAsNh7bgDwZWzCexIYbGbtYieFwbFMRKS0VVTA+PFhfffdCyoJQWE0zZUDD5kZhHjucvcnzGwqcK+ZnQzMBX4e958AHAZUA8uBEwHcfbGZ/Q6YGvf7b3dfnL9qiIgUsE6dwusnn2x4vwQknojcfTawS5ryz4ED05Q7MKKOc40BxmQ7RhGRRq889vdK03EhaYk3zYmISB506BDmJlq8GL77Lulo1qFEJCJSCsrKYKutwvqnnyYbSy1KRCIipaLmPlGBNc8pEYmIlIoCvU+kRCQiUiqUiEREJFEF2oVbiUhEpFToikhERBKlRCQiIomqSURqmhMRkUSo+7aIiCRKTXMiIpKomiuiTz+FNWuSjSWFEpGISKlo3hzatYPVq8OYcwVCiUhEpJQU4H0iJSIRkVJSgPeJlIhEREpJAXbhViISESklapoTEZFEqWlOREQSpUQkIiKJ0j0iERFJlO4RiYhIotQ0JyIiiUptmnNPNpao6BKRmQ0xs3fNrNrMLkw6HhGRgvLww2AGK1ZA9+5QVZV0RMWViMysDPgLcCjQBzjWzPokG5WISIGoqoLKyrVXQvPnh/cJJ6OiSkTAHkC1u89292+B8cDQhGMSESkMI0fC8uXrli1fHsoTZF4gbYTZYGZHAUPc/ZT4/nhgT3c/s9Z+lUAlQHl5eb/x48fnPdb6LFu2jNatWycdRl6orsWnVOoJjauu+x9wAJbmd76b8dyzz27w2Nr1HDRo0Cvu3j8bcTXNxkkaG3cfDYwG6N+/vw8cODDZgNKYPHkyhRhXLqiuxadU6gmNrK7dusHcuesVW7du9dYhl/Ustqa5hUDXlPddYpmIiIwaBa1arVvWqlUoT1CxJaKpQG8z62lmzYFjgEcTjklEpDBUVMDo0aG3nFl4HT06lCeoqJrm3H2VmZ0JPAmUAWPcfUbCYYmIFI6KisQTT21FlYgA3H0CMCHpOEREpGGKrWlOREQaGSUiERFJlBKRiIgkSolIREQSVVQjK2wMM/sUWP8Jr+R1BD5LOog8UV2LT6nUE0qnrrXr2d3dt8rGiUs+ERUqM5uWreEzCp3qWnxKpZ5QOnXNZT3VNCciIolSIhIRkUQpERWu0UkHkEeqa/EplXpC6dQ1Z/XUPSIREUmUrohERCRRSkQiIpIoJaI8MrPfmdmbZva6mT1lZtvGcjOz682sOm7vm3LMcDObFZfhKeX9zOyteMz1ZmaxvL2ZTYz7TzSzdgnU8yozeyfW5SEza5uy7aIY87tmdkhK+ZBYVm1mF6aU9zSzl2L5PXF6D8ysRXxfHbf3yGcdU+I72sxmmNkaM+tfa1tR1bWh6qpfITOzMWb2iZlNTylL+13K5vc138ysq5lNMrO34//bs2N5snV1dy15WoA2KetnATfF9cOAxwEDBgAvxfL2wOz42i6ut4vbXo77Wjz20Fj+R+DCuH4hcGUC9RwMNI3rV9bEAPQB3gBaAD2B9wnTdZTF9V5A87hPn3jMvcAxcf0m4PS4fkbKv98xwD0J/Ux3ALYHJgP9U8qLrq4N/Peos36FvAD7AX2B6Sllab9L2fy+JlDPbYC+cX0L4L34fzXRuuqKKI/cfWnK282Bmp4iQ4E7PZgCtDWzbYBDgInuvtjdlwATgSFxWxt3n+LhJ38ncETKucbG9bEp5Xnj7k+5+6r4dgphplwIsY1395Xu/gFQDewRl2p3n+3u3wLjgaHxL6kDgPvj8an1Sa3n/cCBSfyV6e4z3f3dNJuKrq4NlLZ+CcdUL3d/Hlhcq7iu71I2v6955e4fufurcf0rYCbQmYTrqkSUZ2Y2yszmAxXAJbG4MzA/ZbcFsWxD5QvSlAOUu/tHcf1joDyrFcjcSYS/iiDzenYAvkhJaqn1/PcxcfuXcf9CUUp1TVVX/Rqjur5L2fy+JiY28e4GvETCdS26ifGSZmZPA1un2TTS3R9x95HASDO7CDgTuDRXsbi7m1lO+ufXV8+4z0hgFVCVixjypSF1leKWy+9SEsysNfAAcI67L029wE6irkpEWebuBzVw1yrCTLKXAguBrinbusSyhcDAWuWTY3mXNPsDLDKzbdz9o3iZ/EmGVWiQ+uppZicAPwYOjJfoUHc9qaP8c0JTQNN4JZC6f825FphZU2DLuH/WZfAzTdUo65oFG6p3Y1PXdymb39e8M7NmhCRU5e4PxuJE66qmuTwys94pb4cC78T1R4FhsYfKAODLeJn8JDDYzNrFXiyDgSfjtqVmNiDeKxgGPJJyrpoeLMNTyvPGzIYAFwA/cfflKZseBY6JvcB6Ar0JNzanAr1jr7HmhBvyj8YENgk4Kh6fWp/Ueh4FPJuS8ApBKdU1Vdr6JRzTxqrru5TN72texc+/DZjp7tekbEq2rrnomaGlzh4rDwDTgTeBvwOdY7kBfyH0NnqLdXtfnUS40V0NnJhS3j+e633gBtaOktEBeAaYBTwNtE+gntWE9uPX43JTyraRMeZ3SelNQ+id817cNjKlvBfhF3g1cB/QIpZvFt9Xx+29EvqZ/pTQDr4SWBS/jEVZ1wz+TdLWr5AX4G7gI+C7+PM8ua7vUja/rwnUcx9CJ6k3U76fhyVdVw3xIyIiiVLTnIiIJEqJSEREEqVEJCIiiVIiEhGRRCkRiYhIopSIREQkUUpEIgkws8vM7Ddx/f828hxtzeyMjTiuh5l9Y2E6kh6WMvXBRpyrZTzPt2bWcWPPI6VNiUgkx+JT6XV+19z9hxt56raEKSI2xvvuvutGHvtv7v5NPM+Hm3ouKV1KRFJSLEwKdnBcv8LM/jfNPsPiJGBvmNnfYtm5ZjY9Luek7FtXeQ8Lk8PdSXjKvKuZjTSz98zsRcIcRjX7Lks5ZqaZ3WJh0rKnzKxl3Pawmb0SyyvjoX8AvhevSK6K+/3SzF6OZTebWVkG/za9zOw1M9s9xvKOmd0RY64ys4PM7J8WJkLbo6HnFalX0kNraNGSz4UwAdpkwjQc/wDKam3fkTA8Tcf4vj3QjzC8yeZAa2AGYfj8tOXxuB7AGmBAfF+zbyugDWFYlN/EbctSjlkF7Brf3wv8siaO+NqSkNg6xP1TJ3LbgTB0VLP4/q/AsDT/Bv8+rmadkBhfA3apFcsPCH+wvgKMIQz5MhR4uNY559T8m2nRkumi0belpLj783EwxnOBge6+utYuBwD3uftncf/FZnY88JC7fw1gZg8C+xJ+Kacrfy2ea66HycSI5Q95HATWzOoaCPQDd389rr9CSAgAZ5nZT+N6V8Igqh/XOvZAQsKbGqpISxo2+vpWhIEpj3T3t2vF8laMdwbwjLu7mb2VEpfIJlMikpJiZj8gTJf8uYcZKnPp6404ZmXK+mqgpZkNBA4C9nL35WY2mTAQam0GjHX3izL8zC+BeYQBMVMTUWosa1Ler0G/OySLdI9ISoaFeVaqCE1Ly+J0FbU9CxxtZh3iMe2BF4AjzKyVmW1OGHH7hQ2Up/N83LelmW0BHJ5B6FsCS2IS+j4wIJZ/BWyRst8zwFFm1qkmdjPr3oDzfxtjH2Zmx2UQl0hW6K8aKQlm1gp4EDjP3Wea2e+AK4EnUvdz9xlmNgp4zsxWA6+5+wlmdgdhCgaAW939tXjetOW1ufurZnYP8AahuWxqBuE/AfzKzGYSppSYEs/5eew8MB143N3PN7P/BJ6KvfS+A0YAc+v7AHf/2sx+DEyMnSfezCA+kU2iaSBESoyZ9QAec/edsnjOOYS5aj7L1jmldKhpTqT0rAa2NLPXN/VENQ+0As0I945EMqYrIhERSZSuiEREJFFKRCIikiglIhERSZQSkYiIJEqJSEREEqVEJCIiiVIiEhGRRCkRiYhIov4/QtIXIVY8j5QAAAAASUVORK5CYII=",
       "text/plain": [
        "<Figure size 432x288 with 1 Axes>"
       ]
@@ -2553,7 +2573,7 @@
   "deepnote_execution_queue": [],
   "deepnote_notebook_id": "568ec0db-8924-4d77-814a-ffc3d2455285",
   "kernelspec": {
-   "display_name": "Python 3",
+   "display_name": "Python 3 (ipykernel)",
    "language": "python",
    "name": "python3"
   },
@@ -2567,7 +2587,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.6.8"
+   "version": "3.9.2"
   },
   "toc": {
    "colors": {
diff --git a/material/overview-python.ipynb b/material/overview-python.ipynb
index b6f8fed..b5aa9d0 100644
--- a/material/overview-python.ipynb
+++ b/material/overview-python.ipynb
@@ -24,13 +24,11 @@
     }
    },
    "source": [
-    "<div style=\"float: right;\" markdown=\"1\">\n",
+    "<div style=\"float: right;\">\n",
     "    <img src=\"https://www.python.org/static/community_logos/python-logo-master-v3-TM.png\">\n",
     "</div>\n",
     "\n",
-    "PYTHON\n",
-    "==\n",
-    "<a href=\"https://colab.research.google.com/github/restrepo/ComputationalMethods/blob/master/material/overview-python.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>\n",
+    "# Python Overview\n",
     "\n",
     "Python is an interpreted programming language oriented to easy-readable coding, unlike compiled languages like C/C++ and Fortran, where the syntax usually does not favor the readability. This feature makes Python very interesting when we want to focus on something different than the program structure itself, e.g. on Computational Methods, thereby allowing to optimize our time, to debug syntax errors easily, etc.\n",
     "\n",
@@ -39,8 +37,7 @@
     "\n",
     "[Wikipedia](http://en.wikipedia.org/wiki/Python)\n",
     "\n",
-    "Python Philosophy\n",
-    "--\n",
+    "## Python Philosophy\n",
     "1. Beautiful is better than ugly.\n",
     "2. Explicit is better than implicit.\n",
     "3. Simple is better than complex.\n",
@@ -115,7 +112,7 @@
     }
    },
    "source": [
-    "# String, Integer, Float\n",
+    "## String, Integer, Float\n",
     "The basic types of variables in Python are:\n"
    ]
   },
@@ -244,10 +241,10 @@
     "id": "wTzMkndjJFZp"
    },
    "source": [
-    "# Functions I\n",
+    "## Functions I\n",
     "Python includes a battery of predefined functions which takes an input and generates an output. For example, to check the type of variable we can used the\n",
     "predefined function\n",
-    "## `isinistance`:"
+    "### `isinistance`:"
    ]
   },
   {
@@ -314,7 +311,7 @@
     "id": "RQJLzZTxNxGb"
    },
    "source": [
-    "## `print`"
+    "### `print`"
    ]
   },
   {
@@ -344,10 +341,12 @@
     "cell_id": "00019-517c05f6-21d9-426a-a296-d9dde08cab22",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "PJOXAlbAF_yv"
+    "id": "PJOXAlbAF_yv",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "# Hello World!"
+    "## Hello World!"
    ]
   },
   {
@@ -521,10 +520,12 @@
     "colab": {},
     "colab_type": "code",
     "deepnote_cell_type": "markdown",
-    "id": "lckGJ4UsPBSR"
+    "id": "lckGJ4UsPBSR",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "# Functions"
+    "## Functions II"
    ]
   },
   {
@@ -543,10 +544,12 @@
    "cell_type": "markdown",
    "metadata": {
     "cell_id": "00028-7b74f015-b720-4b86-b279-41314b94db49",
-    "deepnote_cell_type": "markdown"
+    "deepnote_cell_type": "markdown",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## Implicit functions"
+    "### Implicit functions"
    ]
   },
   {
@@ -646,10 +649,12 @@
     "cell_id": "00033-61dd30ec-2a6d-4012-b387-d080d1fd8989",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "R3j70bwqF_1Y"
+    "id": "R3j70bwqF_1Y",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## Explicit functions"
+    "### Explicit functions"
    ]
   },
   {
@@ -900,7 +905,7 @@
     "id": "PYyDcKZ5F_1e"
    },
    "source": [
-    "## Implicit functions of several arguments"
+    "### Implicit functions of several arguments"
    ]
   },
   {
@@ -948,9 +953,12 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
+   },
    "source": [
-    "## Nested functions"
+    "### Nested functions"
    ]
   },
   {
@@ -997,9 +1005,12 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
+   },
    "source": [
-    "## Inner functions with returned internal function"
+    "### Inner functions with returned internal function"
    ]
   },
   {
@@ -1071,9 +1082,12 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
+   },
    "source": [
-    "## Decorators\n",
+    "### Decorators\n",
     "From: [[@]](https://realpython.com/primer-on-python-decorators/) https://realpython.com/primer-on-python-decorators/\n",
     "> By definition, a decorator is a function that takes another function and extends the behavior of the latter function without explicitly modifying it.\n",
     "\n",
@@ -1158,10 +1172,12 @@
     "cell_id": "00049-e128fdf5-6d40-4e6d-a8a2-25bd944de81c",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "tmc45kxUF_y4"
+    "id": "tmc45kxUF_y4",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "# Arithmetics"
+    "## Arithmetics"
    ]
   },
   {
@@ -1170,10 +1186,12 @@
     "cell_id": "00050-56a76643-a756-4bd2-946f-8dbd126ebf74",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "ozwFot0rF_y4"
+    "id": "ozwFot0rF_y4",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## Sum"
+    "### Sum"
    ]
   },
   {
@@ -1302,10 +1320,12 @@
     "cell_id": "00058-51897368-a994-44cc-bbac-d59edcaf405d",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "-fCXEiSsF_y8"
+    "id": "-fCXEiSsF_y8",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## Multiplication"
+    "### Multiplication"
    ]
   },
   {
@@ -1380,10 +1400,12 @@
     "cell_id": "00062-85e7cd59-82e0-4a8f-927e-8c4de20890bf",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "jtskAuqQF_zA"
+    "id": "jtskAuqQF_zA",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## **Division**"
+    "### Division"
    ]
   },
   {
@@ -1483,10 +1505,12 @@
     "cell_id": "00067-a2553e7c-3180-49bd-ae9d-b91598376b9b",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "x7xHh9tjF_zJ"
+    "id": "x7xHh9tjF_zJ",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## **Module**"
+    "### Module"
    ]
   },
   {
@@ -1553,10 +1577,12 @@
     "cell_id": "00070-8ad8abec-b278-45e3-b3a4-aafc43b797cf",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "2iWjCW7tF_zF"
+    "id": "2iWjCW7tF_zF",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## **Power**"
+    "### Power"
    ]
   },
   {
@@ -1592,10 +1618,12 @@
     "cell_id": "00072-40ada915-f658-4795-8cdb-481dc6f2b7d7",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "8_EWbqIzF_zQ"
+    "id": "8_EWbqIzF_zQ",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## **Scientific notation**"
+    "### Scientific notation"
    ]
   },
   {
@@ -1990,10 +2018,12 @@
     "cell_id": "00091-d49ebd94-56ff-4f1f-a968-52e4efe9dad3",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "rf3I0p_lnix5"
+    "id": "rf3I0p_lnix5",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## Complex numbers"
+    "### Complex numbers"
    ]
   },
   {
@@ -2505,7 +2535,7 @@
     "id": "WNGKkoW4F_zT"
    },
    "source": [
-    "# Lists, Tuples, Dictionaries and Sets"
+    "## Lists, Tuples, Dictionaries and Sets"
    ]
   },
   {
@@ -2514,10 +2544,12 @@
     "cell_id": "00113-6ca5b993-d2b8-48df-95e5-92e102a819fc",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "ACaVyEIMF_zU"
+    "id": "ACaVyEIMF_zU",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## Lists"
+    "### Lists"
    ]
   },
   {
@@ -2974,10 +3006,12 @@
     "cell_id": "00123-57f91b00-6a13-4568-a840-c95e04b50c68",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "pfNBq-Bj40ZQ"
+    "id": "pfNBq-Bj40ZQ",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "### Slicing: \n",
+    "#### Slicing: \n",
     "Extract elements from a list, `l` from one given index to another given index. We pass slice instead of index like this: \n",
     "```python3\n",
     "l[start:end]\n",
@@ -3259,10 +3293,12 @@
     "cell_id": "00130-1315bac9-04c4-4e12-bd71-e54c9c188e09",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "-nwqd5OH40Zh"
+    "id": "-nwqd5OH40Zh",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "### Embedded lists"
+    "#### Embedded lists"
    ]
   },
   {
@@ -3426,10 +3462,12 @@
     "cell_id": "00136-243850cf-39b5-4c5d-842f-72e235c72269",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "xou55DH140Zs"
+    "id": "xou55DH140Zs",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "### Sum of lists"
+    "#### Sum of lists"
    ]
   },
   {
@@ -3787,7 +3825,7 @@
     "id": "n5PCnFO3F_z1"
    },
    "source": [
-    "### List Comprehension"
+    "#### List Comprehension"
    ]
   },
   {
@@ -4077,7 +4115,7 @@
     "id": "J0-SP0qP40aB"
    },
    "source": [
-    "### Reversed comprehension\n",
+    "#### Reversed comprehension\n",
     "We can use the method `reversed(...)` to generate an iterator with the revesersed list so that the original list is kept."
    ]
   },
@@ -4147,10 +4185,12 @@
     "cell_id": "00153-98fee61f-e155-45ea-ae9a-59845ba49a0a",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "txgjCoaIF_z7"
+    "id": "txgjCoaIF_z7",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## Tuples"
+    "### Tuples"
    ]
   },
   {
@@ -4484,9 +4524,12 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
+   },
    "source": [
-    "### Some list functions"
+    "#### Some list functions"
    ]
   },
   {
@@ -4677,10 +4720,12 @@
     "cell_id": "00175-acb86315-216f-43fd-b3f1-6b1d62f8e1ee",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "rafwug2SF_0F"
+    "id": "rafwug2SF_0F",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## Dictionaries"
+    "### Dictionaries"
    ]
   },
   {
@@ -5239,10 +5284,11 @@
    "metadata": {
     "cell_id": "00206-924ed65c-1523-4caa-a508-2189973fbfb5",
     "deepnote_cell_type": "markdown",
+    "jp-MarkdownHeadingCollapsed": true,
     "tags": []
    },
    "source": [
-    "## JSON object\n",
+    "### JSON object\n",
     "A list of dictionaries is simple example of a non-relational database which can be represented as a JSON object. Their main advantage is that with the proper glosary for the keys of the dictionaries the data analysis is well expressed through the impledmented code"
    ]
   },
@@ -5250,7 +5296,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Two basic concepts"
+    "#### Two basic concepts"
    ]
   },
   {
@@ -5311,9 +5357,12 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
+   },
    "source": [
-    "### They can be nested!"
+    "#### They can be nested!"
    ]
   },
   {
@@ -5514,9 +5563,12 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
+   },
    "source": [
-    "### Example:\n",
+    "#### Example:\n",
     "Extract information from a JSON object"
    ]
   },
@@ -5629,6 +5681,7 @@
    "metadata": {
     "cell_id": "00194-4b3af145-7feb-45c8-8c51-c7ffa719210f",
     "deepnote_cell_type": "markdown",
+    "jp-MarkdownHeadingCollapsed": true,
     "tags": []
    },
    "source": [
@@ -5810,7 +5863,7 @@
     "id": "oMhSjbqYF_1C"
    },
    "source": [
-    "# Conditionals"
+    "## Conditionals"
    ]
   },
   {
@@ -5822,7 +5875,7 @@
     "id": "JRx2uDsXF_1C"
    },
    "source": [
-    "## if"
+    "### if"
    ]
   },
   {
@@ -5918,7 +5971,7 @@
     "tags": []
    },
    "source": [
-    "# Loops"
+    "## Loops"
    ]
   },
   {
@@ -5927,10 +5980,12 @@
     "cell_id": "00198-ed773f4e-f08e-43d6-970f-7c4e7019a757",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "RKDoomeWF_1J"
+    "id": "RKDoomeWF_1J",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## for"
+    "### for"
    ]
   },
   {
@@ -6118,10 +6173,12 @@
     "cell_id": "00207-40757f14-9255-462c-b874-9aaab23a721d",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "Vk6CUxqjF_1S"
+    "id": "Vk6CUxqjF_1S",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
-    "## while"
+    "### while"
    ]
   },
   {
@@ -6380,7 +6437,7 @@
     "id": "MpuXWP76PRkT"
    },
    "source": [
-    "# Methods and attributes of objects in Python\n",
+    "## Methods and attributes of objects in Python\n",
     "All the objects in `Python`, including the function and variable types discussed here, are enhanced with special functions called _methods_, which are implemented after a point of the name of the variable, in the format:\n",
     "```python\n",
     "variable.method()\n",
@@ -6399,7 +6456,9 @@
     "cell_id": "00220-5bbd289e-7ad7-4874-b6bc-3be16881cfcd",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "999oCdgfPZOM"
+    "id": "999oCdgfPZOM",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
     "### Methods\n",
@@ -6589,7 +6648,9 @@
     "cell_id": "00228-1201df6f-5487-4fb1-a596-4b1a123eae7f",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "NaWNpAh8PtAD"
+    "id": "NaWNpAh8PtAD",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
     "### Attributes"
@@ -6871,7 +6932,9 @@
     "cell_id": "00241-8546677a-b493-4719-a269-af0277736ecb",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "xJBhkIFf40bz"
+    "id": "xJBhkIFf40bz",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
     "## Unicode\n",
@@ -7002,7 +7065,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
+   },
    "source": [
     "## Stop exection\n",
     "Stop the program if a condition is not satisfied\n",
@@ -7047,7 +7113,9 @@
     "cell_id": "00250-278fdad9-ce65-4221-89fb-ade632e04fc3",
     "colab_type": "text",
     "deepnote_cell_type": "markdown",
-    "id": "9MgMakz7jhml"
+    "id": "9MgMakz7jhml",
+    "jp-MarkdownHeadingCollapsed": true,
+    "tags": []
    },
    "source": [
     "## Final remarks\n",
diff --git a/material/statistics.ipynb b/material/statistics.ipynb
index 1070bdc..15e7150 100644
--- a/material/statistics.ipynb
+++ b/material/statistics.ipynb
@@ -104,7 +104,7 @@
     "id": "WnxpZdlpBsFh"
    },
    "source": [
-    "https://drive.google.com/file/d/12cwfl5i2SVlNF60oUF9-ep3q7NHB4AcH/view?usp=sharing## Data Adjust\n",
+    "https://drive.google.com/file/d/12cwfl5i2SVlNF60oUF9-ep3q7NHB4AcH/view?usp=sharing### Data Adjust\n",
     "See also: \n",
     "* https://www.mathsisfun.com/data/least-squares-regression.html\n",
     "* http://blog.mmast.net/least-squares-fitting-numpy-scipy\n",
@@ -297,7 +297,7 @@
     "id": "fBRt-ri7BsFw"
    },
    "source": [
-    "## Example 1"
+    "### Example 1"
    ]
   },
   {
@@ -988,7 +988,7 @@
     "id": "cOlJ375RBsHF"
    },
    "source": [
-    "## How does this work:\n",
+    "### How does this work:\n",
     "From [Least sqaure method in python](https://stackoverflow.com/a/43623588/2268280):\n",
     "\n",
     "> [curve_fit]() is one of many optimization functions offered by scipy. Given an initial value, the resulting estimated parameters are iteratively refined so that the resulting curve minimizes the residual error, or difference between the fitted line and sampling data. A better guess reduces the number of iterations and speeds up the result. With these estimated parameters for the fitted curve, one can now calculate the specific coefficients for a particular equation (a final exercise left to the OP)."
@@ -1069,7 +1069,7 @@
     "id": "0o0QRuj1BsHd"
    },
    "source": [
-    "## <font color='red'>     **Activity** </font>"
+    "### <font color='red'>     **Activity** </font>"
    ]
   },
   {
@@ -1263,7 +1263,7 @@
     "id": "Unu1o3YMBsHh"
    },
    "source": [
-    "## <font color='red'>     **Activity** </font>\n"
+    "### <font color='red'>     **Activity** </font>\n"
    ]
   },
   {
@@ -1386,12 +1386,12 @@
     }
    ],
    "source": [
-    "# REGRESSION ------------------------------------------------------------------\n",
-    "p0 = [0, -3e-2, 1]                                        # guessed params\n",
+    "## REGRESSION ------------------------------------------------------------------\n",
+    "p0 = [0, -3e-2, 1]                                        ## guessed params\n",
     "w, _ = optimize.curve_fit(func, x_samp, y_samp, p0=p0)     \n",
     "print(\"E]stimated Parameters\", w)  \n",
     "\n",
-    "# Model\n",
+    "## Model\n",
     "x_lin=np.linspace(200,1350)\n",
     "y_model = func(x_lin, *w)"
    ]
@@ -1428,9 +1428,9 @@
     }
    ],
    "source": [
-    "x_lin = np.linspace(0, x_samp.max(), 50)                   # 50 evenly spaced digits between 0 and max\n",
-    "# PLOT ------------------------------------------------------------------------\n",
-    "# Visualize data and fitted curves\n",
+    "x_lin = np.linspace(0, x_samp.max(), 50)                   ## 50 evenly spaced digits between 0 and max\n",
+    "## PLOT ------------------------------------------------------------------------\n",
+    "## Visualize data and fitted curves\n",
     "plt.plot(x_samp, y_samp, \"ko\", label=\"Data\")\n",
     "plt.plot(x_lin, y_model, \"k--\", label=\"Fit\")\n",
     "plt.title(\"Least squares regression\")\n",
@@ -1469,14 +1469,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Example: exponential fit"
+    "#### Example: exponential fit"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "#### Guess the number of pages of a book\n",
+    "##### Guess the number of pages of a book\n",
     "https://docs.google.com/forms/d/e/1FAIpQLSeZJ0QII5JcST-M9_JgGYNzX3GahULVVFc31DQnWjJ4SdUQwg/viewform?fbclid=IwAR0wEadM0ZH-HXmp3lkAM3emCDPxs_6F509BS3EkOvldp-NFzbCOkZVSjR4"
    ]
   },
@@ -1515,20 +1515,20 @@
       "File \u001b[0;32m/usr/local/lib/python3.9/dist-packages/pandas/util/_decorators.py:211\u001b[0m, in \u001b[0;36mdeprecate_kwarg.<locals>._deprecate_kwarg.<locals>.wrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m    209\u001b[0m     \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m    210\u001b[0m         kwargs[new_arg_name] \u001b[38;5;241m=\u001b[39m new_arg_value\n\u001b[0;32m--> 211\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mfunc\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n",
       "File \u001b[0;32m/usr/local/lib/python3.9/dist-packages/pandas/util/_decorators.py:317\u001b[0m, in \u001b[0;36mdeprecate_nonkeyword_arguments.<locals>.decorate.<locals>.wrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m    311\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;28mlen\u001b[39m(args) \u001b[38;5;241m>\u001b[39m num_allow_args:\n\u001b[1;32m    312\u001b[0m     warnings\u001b[38;5;241m.\u001b[39mwarn(\n\u001b[1;32m    313\u001b[0m         msg\u001b[38;5;241m.\u001b[39mformat(arguments\u001b[38;5;241m=\u001b[39marguments),\n\u001b[1;32m    314\u001b[0m         \u001b[38;5;167;01mFutureWarning\u001b[39;00m,\n\u001b[1;32m    315\u001b[0m         stacklevel\u001b[38;5;241m=\u001b[39mfind_stack_level(inspect\u001b[38;5;241m.\u001b[39mcurrentframe()),\n\u001b[1;32m    316\u001b[0m     )\n\u001b[0;32m--> 317\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mfunc\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n",
       "File \u001b[0;32m/usr/local/lib/python3.9/dist-packages/pandas/io/parsers/readers.py:950\u001b[0m, in \u001b[0;36mread_csv\u001b[0;34m(filepath_or_buffer, sep, delimiter, header, names, index_col, usecols, squeeze, prefix, mangle_dupe_cols, dtype, engine, converters, true_values, false_values, skipinitialspace, skiprows, skipfooter, nrows, na_values, keep_default_na, na_filter, verbose, skip_blank_lines, parse_dates, infer_datetime_format, keep_date_col, date_parser, dayfirst, cache_dates, iterator, chunksize, compression, thousands, decimal, lineterminator, quotechar, quoting, doublequote, escapechar, comment, encoding, encoding_errors, dialect, error_bad_lines, warn_bad_lines, on_bad_lines, delim_whitespace, low_memory, memory_map, float_precision, storage_options)\u001b[0m\n\u001b[1;32m    935\u001b[0m kwds_defaults \u001b[38;5;241m=\u001b[39m _refine_defaults_read(\n\u001b[1;32m    936\u001b[0m     dialect,\n\u001b[1;32m    937\u001b[0m     delimiter,\n\u001b[0;32m   (...)\u001b[0m\n\u001b[1;32m    946\u001b[0m     defaults\u001b[38;5;241m=\u001b[39m{\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mdelimiter\u001b[39m\u001b[38;5;124m\"\u001b[39m: \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124m,\u001b[39m\u001b[38;5;124m\"\u001b[39m},\n\u001b[1;32m    947\u001b[0m )\n\u001b[1;32m    948\u001b[0m kwds\u001b[38;5;241m.\u001b[39mupdate(kwds_defaults)\n\u001b[0;32m--> 950\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43m_read\u001b[49m\u001b[43m(\u001b[49m\u001b[43mfilepath_or_buffer\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mkwds\u001b[49m\u001b[43m)\u001b[49m\n",
-      "File \u001b[0;32m/usr/local/lib/python3.9/dist-packages/pandas/io/parsers/readers.py:605\u001b[0m, in \u001b[0;36m_read\u001b[0;34m(filepath_or_buffer, kwds)\u001b[0m\n\u001b[1;32m    602\u001b[0m _validate_names(kwds\u001b[38;5;241m.\u001b[39mget(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mnames\u001b[39m\u001b[38;5;124m\"\u001b[39m, \u001b[38;5;28;01mNone\u001b[39;00m))\n\u001b[1;32m    604\u001b[0m \u001b[38;5;66;03m# Create the parser.\u001b[39;00m\n\u001b[0;32m--> 605\u001b[0m parser \u001b[38;5;241m=\u001b[39m \u001b[43mTextFileReader\u001b[49m\u001b[43m(\u001b[49m\u001b[43mfilepath_or_buffer\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwds\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    607\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m chunksize \u001b[38;5;129;01mor\u001b[39;00m iterator:\n\u001b[1;32m    608\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m parser\n",
+      "File \u001b[0;32m/usr/local/lib/python3.9/dist-packages/pandas/io/parsers/readers.py:605\u001b[0m, in \u001b[0;36m_read\u001b[0;34m(filepath_or_buffer, kwds)\u001b[0m\n\u001b[1;32m    602\u001b[0m _validate_names(kwds\u001b[38;5;241m.\u001b[39mget(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mnames\u001b[39m\u001b[38;5;124m\"\u001b[39m, \u001b[38;5;28;01mNone\u001b[39;00m))\n\u001b[1;32m    604\u001b[0m \u001b[38;5;66;03m## Create the parser.\u001b[39;00m\n\u001b[0;32m--> 605\u001b[0m parser \u001b[38;5;241m=\u001b[39m \u001b[43mTextFileReader\u001b[49m\u001b[43m(\u001b[49m\u001b[43mfilepath_or_buffer\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwds\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    607\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m chunksize \u001b[38;5;129;01mor\u001b[39;00m iterator:\n\u001b[1;32m    608\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m parser\n",
       "File \u001b[0;32m/usr/local/lib/python3.9/dist-packages/pandas/io/parsers/readers.py:1442\u001b[0m, in \u001b[0;36mTextFileReader.__init__\u001b[0;34m(self, f, engine, **kwds)\u001b[0m\n\u001b[1;32m   1439\u001b[0m     \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39moptions[\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mhas_index_names\u001b[39m\u001b[38;5;124m\"\u001b[39m] \u001b[38;5;241m=\u001b[39m kwds[\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mhas_index_names\u001b[39m\u001b[38;5;124m\"\u001b[39m]\n\u001b[1;32m   1441\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mhandles: IOHandles \u001b[38;5;241m|\u001b[39m \u001b[38;5;28;01mNone\u001b[39;00m \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;01mNone\u001b[39;00m\n\u001b[0;32m-> 1442\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_engine \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_make_engine\u001b[49m\u001b[43m(\u001b[49m\u001b[43mf\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mengine\u001b[49m\u001b[43m)\u001b[49m\n",
       "File \u001b[0;32m/usr/local/lib/python3.9/dist-packages/pandas/io/parsers/readers.py:1729\u001b[0m, in \u001b[0;36mTextFileReader._make_engine\u001b[0;34m(self, f, engine)\u001b[0m\n\u001b[1;32m   1727\u001b[0m     is_text \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;01mFalse\u001b[39;00m\n\u001b[1;32m   1728\u001b[0m     mode \u001b[38;5;241m=\u001b[39m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mrb\u001b[39m\u001b[38;5;124m\"\u001b[39m\n\u001b[0;32m-> 1729\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mhandles \u001b[38;5;241m=\u001b[39m \u001b[43mget_handle\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m   1730\u001b[0m \u001b[43m    \u001b[49m\u001b[43mf\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m   1731\u001b[0m \u001b[43m    \u001b[49m\u001b[43mmode\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m   1732\u001b[0m \u001b[43m    \u001b[49m\u001b[43mencoding\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43moptions\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mget\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43mencoding\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;28;43;01mNone\u001b[39;49;00m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m   1733\u001b[0m \u001b[43m    \u001b[49m\u001b[43mcompression\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43moptions\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mget\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43mcompression\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;28;43;01mNone\u001b[39;49;00m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m   1734\u001b[0m \u001b[43m    \u001b[49m\u001b[43mmemory_map\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43moptions\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mget\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43mmemory_map\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;28;43;01mFalse\u001b[39;49;00m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m   1735\u001b[0m \u001b[43m    \u001b[49m\u001b[43mis_text\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mis_text\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m   1736\u001b[0m \u001b[43m    \u001b[49m\u001b[43merrors\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43moptions\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mget\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43mencoding_errors\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43mstrict\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m   1737\u001b[0m \u001b[43m    \u001b[49m\u001b[43mstorage_options\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43moptions\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mget\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43mstorage_options\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;28;43;01mNone\u001b[39;49;00m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m   1738\u001b[0m \u001b[43m\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m   1739\u001b[0m \u001b[38;5;28;01massert\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mhandles \u001b[38;5;129;01mis\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m\n\u001b[1;32m   1740\u001b[0m f \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mhandles\u001b[38;5;241m.\u001b[39mhandle\n",
-      "File \u001b[0;32m/usr/local/lib/python3.9/dist-packages/pandas/io/common.py:714\u001b[0m, in \u001b[0;36mget_handle\u001b[0;34m(path_or_buf, mode, encoding, compression, memory_map, is_text, errors, storage_options)\u001b[0m\n\u001b[1;32m    711\u001b[0m     codecs\u001b[38;5;241m.\u001b[39mlookup_error(errors)\n\u001b[1;32m    713\u001b[0m \u001b[38;5;66;03m# open URLs\u001b[39;00m\n\u001b[0;32m--> 714\u001b[0m ioargs \u001b[38;5;241m=\u001b[39m \u001b[43m_get_filepath_or_buffer\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m    715\u001b[0m \u001b[43m    \u001b[49m\u001b[43mpath_or_buf\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    716\u001b[0m \u001b[43m    \u001b[49m\u001b[43mencoding\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mencoding\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    717\u001b[0m \u001b[43m    \u001b[49m\u001b[43mcompression\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mcompression\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    718\u001b[0m \u001b[43m    \u001b[49m\u001b[43mmode\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mmode\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    719\u001b[0m \u001b[43m    \u001b[49m\u001b[43mstorage_options\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mstorage_options\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    720\u001b[0m \u001b[43m\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    722\u001b[0m handle \u001b[38;5;241m=\u001b[39m ioargs\u001b[38;5;241m.\u001b[39mfilepath_or_buffer\n\u001b[1;32m    723\u001b[0m handles: \u001b[38;5;28mlist\u001b[39m[BaseBuffer]\n",
-      "File \u001b[0;32m/usr/local/lib/python3.9/dist-packages/pandas/io/common.py:364\u001b[0m, in \u001b[0;36m_get_filepath_or_buffer\u001b[0;34m(filepath_or_buffer, encoding, compression, mode, storage_options)\u001b[0m\n\u001b[1;32m    362\u001b[0m \u001b[38;5;66;03m# assuming storage_options is to be interpreted as headers\u001b[39;00m\n\u001b[1;32m    363\u001b[0m req_info \u001b[38;5;241m=\u001b[39m urllib\u001b[38;5;241m.\u001b[39mrequest\u001b[38;5;241m.\u001b[39mRequest(filepath_or_buffer, headers\u001b[38;5;241m=\u001b[39mstorage_options)\n\u001b[0;32m--> 364\u001b[0m \u001b[38;5;28;01mwith\u001b[39;00m \u001b[43murlopen\u001b[49m\u001b[43m(\u001b[49m\u001b[43mreq_info\u001b[49m\u001b[43m)\u001b[49m \u001b[38;5;28;01mas\u001b[39;00m req:\n\u001b[1;32m    365\u001b[0m     content_encoding \u001b[38;5;241m=\u001b[39m req\u001b[38;5;241m.\u001b[39mheaders\u001b[38;5;241m.\u001b[39mget(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mContent-Encoding\u001b[39m\u001b[38;5;124m\"\u001b[39m, \u001b[38;5;28;01mNone\u001b[39;00m)\n\u001b[1;32m    366\u001b[0m     \u001b[38;5;28;01mif\u001b[39;00m content_encoding \u001b[38;5;241m==\u001b[39m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mgzip\u001b[39m\u001b[38;5;124m\"\u001b[39m:\n\u001b[1;32m    367\u001b[0m         \u001b[38;5;66;03m# Override compression based on Content-Encoding header\u001b[39;00m\n",
+      "File \u001b[0;32m/usr/local/lib/python3.9/dist-packages/pandas/io/common.py:714\u001b[0m, in \u001b[0;36mget_handle\u001b[0;34m(path_or_buf, mode, encoding, compression, memory_map, is_text, errors, storage_options)\u001b[0m\n\u001b[1;32m    711\u001b[0m     codecs\u001b[38;5;241m.\u001b[39mlookup_error(errors)\n\u001b[1;32m    713\u001b[0m \u001b[38;5;66;03m## open URLs\u001b[39;00m\n\u001b[0;32m--> 714\u001b[0m ioargs \u001b[38;5;241m=\u001b[39m \u001b[43m_get_filepath_or_buffer\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m    715\u001b[0m \u001b[43m    \u001b[49m\u001b[43mpath_or_buf\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    716\u001b[0m \u001b[43m    \u001b[49m\u001b[43mencoding\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mencoding\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    717\u001b[0m \u001b[43m    \u001b[49m\u001b[43mcompression\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mcompression\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    718\u001b[0m \u001b[43m    \u001b[49m\u001b[43mmode\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mmode\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    719\u001b[0m \u001b[43m    \u001b[49m\u001b[43mstorage_options\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mstorage_options\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    720\u001b[0m \u001b[43m\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    722\u001b[0m handle \u001b[38;5;241m=\u001b[39m ioargs\u001b[38;5;241m.\u001b[39mfilepath_or_buffer\n\u001b[1;32m    723\u001b[0m handles: \u001b[38;5;28mlist\u001b[39m[BaseBuffer]\n",
+      "File \u001b[0;32m/usr/local/lib/python3.9/dist-packages/pandas/io/common.py:364\u001b[0m, in \u001b[0;36m_get_filepath_or_buffer\u001b[0;34m(filepath_or_buffer, encoding, compression, mode, storage_options)\u001b[0m\n\u001b[1;32m    362\u001b[0m \u001b[38;5;66;03m## assuming storage_options is to be interpreted as headers\u001b[39;00m\n\u001b[1;32m    363\u001b[0m req_info \u001b[38;5;241m=\u001b[39m urllib\u001b[38;5;241m.\u001b[39mrequest\u001b[38;5;241m.\u001b[39mRequest(filepath_or_buffer, headers\u001b[38;5;241m=\u001b[39mstorage_options)\n\u001b[0;32m--> 364\u001b[0m \u001b[38;5;28;01mwith\u001b[39;00m \u001b[43murlopen\u001b[49m\u001b[43m(\u001b[49m\u001b[43mreq_info\u001b[49m\u001b[43m)\u001b[49m \u001b[38;5;28;01mas\u001b[39;00m req:\n\u001b[1;32m    365\u001b[0m     content_encoding \u001b[38;5;241m=\u001b[39m req\u001b[38;5;241m.\u001b[39mheaders\u001b[38;5;241m.\u001b[39mget(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mContent-Encoding\u001b[39m\u001b[38;5;124m\"\u001b[39m, \u001b[38;5;28;01mNone\u001b[39;00m)\n\u001b[1;32m    366\u001b[0m     \u001b[38;5;28;01mif\u001b[39;00m content_encoding \u001b[38;5;241m==\u001b[39m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mgzip\u001b[39m\u001b[38;5;124m\"\u001b[39m:\n\u001b[1;32m    367\u001b[0m         \u001b[38;5;66;03m# Override compression based on Content-Encoding header\u001b[39;00m\n",
       "File \u001b[0;32m/usr/local/lib/python3.9/dist-packages/pandas/io/common.py:266\u001b[0m, in \u001b[0;36murlopen\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m    260\u001b[0m \u001b[38;5;124;03m\"\"\"\u001b[39;00m\n\u001b[1;32m    261\u001b[0m \u001b[38;5;124;03mLazy-import wrapper for stdlib urlopen, as that imports a big chunk of\u001b[39;00m\n\u001b[1;32m    262\u001b[0m \u001b[38;5;124;03mthe stdlib.\u001b[39;00m\n\u001b[1;32m    263\u001b[0m \u001b[38;5;124;03m\"\"\"\u001b[39;00m\n\u001b[1;32m    264\u001b[0m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;21;01murllib\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mrequest\u001b[39;00m\n\u001b[0;32m--> 266\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43murllib\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mrequest\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43murlopen\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n",
       "File \u001b[0;32m/usr/lib/python3.9/urllib/request.py:214\u001b[0m, in \u001b[0;36murlopen\u001b[0;34m(url, data, timeout, cafile, capath, cadefault, context)\u001b[0m\n\u001b[1;32m    212\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m    213\u001b[0m     opener \u001b[38;5;241m=\u001b[39m _opener\n\u001b[0;32m--> 214\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mopener\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mopen\u001b[49m\u001b[43m(\u001b[49m\u001b[43murl\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mdata\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mtimeout\u001b[49m\u001b[43m)\u001b[49m\n",
       "File \u001b[0;32m/usr/lib/python3.9/urllib/request.py:523\u001b[0m, in \u001b[0;36mOpenerDirector.open\u001b[0;34m(self, fullurl, data, timeout)\u001b[0m\n\u001b[1;32m    521\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m processor \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mprocess_response\u001b[38;5;241m.\u001b[39mget(protocol, []):\n\u001b[1;32m    522\u001b[0m     meth \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mgetattr\u001b[39m(processor, meth_name)\n\u001b[0;32m--> 523\u001b[0m     response \u001b[38;5;241m=\u001b[39m \u001b[43mmeth\u001b[49m\u001b[43m(\u001b[49m\u001b[43mreq\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mresponse\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    525\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m response\n",
-      "File \u001b[0;32m/usr/lib/python3.9/urllib/request.py:632\u001b[0m, in \u001b[0;36mHTTPErrorProcessor.http_response\u001b[0;34m(self, request, response)\u001b[0m\n\u001b[1;32m    629\u001b[0m \u001b[38;5;66;03m# According to RFC 2616, \"2xx\" code indicates that the client's\u001b[39;00m\n\u001b[1;32m    630\u001b[0m \u001b[38;5;66;03m# request was successfully received, understood, and accepted.\u001b[39;00m\n\u001b[1;32m    631\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m (\u001b[38;5;241m200\u001b[39m \u001b[38;5;241m<\u001b[39m\u001b[38;5;241m=\u001b[39m code \u001b[38;5;241m<\u001b[39m \u001b[38;5;241m300\u001b[39m):\n\u001b[0;32m--> 632\u001b[0m     response \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mparent\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43merror\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m    633\u001b[0m \u001b[43m        \u001b[49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[38;5;124;43mhttp\u001b[39;49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mrequest\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mresponse\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mcode\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mmsg\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mhdrs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    635\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m response\n",
+      "File \u001b[0;32m/usr/lib/python3.9/urllib/request.py:632\u001b[0m, in \u001b[0;36mHTTPErrorProcessor.http_response\u001b[0;34m(self, request, response)\u001b[0m\n\u001b[1;32m    629\u001b[0m \u001b[38;5;66;03m## According to RFC 2616, \"2xx\" code indicates that the client's\u001b[39;00m\n\u001b[1;32m    630\u001b[0m \u001b[38;5;66;03m# request was successfully received, understood, and accepted.\u001b[39;00m\n\u001b[1;32m    631\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m (\u001b[38;5;241m200\u001b[39m \u001b[38;5;241m<\u001b[39m\u001b[38;5;241m=\u001b[39m code \u001b[38;5;241m<\u001b[39m \u001b[38;5;241m300\u001b[39m):\n\u001b[0;32m--> 632\u001b[0m     response \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mparent\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43merror\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m    633\u001b[0m \u001b[43m        \u001b[49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[38;5;124;43mhttp\u001b[39;49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mrequest\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mresponse\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mcode\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mmsg\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mhdrs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    635\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m response\n",
       "File \u001b[0;32m/usr/lib/python3.9/urllib/request.py:555\u001b[0m, in \u001b[0;36mOpenerDirector.error\u001b[0;34m(self, proto, *args)\u001b[0m\n\u001b[1;32m    553\u001b[0m     http_err \u001b[38;5;241m=\u001b[39m \u001b[38;5;241m0\u001b[39m\n\u001b[1;32m    554\u001b[0m args \u001b[38;5;241m=\u001b[39m (\u001b[38;5;28mdict\u001b[39m, proto, meth_name) \u001b[38;5;241m+\u001b[39m args\n\u001b[0;32m--> 555\u001b[0m result \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_call_chain\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    556\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m result:\n\u001b[1;32m    557\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m result\n",
       "File \u001b[0;32m/usr/lib/python3.9/urllib/request.py:494\u001b[0m, in \u001b[0;36mOpenerDirector._call_chain\u001b[0;34m(self, chain, kind, meth_name, *args)\u001b[0m\n\u001b[1;32m    492\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m handler \u001b[38;5;129;01min\u001b[39;00m handlers:\n\u001b[1;32m    493\u001b[0m     func \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mgetattr\u001b[39m(handler, meth_name)\n\u001b[0;32m--> 494\u001b[0m     result \u001b[38;5;241m=\u001b[39m \u001b[43mfunc\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    495\u001b[0m     \u001b[38;5;28;01mif\u001b[39;00m result \u001b[38;5;129;01mis\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m:\n\u001b[1;32m    496\u001b[0m         \u001b[38;5;28;01mreturn\u001b[39;00m result\n",
       "File \u001b[0;32m/usr/lib/python3.9/urllib/request.py:747\u001b[0m, in \u001b[0;36mHTTPRedirectHandler.http_error_302\u001b[0;34m(self, req, fp, code, msg, headers)\u001b[0m\n\u001b[1;32m    744\u001b[0m fp\u001b[38;5;241m.\u001b[39mread()\n\u001b[1;32m    745\u001b[0m fp\u001b[38;5;241m.\u001b[39mclose()\n\u001b[0;32m--> 747\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mparent\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mopen\u001b[49m\u001b[43m(\u001b[49m\u001b[43mnew\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mtimeout\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mreq\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mtimeout\u001b[49m\u001b[43m)\u001b[49m\n",
       "File \u001b[0;32m/usr/lib/python3.9/urllib/request.py:523\u001b[0m, in \u001b[0;36mOpenerDirector.open\u001b[0;34m(self, fullurl, data, timeout)\u001b[0m\n\u001b[1;32m    521\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m processor \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mprocess_response\u001b[38;5;241m.\u001b[39mget(protocol, []):\n\u001b[1;32m    522\u001b[0m     meth \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mgetattr\u001b[39m(processor, meth_name)\n\u001b[0;32m--> 523\u001b[0m     response \u001b[38;5;241m=\u001b[39m \u001b[43mmeth\u001b[49m\u001b[43m(\u001b[49m\u001b[43mreq\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mresponse\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    525\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m response\n",
-      "File \u001b[0;32m/usr/lib/python3.9/urllib/request.py:632\u001b[0m, in \u001b[0;36mHTTPErrorProcessor.http_response\u001b[0;34m(self, request, response)\u001b[0m\n\u001b[1;32m    629\u001b[0m \u001b[38;5;66;03m# According to RFC 2616, \"2xx\" code indicates that the client's\u001b[39;00m\n\u001b[1;32m    630\u001b[0m \u001b[38;5;66;03m# request was successfully received, understood, and accepted.\u001b[39;00m\n\u001b[1;32m    631\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m (\u001b[38;5;241m200\u001b[39m \u001b[38;5;241m<\u001b[39m\u001b[38;5;241m=\u001b[39m code \u001b[38;5;241m<\u001b[39m \u001b[38;5;241m300\u001b[39m):\n\u001b[0;32m--> 632\u001b[0m     response \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mparent\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43merror\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m    633\u001b[0m \u001b[43m        \u001b[49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[38;5;124;43mhttp\u001b[39;49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mrequest\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mresponse\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mcode\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mmsg\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mhdrs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    635\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m response\n",
+      "File \u001b[0;32m/usr/lib/python3.9/urllib/request.py:632\u001b[0m, in \u001b[0;36mHTTPErrorProcessor.http_response\u001b[0;34m(self, request, response)\u001b[0m\n\u001b[1;32m    629\u001b[0m \u001b[38;5;66;03m## According to RFC 2616, \"2xx\" code indicates that the client's\u001b[39;00m\n\u001b[1;32m    630\u001b[0m \u001b[38;5;66;03m# request was successfully received, understood, and accepted.\u001b[39;00m\n\u001b[1;32m    631\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m (\u001b[38;5;241m200\u001b[39m \u001b[38;5;241m<\u001b[39m\u001b[38;5;241m=\u001b[39m code \u001b[38;5;241m<\u001b[39m \u001b[38;5;241m300\u001b[39m):\n\u001b[0;32m--> 632\u001b[0m     response \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mparent\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43merror\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m    633\u001b[0m \u001b[43m        \u001b[49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[38;5;124;43mhttp\u001b[39;49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mrequest\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mresponse\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mcode\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mmsg\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mhdrs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    635\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m response\n",
       "File \u001b[0;32m/usr/lib/python3.9/urllib/request.py:561\u001b[0m, in \u001b[0;36mOpenerDirector.error\u001b[0;34m(self, proto, *args)\u001b[0m\n\u001b[1;32m    559\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m http_err:\n\u001b[1;32m    560\u001b[0m     args \u001b[38;5;241m=\u001b[39m (\u001b[38;5;28mdict\u001b[39m, \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mdefault\u001b[39m\u001b[38;5;124m'\u001b[39m, \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mhttp_error_default\u001b[39m\u001b[38;5;124m'\u001b[39m) \u001b[38;5;241m+\u001b[39m orig_args\n\u001b[0;32m--> 561\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_call_chain\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m)\u001b[49m\n",
       "File \u001b[0;32m/usr/lib/python3.9/urllib/request.py:494\u001b[0m, in \u001b[0;36mOpenerDirector._call_chain\u001b[0;34m(self, chain, kind, meth_name, *args)\u001b[0m\n\u001b[1;32m    492\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m handler \u001b[38;5;129;01min\u001b[39;00m handlers:\n\u001b[1;32m    493\u001b[0m     func \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mgetattr\u001b[39m(handler, meth_name)\n\u001b[0;32m--> 494\u001b[0m     result \u001b[38;5;241m=\u001b[39m \u001b[43mfunc\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    495\u001b[0m     \u001b[38;5;28;01mif\u001b[39;00m result \u001b[38;5;129;01mis\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m:\n\u001b[1;32m    496\u001b[0m         \u001b[38;5;28;01mreturn\u001b[39;00m result\n",
       "File \u001b[0;32m/usr/lib/python3.9/urllib/request.py:641\u001b[0m, in \u001b[0;36mHTTPDefaultErrorHandler.http_error_default\u001b[0;34m(self, req, fp, code, msg, hdrs)\u001b[0m\n\u001b[1;32m    640\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mhttp_error_default\u001b[39m(\u001b[38;5;28mself\u001b[39m, req, fp, code, msg, hdrs):\n\u001b[0;32m--> 641\u001b[0m     \u001b[38;5;28;01mraise\u001b[39;00m HTTPError(req\u001b[38;5;241m.\u001b[39mfull_url, code, msg, hdrs, fp)\n",
@@ -1623,13 +1623,6 @@
     "cv=index_field(cva,\"Country/Region\",filter=d)"
    ]
   },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
-  },
   {
    "cell_type": "markdown",
    "metadata": {
@@ -1637,7 +1630,7 @@
     "id": "PadKA9WIBsIC"
    },
    "source": [
-    "# Random Numbers "
+    "## Random Numbers "
    ]
   },
   {
@@ -1747,7 +1740,7 @@
    ],
    "source": [
     "import numpy as np\n",
-    "# Randon number function \n",
+    "## Randon number function \n",
     "def random_number(seed,a=4,b=1,N=9):\n",
     "    return (a*seed + b)%N\n",
     "\n",
@@ -2242,7 +2235,7 @@
     "id": "HQQ3Srq-BsI7"
    },
    "source": [
-    "## Example 2"
+    "### Example 2"
    ]
   },
   {
@@ -2252,7 +2245,7 @@
     "id": "Jnee7IWbBsI8"
    },
    "source": [
-    "### Random walk\n",
+    "#### Random walk\n",
     "\n",
     "Start at the origin and take a 2D random walk. It is chosen values for $\\Delta x'$ and $\\Delta y'$ in the range [-1,1].\n",
     "They are normalized so each step is of unit length. "
@@ -2392,7 +2385,7 @@
     "id": "pq9z8GhHBsI_"
    },
    "source": [
-    "## <font color='red'>     </font>\n",
+    "### <font color='red'>     </font>\n",
     "**Activity** \n"
    ]
   },
@@ -2718,7 +2711,7 @@
    },
    "outputs": [],
    "source": [
-    "# Finding adjusting parameters \n",
+    "## Finding adjusting parameters \n",
     "def Linear_least_square( x,y ):\n",
     "    \n",
     "    #Finding coefficients \n",
@@ -2767,7 +2760,7 @@
    "source": [
     "xrange=range\n",
     "#========================================================\n",
-    "# Adjusting to a first order polynomy the data set v \n",
+    "## Adjusting to a first order polynomy the data set v \n",
     "#========================================================\n",
     "#Setting figure\n",
     "plt.figure( figsize = (8,5) )\n",
@@ -2807,7 +2800,7 @@
     "id": "6xROQHWYBsHV"
    },
    "source": [
-    "## <font color='red'>     **Activity** </font>\n",
+    "### <font color='red'>     **Activity** </font>\n",
     "http://scipy-cookbook.readthedocs.io/items/robust_regression.html"
    ]
   },