From 66c57247e4f1094e8b80394454021d5ddadcd98b Mon Sep 17 00:00:00 2001 From: "github-actions[bot]" Date: Thu, 13 Aug 2020 00:23:50 +0000 Subject: [PATCH] Rebuild introduction/01-ode_introduction.jmd --- html/introduction/01-ode_introduction.html | 433 +++++++++--------- markdown/introduction/01-ode_introduction.md | 415 +++++++++-------- .../introduction/01-ode_introduction.ipynb | 8 +- pdf/introduction/01-ode_introduction.pdf | Bin 532361 -> 532362 bytes 4 files changed, 428 insertions(+), 428 deletions(-) diff --git a/html/introduction/01-ode_introduction.html b/html/introduction/01-ode_introduction.html index 84af8649..2d8165a8 100644 --- a/html/introduction/01-ode_introduction.html +++ b/html/introduction/01-ode_introduction.html @@ -663,7 +663,7 @@
Chris Rackauckas

Basic Introduction Via Ordinary Differential Equations

-

This notebook will get you started with DifferentialEquations.jl by introducing you to the functionality for solving ordinary differential equations (ODEs). The corresponding documentation page is the ODE tutorial. While some of the syntax may be different for other types of equations, the same general principles hold in each case. Our goal is to give a gentle and thorough introduction that highlights these principles in a way that will help you generalize what you have learned.

+

This notebook will get you started with DifferentialEquations.jl by introducing you to the functionality for solving ordinary differential equations (ODEs). The corresponding documentation page is the ODE tutorial. While some of the syntax may be different for other types of equations, the same general principles hold in each case. Our goal is to give a gentle and thorough introduction that highlights these principles in a way that will help you generalize what you have learned.

Background

If you are new to the study of differential equations, it can be helpful to do a quick background read on the definition of ordinary differential equations. We define an ordinary differential equation as an equation which describes the way that a variable $u$ changes, that is

\[ @@ -743,7 +743,7 @@

First Model: Exponential Growth

and that's it: we have succesfully solved our first ODE!

Analyzing the Solution

-

Of course, the solution type is not interesting in and of itself. We want to understand the solution! The documentation page which explains in detail the functions for analyzing the solution is the Solution Handling page. Here we will describe some of the basics. You can plot the solution using the plot recipe provided by Plots.jl:

+

Of course, the solution type is not interesting in and of itself. We want to understand the solution! The documentation page which explains in detail the functions for analyzing the solution is the Solution Handling page. Here we will describe some of the basics. You can plot the solution using the plot recipe provided by Plots.jl:

@@ -752,7 +752,7 @@ 
Analyzing the Solution
- +

From the picture we see that the solution is an exponential curve, which matches our intuition. As a plot recipe, we can annotate the result using any of the Plots.jl attributes. For example:

@@ -763,7 +763,7 @@

Analyzing the Solution

- +

Using the mutating plot! command we can add other pieces to our plot. For this ODE we know that the true solution is $u(t) = u_0 exp(at)$, so let's add some of the true solution to our plot:

@@ -773,7 +773,7 @@

Analyzing the Solution

- +

In the previous command I demonstrated sol.t, which grabs the array of time points that the solution was saved at:

@@ -869,7 +869,7 @@

Analyzing the Solution

Controlling the Solver

-

DifferentialEquations.jl has a common set of solver controls among its algorithms which can be found at the Common Solver Options page. We will detail some of the most widely used options.

+

DifferentialEquations.jl has a common set of solver controls among its algorithms which can be found at the Common Solver Options page. We will detail some of the most widely used options.

The most useful options are the tolerances abstol and reltol. These tell the internal adaptive time stepping engine how precise of a solution you want. Generally, reltol is the relative accuracy while abstol is the accuracy when u is near zero. These tolerances are local tolerances and thus are not global guarantees. However, a good rule of thumb is that the total solution accuracy is 1-2 digits less than the relative tolerances. Thus for the defaults abstol=1e-6 and reltol=1e-3, you can expect a global accuracy of about 1-2 digits. If we want to get around 6 digits of accuracy, we can use the commands:

@@ -913,7 +913,7 @@

Controlling the Solver

- +

Notice that by decreasing the tolerance, the number of steps the solver had to take was 9 instead of the previous 5. There is a trade off between accuracy and speed, and it is up to you to determine what is the right balance for your problem.

Another common option is to use saveat to make the solver save at specific time points. For example, if we want the solution at an even grid of t=0.1k for integers k, we would use the command:

@@ -1041,7 +1041,7 @@

Controlling the Solver

Note that similarly on the other side there is save_end=false.

-

More advanced saving behaviors, such as saving functionals of the solution, are handled via the SavingCallback in the Callback Library which will be addressed later in the tutorial.

+

More advanced saving behaviors, such as saving functionals of the solution, are handled via the SavingCallback in the Callback Library which will be addressed later in the tutorial.

Choosing Solver Algorithms

There is no best algorithm for numerically solving a differential equation. When you call solve(prob), DifferentialEquations.jl makes a guess at a good algorithm for your problem, given the properties that you ask for (the tolerances, the saving information, etc.). However, in many cases you may want more direct control. A later notebook will help introduce the various algorithms in DifferentialEquations.jl, but for now let's introduce the syntax.

The most crucial determining factor in choosing a numerical method is the stiffness of the model. Stiffness is roughly characterized by a Jacobian f with large eigenvalues. That's quite mathematical, and we can think of it more intuitively: if you have big numbers in f (like parameters of order 1e5), then it's probably stiff. Or, as the creator of the MATLAB ODE Suite, Lawrence Shampine, likes to define it, if the standard algorithms are slow, then it's stiff. We will go into more depth about diagnosing stiffness in a later tutorial, but for now note that if you believe your model may be stiff, you can hint this to the algorithm chooser via alg_hints = [:stiff].

@@ -1293,7 +1293,7 @@

Systems of ODEs: The Lorenz Equation

- +

If we instead want to plot values against each other, we can use the vars command. Let's plot variable 1 against variable 2 against variable 3:

@@ -1303,7 +1303,7 @@

Systems of ODEs: The Lorenz Equation

- +

This is the classic Lorenz attractor plot, where the x axis is u[1], the y axis is u[2], and the z axis is u[3]. Note that the plot recipe by default uses the interpolation, but we can turn this off:

@@ -1313,7 +1313,7 @@

Systems of ODEs: The Lorenz Equation

- +

Yikes! This shows how calculating the continuous solution has saved a lot of computational effort by computing only a sparse solution and filling in the values! Note that in vars, 0=time, and thus we can plot the time series of a single component like:

@@ -1323,7 +1323,7 @@

Systems of ODEs: The Lorenz Equation

- +

Internal Types

The last basic user-interface feature to explore is the choice of types. DifferentialEquations.jl respects your input types to determine the internal types that are used. Thus since in the previous cases, when we used Float64 values for the initial condition, this meant that the internal values would be solved using Float64. We made sure that time was specified via Float64 values, meaning that time steps would utilize 64-bit floats as well. But, by simply changing these types we can change what is used internally.

@@ -1346,52 +1346,48 @@

Internal Types

 retcode: Success
 Interpolation: automatic order switching interpolation
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@@ -1405,10 +1401,10 @@

Internal Types

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@@ -1422,10 +1418,10 @@

Internal Types

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@@ -1441,65 +1437,55 @@

Internal Types

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 Interpolation: automatic order switching interpolation
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Internal Types

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@@ -1527,69 +1513,58 @@

Internal Types

 retcode: Success
 Interpolation: automatic order switching interpolation
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@@ -1615,46 +1590,46 @@

Internal Types

Interpolation: automatic order switching interpolation t: 10-element Array{Float64,1}: 0.0 - 0.04787147906142052 - 0.12539802723603002 - 0.21507571783207402 - 0.325846259176126 - 0.4460780571742323 - 0.5872688137412091 - 0.7254883852950837 - 0.8746888396672391 + 0.048006104540199344 + 0.12508535396737538 + 0.2128933453419654 + 0.32374584194942524 + 0.4446150526079904 + 0.5833187308680358 + 0.7299931730760287 + 0.8834638751632482 1.0 u: 10-element Array{StaticArrays.SArray{Tuple{4,2},Float64,2,8},1}: - [0.4874233029564281 0.3982867450807297; 0.6772478849572932 0.6959236614196 -576; 0.6085330252525174 0.9374145300523831; 0.23536295170345212 0.995082846 -2618915] - [0.43638242256861465 0.14301977844655916; 0.7621138510001858 0.69894050777 -71581; 0.5341686470849878 0.9381431319946251; 0.37586987888505197 1.2442550 -154598546] - [0.27779815214551834 -0.4174290891948917; 0.793141527335821 0.540624562039 -3191; 0.45840817411154205 1.086355879293604; 0.5833150951434182 1.583199145 -06541] - [-0.016829347389339444 -1.2653846282178236; 0.6870445133418367 0.174608873 -34331776; 0.46964627496784694 1.53721689113887; 0.7769949328810547 1.841512 -2300669928] - [-0.5206740041543536 -2.524411407968275; 0.40969849906855377 -0.3753253293 -3379414; 0.6788138558288686 2.5806036031435466; 0.9187098123407041 1.898433 -2366280396] - [-1.1815959989375062 -3.974778988372509; 0.051100745340523335 -0.786059412 -6151734; 1.1970019209323275 4.356707097792163; 0.9105784263360842 1.5469023 -68592912] - [-1.9680054748106561 -5.402860864133682; -0.20940321655428795 -0.482147548 -12639904; 2.20298262314218 7.182542513561056; 0.6319484020283754 0.48019898 -98343846] - [-2.5327620620005526 -5.9585953516559895; -0.005926578426302509 1.24124998 -10927173; 3.5124964485205914 10.327992534124597; 0.035749747961268175 -1.30 -498695854531] - [-2.596994248611811 -4.859412994319518; 1.0430262489490545 5.2527113808371 -89; 5.014706361397094 13.277481617574193; -0.9754492789048056 -3.9954590891 -753945] - [-1.9434568050318903 -1.9782594583345565; 2.759616135186503 10.52423871589 -5834; 5.995676470583454 14.410346738576262; -2.0761188513245865 -6.67750766 -9090453] + [0.6470268048115251 0.42689239938470225; 0.4468712596313109 0.930792622748 +21; 0.8995497299530391 0.4955864508508838; 0.19897099052479028 0.7955219510 +043468] + [0.602392813583938 0.2316510143444062; 0.6326231722354123 0.87331931673791 +55; 0.7936623605638815 0.4738199645556346; 0.4236254028720524 0.96137318774 +05786] + [0.41258139268512584 -0.1804019440562041; 0.769282141598299 0.668409010558 +5981; 0.6794213884975377 0.5458882638846405; 0.7637205865686998 1.180165833 +722606] + [0.02194072683165854 -0.7782737762342137; 0.7054948276902163 0.31552386775 +256436; 0.6800307037504818 0.8220189345947837; 1.0966484744079248 1.3372547 +190895772] + [-0.7101938625822555 -1.6694784653940866; 0.3716562266183081 -0.1881217526 +428317; 0.9628967051367382 1.5119594167855097; 1.3868615058813365 1.3554355 +28105561] + [-1.7332757050882404 -2.6890904446257506; -0.13852866977255995 -0.59266048 +01359094; 1.721984075196057 2.718169853049194; 1.4690378922944827 1.0921911 +459488223] + [-3.0034987304271112 -3.666588030802566; -0.5701335090616335 -0.5102704207 +897966; 3.2257135247869404 4.604820390072593; 1.1673356535511101 0.36304997 +29154003] + [-4.10323291282484 -4.066217941561766; -0.34476360963855546 0.661994078047 +2835; 5.448360327248404 6.881372397693559; 0.2934308635954497 -0.9404065360 +691877] + [-4.437996394845046 -3.226698038570903; 1.2302814174986403 3.4921094452365 +8; 8.06195674098168 8.917139630712603; -1.2783527190357418 -2.8386164247715 +86] + [-3.726546412830267 -1.367370783576439; 3.63590816493634 6.832838070121893 +; 9.765012397461916 9.606747074260191; -2.8823394142024394 -4.5229393664650 +89] @@ -1667,10 +1642,10 @@

Internal Types

 4×2 StaticArrays.SArray{Tuple{4,2},Float64,2,8} with indices SOneTo(4)×SOne
 To(2):
- 0.277798  -0.417429
- 0.793142   0.540625
- 0.458408   1.08636
- 0.583315   1.5832
+ 0.412581  -0.180402
+ 0.769282   0.668409
+ 0.679421   0.545888
+ 0.763721   1.18017
@@ -1679,7 +1654,7 @@

Conclusion

Appendix

-

This tutorial is part of the SciMLTutorials.jl repository, found at: https://github.com/SciMLTutorials/SciMLTutorials.jl. For more information on doing scientific machine learning (SciML) with open source software, check out https://sciml.ai/.

+

This tutorial is part of the SciMLTutorials.jl repository, found at: https://github.com/SciML/SciMLTutorials.jl. For more information on doing scientific machine learning (SciML) with open source software, check out https://sciml.ai/.

To locally run this tutorial, do the following commands:

using SciMLTutorials
@@ -1708,7 +1683,7 @@ 
Conclusion

 [6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf] BenchmarkTools 0.5.0
 [0c46a032-eb83-5123-abaf-570d42b7fbaa] DifferentialEquations 6.15.0
 [65888b18-ceab-5e60-b2b9-181511a3b968] ParameterizedFunctions 5.4.0
-[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.6
+[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.8
 [90137ffa-7385-5640-81b9-e52037218182] StaticArrays 0.12.4
 [c3572dad-4567-51f8-b174-8c6c989267f4] Sundials 4.2.5
 [37e2e46d-f89d-539d-b4ee-838fcccc9c8e] LinearAlgebra
@@ -1719,7 +1694,7 @@

Conclusion

Published from 01-ode_introduction.jmd - using Weave.jl v0.10.2 on 2020-07-28. + using Weave.jl v0.10.2 on 2020-08-13.

diff --git a/markdown/introduction/01-ode_introduction.md b/markdown/introduction/01-ode_introduction.md index 51fba6c4..8c3c595b 100644 --- a/markdown/introduction/01-ode_introduction.md +++ b/markdown/introduction/01-ode_introduction.md @@ -6,7 +6,7 @@ title: "An Intro to DifferentialEquations.jl" ## Basic Introduction Via Ordinary Differential Equations -This notebook will get you started with DifferentialEquations.jl by introducing you to the functionality for solving ordinary differential equations (ODEs). The corresponding documentation page is the [ODE tutorial](https://docs.juliadiffeq.org/dev/tutorials/ode_example/). While some of the syntax may be different for other types of equations, the same general principles hold in each case. Our goal is to give a gentle and thorough introduction that highlights these principles in a way that will help you generalize what you have learned. +This notebook will get you started with DifferentialEquations.jl by introducing you to the functionality for solving ordinary differential equations (ODEs). The corresponding documentation page is the [ODE tutorial](https://docs.sciml.ai/dev/tutorials/ode_example/). While some of the syntax may be different for other types of equations, the same general principles hold in each case. Our goal is to give a gentle and thorough introduction that highlights these principles in a way that will help you generalize what you have learned. ### Background @@ -98,7 +98,7 @@ and that's it: we have succesfully solved our first ODE! #### Analyzing the Solution -Of course, the solution type is not interesting in and of itself. We want to understand the solution! The documentation page which explains in detail the functions for analyzing the solution is the [Solution Handling](https://docs.juliadiffeq.org/dev/basics/solution/) page. Here we will describe some of the basics. You can plot the solution using the plot recipe provided by [Plots.jl](http://docs.juliaplots.org/dev/): +Of course, the solution type is not interesting in and of itself. We want to understand the solution! The documentation page which explains in detail the functions for analyzing the solution is the [Solution Handling](https://docs.sciml.ai/dev/basics/solution/) page. Here we will describe some of the basics. You can plot the solution using the plot recipe provided by [Plots.jl](http://docs.juliaplots.org/dev/): ````julia using Plots; gr() @@ -238,7 +238,7 @@ sol(0.45) #### Controlling the Solver -DifferentialEquations.jl has a common set of solver controls among its algorithms which can be found [at the Common Solver Options](https://docs.juliadiffeq.org/dev/basics/common_solver_opts/) page. We will detail some of the most widely used options. +DifferentialEquations.jl has a common set of solver controls among its algorithms which can be found [at the Common Solver Options](https://docs.sciml.ai/dev/basics/common_solver_opts/) page. We will detail some of the most widely used options. The most useful options are the tolerances `abstol` and `reltol`. These tell the internal adaptive time stepping engine how precise of a solution you want. Generally, `reltol` is the relative accuracy while `abstol` is the accuracy when `u` is near zero. These tolerances are local tolerances and thus are not global guarantees. However, a good rule of thumb is that the total solution accuracy is 1-2 digits less than the relative tolerances. Thus for the defaults `abstol=1e-6` and `reltol=1e-3`, you can expect a global accuracy of about 1-2 digits. If we want to get around 6 digits of accuracy, we can use the commands: @@ -427,7 +427,7 @@ u: 1-element Array{Float64,1}: Note that similarly on the other side there is `save_end=false`. -More advanced saving behaviors, such as saving functionals of the solution, are handled via the `SavingCallback` in the [Callback Library](https://docs.juliadiffeq.org/dev/features/callback_library/#saving_callback-1) which will be addressed later in the tutorial. +More advanced saving behaviors, such as saving functionals of the solution, are handled via the `SavingCallback` in the [Callback Library](https://docs.sciml.ai/dev/features/callback_library/#saving_callback-1) which will be addressed later in the tutorial. #### Choosing Solver Algorithms @@ -767,46 +767,46 @@ retcode: Success Interpolation: automatic order switching interpolation t: 10-element Array{Float64,1}: 0.0 - 0.04391849892224981 - 0.11616653082165809 - 0.19658120398062442 - 0.2952629698711988 - 0.41733702034155395 - 0.5506712825781492 - 0.7166461355644652 - 0.8883937134768247 + 0.035837145273206604 + 0.09398941811515005 + 0.1711376760453636 + 0.268191831977834 + 0.379912344355524 + 0.5067002136400818 + 0.6591129107963655 + 0.8295266708196971 1.0 u: 10-element Array{Array{Float64,2},1}: - [0.19547760575940165 0.47247399507044374; 0.9947932470272081 0.15631428237 -218148; 0.3573489500076166 0.23763371724112692; 0.03808030495696069 0.31982 -31726743066] - [0.1967588904438102 0.4049692637996855; 0.9971930547885663 0.2018872361416 -5527; 0.32439425488809615 0.17741325228186972; 0.02792721637674083 0.470989 -78156002314] - [0.20539607624568004 0.21469182945001108; 0.9933207532448118 0.17931771064 -2172; 0.2676854653186985 0.12814765402639164; 0.0026932230530117354 0.70609 -82198891923] - [0.22908519743536082 -0.11156461673042634; 0.9826081603369168 0.0274918524 -95605457; 0.19700705908618227 0.17463789252360576; -0.035781566166134156 0. -9339463609707335] - [0.28559261346488707 -0.6637539057836184; 0.9671250650391829 -0.2944838280 -7270324; 0.09025850576027471 0.42558522498464146; -0.09262120131943437 1.13 -79397125539228] - [0.4060667968976291 -1.5276892713937908; 0.9493118233141383 -0.77656521243 -56482; -0.09198482510918725 1.1005046932277391; -0.16460909725367076 1.2243 -954837465991] - [0.6033391768554422 -2.57278044905076; 0.9220470903483592 -1.1706998836767 -41; -0.38443862821019276 2.3476929618145768; -0.2211142385015843 1.03543600 -71645454] - [0.9160020947639793 -3.6864769046682495; 0.8208246732749581 -0.97882022715 -92469; -0.9237891497332187 4.56751278018785; -0.21007136743282612 0.2769593 -8201657266] - [1.2214963278777484 -4.021425675138098; 0.5093406799523311 0.6643799209382 -052; -1.6874968552130294 7.215322945108149; -0.03812101279378505 -1.1961215 -559740883] - [1.327290834745476 -3.422361319155158; 0.10870253255667689 2.7861791915498 -078; -2.2531847418271047 8.702214410706276; 0.18892900683270336 -2.50756160 -184843] + [0.18084059433736632 0.6735843499165595; 0.3919291156743039 0.115742286780 +54488; 0.40979783636057543 0.05395672549875652; 0.020631558005682837 0.0982 +4514959105457] + [0.18046325993955636 0.668686753580626; 0.4408350224439239 0.1850137351973 +129; 0.3852002493779446 -0.036721323047792714; 0.05579205524922205 0.225872 +4549433267] + [0.16655433987985668 0.6102374663148857; 0.4975366228736487 0.228314853566 +535; 0.3493408640369401 -0.167515990426645; 0.1089016604415104 0.4339295839 +870391] + [0.12386822230553543 0.4314730648443703; 0.5325623530093845 0.156577054425 +4565; 0.31465828523555234 -0.28733211161914185; 0.1691757802961833 0.702842 +1494344252] + [0.03560974297215906 0.03857849082803655; 0.5220504563429157 -0.1229849351 +6288994; 0.30183315737344385 -0.3011683697965832; 0.22302549836951152 1.008 +3219085764692] + [-0.10101207299364791 -0.6357986239205559; 0.45967491107086084 -0.64175248 +33350452; 0.34203420153287106 -0.047480516997589206; 0.24639689588530547 1. +2751457854543928] + [-0.2739142208109207 -1.6384011625137633; 0.3715690742276302 -1.3342620235 +00564; 0.46690336065353466 0.6924903842987244; 0.21270864269720302 1.402777 +4296771829] + [-0.44714881115771116 -3.0119983730557074; 0.329338443182707 -1.9715608166 +3602; 0.7140441613724511 2.311174601725102; 0.07752908941628911 1.198490978 +9188136] + [-0.4887018088262447 -4.360217640724318; 0.48882107525514373 -1.8128860253 +65758; 1.037498127480093 4.994001467744241; -0.19453130773147526 0.35527938 +740192766] + [-0.2375674812350816 -4.836668317759423; 0.9640684228881622 0.012009728048 +617108; 1.2418221688721471 8.13749460005548; -0.5631971391401789 -1.2496559 +97163155] ```` @@ -822,10 +822,10 @@ sol[3] ```` 4×2 Array{Float64,2}: - 0.205396 0.214692 - 0.993321 0.179318 - 0.267685 0.128148 - 0.00269322 0.706098 + 0.166554 0.610237 + 0.497537 0.228315 + 0.349341 -0.167516 + 0.108902 0.43393 ```` @@ -841,10 +841,10 @@ big_u0 = big.(u0) ```` 4×2 Array{BigFloat,2}: - 0.195478 0.472474 - 0.994793 0.156314 - 0.357349 0.237634 - 0.0380803 0.319823 + 0.180841 0.673584 + 0.391929 0.115742 + 0.409798 0.0539567 + 0.0206316 0.0982451 ```` @@ -862,55 +862,65 @@ sol = solve(prob) ```` retcode: Success Interpolation: automatic order switching interpolation -t: 5-element Array{Float64,1}: +t: 6-element Array{Float64,1}: 0.0 - 0.16431365291680158 - 0.42772799694479635 - 0.7429273250346953 + 0.03583714527320661 + 0.194991627996516 + 0.4630908878992731 + 0.7738689450854899 1.0 -u: 5-element Array{Array{BigFloat,2},1}: - [0.195477605759401651397411114885471761226654052734375 0.47247399507044374 -189490554272197186946868896484375; 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-0.19600798255877244632288902 -18995307137628399176768369382345035194959374446721225 0.0975410926106817946 -0997044065474665955596688167341843947515276845093994928158256] - [1.32729326241134006287635767311000464401606204689510237934020374199800825 -9469066 -3.4223601333714531089671639954023511016040980682317218542666375014 -39773870896201; 0.108698691255617172263659937892566078017278539155819006244 -037210103529069167236 2.786202813391037818359936976347279706879791521774288 -863680011651198991729051208; -2.2531929692789063074567968481220780504014682 -51957243224323051431841301561602982 8.7022373460083854198385569651104452783 -58232039893859938841147476377781683039161; 0.188931257809458908993136336202 -8892142683421865168275312074040879641080749853423 -2.5075752950901654872209 -05037893847063468506431249839324076778500700819963307878] +u: 6-element Array{Array{BigFloat,2},1}: + [0.180840594337366322719162781140767037868499755859375 0.67358434991655946 +61093336981139145791530609130859375; 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0.9640742203046870982916028247124909723589491965795135469 +142904744642675147665146 0.012026382514392620743978690885045373060404761873 +16632796454241152766663517092376; 1.241824630195814033553847556076939963741 +099321798557522891495291539848825475431 8.137527291010752967371510077875190 +577899551994606589389989645911041838467192255; -0.5632001619743111957736049 +762644356956324697112698587854590775045087184782546935 -1.24966846338776872 +5591003859894959509238124974757938359952696079488339402465239] ```` @@ -921,8 +931,8 @@ sol[1,3] ```` -0.4190519131376829214900992052853267706081280804459160189690813640947013103 -54188 +0.1054770303417736243935025500744173133506311201067088331688609777284457340 +603811 ```` @@ -940,58 +950,69 @@ sol = solve(prob) ```` retcode: Success Interpolation: automatic order switching interpolation -t: 5-element Array{BigFloat,1}: +t: 6-element Array{BigFloat,1}: 0.0 - 0.164313652916801585626492896583952165578411155101951286067465557611047203 -1138849 - 0.427727996944796294527843944757292385903083568773961565170119276588554755 -9741266 - 0.742927325034695136401447040365724856273947169914597190904776231270937791 -2250609 + 0.035837145273206608178723189167339137857095534903468664917505359195347374 +23204982 + 0.194991627996515979217804394973535599921567308095355171016973940049211908 +9884716 + 0.463090887899273020530156824014946889035006382941401467731341887223824925 +6842297 + 0.773868945085489624549045514310018071694747329429624922220186895675646448 +2974645 1.0 -u: 5-element Array{Array{BigFloat,2},1}: - [0.195477605759401651397411114885471761226654052734375 0.47247399507044374 -189490554272197186946868896484375; 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-0.196007982558772479496696254 -0685288604846133832860236083107591084403132548498308 0.09754109261068218241 -424080204937089362898727370318886651500308268935613813062462] - [1.32729326241134009473729119383123162906909567782678095071089021205217968 -2879939 -3.4223601333714523499479740294823141308942481859279460784346289160 -41779415048734; 0.108698691255616798580082746990476355754390152450214892872 -3511786860842125377951 2.78620281339103973930908667389844392818958923470920 -7369001012163869895613773576; -2.253192969278906733802575017310885364302372 -826646261931733149401512713203984502 8.702237346008386350915901958643353084 -280821640119173339345533706381427717014159; 0.18893125780945911545104970108 -41252970547558183373656262026927812003675526130454 -2.507575295090166553239 -637180652369270880869599031441813099243322060728415495143] +u: 6-element Array{Array{BigFloat,2},1}: + [0.180840594337366322719162781140767037868499755859375 0.67358434991655946 +61093336981139145791530609130859375; 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0.53504762541413646821294797137891029476293620166061606844 +79276648493804408495446 0.1061517677316855471489357946409619570751939406063 +065166698026943688107711988088; 0.30792452730653374763960014155661966819808 +49202039011509984942656173272997454979 -0.307094477860558215750222920176060 +5518897008164779362686960152697734571150537167; 0.1849158865576971139715019 +717657115156632348880362748243228332253214988431866827 0.782311740584947638 +8764273661869517449477590933250834858133726536677916324326434] + [-0.2146119388544680526499783059115886830471181037400863945339231680797528 +011617107 -1.27061491587453683957468371948949588237502762188086545121029150 +7582922781830084; 0.4002798478015793843310604305141846004576684168117035267 +883718652821936310438161 -1.09669014271146961257414559087044939721570022451 +6480861847891670770450405681655; 0.4145423175382235193934039266927032516183 +471541035136210020400652880132436128887 0.378175837600582638217151796514327 +2538983778336510833096915194273958574596751799; 0.2320743638792340509899185 +204826100253512487010923767020456986416738900304348777 1.384610241749128245 +706808266384673973812180557026050828247699762142528773573833] + [-0.5007388901746559409006362313182505067290462865994494904448966585591709 +028928521 -3.97761001712051933830887569347964167007444388182066265352816133 +0546840323732766; 0.4054437292182932751423385962833642983697728370960068813 +59815635683257663056588 -2.011353006145746328676274472241944655047177600404 +328052564860805146820706463074; 0.93476338820963439234872830995177945763420 +47952204520983261087211883504419678039 4.0344030920550600961857278545471264 +09573172930223148788851170379316701047075139; -0.09274585381337549554031239 +546365706947304498393809063158412877404364447107910032 0.710824631808655567 +8413460606867351183645296484721313404784406029032226857640423] + [-0.2375655989766288635623445937698041316088398338324418675676537220259683 +109363916 -4.83667706389088247932421890526157374482068130187471835745441467 +5036155870062889; 0.9640742203046872031657291038113834457202187672024569551 +47720536005092765000437 0.0120263825143930906000951047150891152985436824793 +0366545333596687897331867319626; 1.2418246301958140442970293004983402005682 +41217176261307562452032512373044316663 8.1375272910107534696654638963324004 +11885665267029395648207824836185704394115608; -0.56320016197431126022005696 +0096451776085620357944243292217147412965006226427203 -1.2496684633877690498 +1468517109509246001332365789552498117853757303889859295492] ```` @@ -1017,48 +1038,52 @@ sol = solve(prob) ```` retcode: Success Interpolation: automatic order switching interpolation -t: 10-element Array{Float64,1}: +t: 11-element Array{Float64,1}: 0.0 - 0.044869042977834686 - 0.12047218467423944 - 0.2070300915623275 - 0.31503114913992514 - 0.43779140558809754 - 0.575815544150443 - 0.7266748899317822 - 0.8810312961677973 + 0.031534234320871164 + 0.0903538209535513 + 0.16273285285168482 + 0.2540631921245806 + 0.3596373654404972 + 0.48240494481565993 + 0.6122298297351463 + 0.7546228200119018 + 0.9245258073639878 1.0 -u: 10-element Array{StaticArrays.SArray{Tuple{4,2},Float64,2,8},1}: - [0.4520230126339584 0.6680268714046109; 0.2368659623450058 0.7924440732034 -83; 0.7620755122991452 0.6781378044136375; 0.3435749209054344 0.69061940762 -27987] - [0.37201258585921493 0.514220246730995; 0.3567220622025925 0.8323975712094 -936; 0.70716699474354 0.6072926012645857; 0.5347450815707021 0.916571942815 -2984] - [0.13231654140001026 0.12666717046882936; 0.42658141445512027 0.7415553632 -985512; 0.6795310365018049 0.5889502442558292; 0.8323524898084189 1.2576502 -664321256] - [-0.29751254646045877 -0.5021177507444233; 0.32951655665886415 0.437385065 -69164215; 0.7883053652050243 0.7713551773106111; 1.1133248314957096 1.56100 -3336327043] - [-1.0354381534450656 -1.515623185783018; 0.02489663619642557 -0.1224110615 -8104331; 1.202648044838969 1.3806948030956114; 1.330924476849903 1.75784699 -5640528] - [-2.0563188091682374 -2.850692116825032; -0.3870142281921569 -0.7539286188 -02889; 2.1230304621864518 2.662083951089823; 1.3314529882288728 1.658053699 -6231647] - [-3.229259238424727 -4.309781058826721; -0.5954576201395906 -1.01886010124 -72134; 3.7487439363241064 4.841695838848294; 0.9283975265347764 1.032035150 -0376173] - [-4.150006301664626 -5.348462541236869; -0.003884115166779778 -0.133020076 -2828313; 6.077973947112321 7.860695646513079; -0.08921447611666489 -0.36892 -328153197584] - [-4.125149934653158 -5.084176300502514; 2.045146402537434 2.70530792575870 -65; 8.563827643197406 10.95309478080606; -1.7790493688595437 -2.58548636531 -9851] - [-3.0152622751531912 -3.443740195116914; 4.873727763649718 6.4960585364530 -665; 9.998639475680433 12.608225285564842; -3.464284909772859 -4.7349967607 -28366] +u: 11-element Array{StaticArrays.SArray{Tuple{4,2},Float64,2,8},1}: + [0.6282568043393855 0.07569384480412289; 0.04863254826785712 0.55609408933 +31805; 0.31211882496530063 0.9136754860465177; 0.8384047176976088 0.9599419 +335396784] + [0.4986928982171224 -0.08534832172191445; 0.06432701968506847 0.5414625992 +000579; 0.27016687131620104 0.9462228296202699; 1.0306218063241135 1.079841 +01075332] + [0.1651626783359869 -0.44738910343408234; -0.011337417684148102 0.44931018 +04563632; 0.26046096335881863 1.0767262597090688; 1.369185538579408 1.27045 +9812784981] + [-0.40594149770977866 -0.9906310561270396; -0.2701642727359141 0.248683491 +38858778; 0.403824275225324 1.3808811481826337; 1.7330447962591307 1.432696 +9038354638] + [-1.3592894238728734 -1.792159817488078; -0.7902177270473212 -0.0662864183 +0804141; 0.8938148073872584 2.0211195946103806; 2.0709810894814216 1.495551 +80432164] + [-2.719797153268021 -2.7913538364585055; -1.5046136671569859 -0.3640986760 +079169; 1.9765519813167867 3.139792985505065; 2.227260778467857 1.327087942 +059268] + [-4.481072011152188 -3.8611293084992315; -2.1684641943714724 -0.3449016776 +0947005; 4.005681152109637 4.9141846701290675; 1.98884828146129 0.747182250 +5878576] + [-6.226827745091287 -4.567945906464105; -2.1798109262526673 0.477796149555 +5902; 7.009832028724327 7.160548035095383; 1.122495415136825 -0.36332239361 +229646] + [-7.4285727227720315 -4.352992416018494; -0.6898836927321059 2.76009030466 +595; 10.994208707479649 9.590290824212643; -0.6663422289601704 -2.164618585 +8788] + [-6.795457540002504 -1.948167355445622; 4.068291748349399 7.75024954461039 +2; 15.615881464483781 11.292130406109363; -3.9656334476726434 -4.9312991163 +23148] + [-5.429150981768089 0.0882010186754072; 7.401221298698479 10.7376323149607 +52; 17.114431832927888 11.167200396917249; -5.775251136289583 -6.2551084200 +70851] ```` @@ -1071,10 +1096,10 @@ sol[3] ```` 4×2 StaticArrays.SArray{Tuple{4,2},Float64,2,8} with indices SOneTo(4)×SOne To(2): - 0.132317 0.126667 - 0.426581 0.741555 - 0.679531 0.58895 - 0.832352 1.25765 + 0.165163 -0.447389 + -0.0113374 0.44931 + 0.260461 1.07673 + 1.36919 1.27046 ```` @@ -1088,7 +1113,7 @@ These are the basic controls in DifferentialEquations.jl. All equations are defi ## Appendix - This tutorial is part of the SciMLTutorials.jl repository, found at: . + This tutorial is part of the SciMLTutorials.jl repository, found at: . For more information on doing scientific machine learning (SciML) with open source software, check out . To locally run this tutorial, do the following commands: @@ -1122,7 +1147,7 @@ Status `/builds/JuliaGPU/DiffEqTutorials.jl/tutorials/introduction/Project.toml` [6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf] BenchmarkTools 0.5.0 [0c46a032-eb83-5123-abaf-570d42b7fbaa] DifferentialEquations 6.15.0 [65888b18-ceab-5e60-b2b9-181511a3b968] ParameterizedFunctions 5.4.0 -[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.6 +[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.8 [90137ffa-7385-5640-81b9-e52037218182] StaticArrays 0.12.4 [c3572dad-4567-51f8-b174-8c6c989267f4] Sundials 4.2.5 [37e2e46d-f89d-539d-b4ee-838fcccc9c8e] LinearAlgebra diff --git a/notebook/introduction/01-ode_introduction.ipynb b/notebook/introduction/01-ode_introduction.ipynb index c37d009e..4998bfb2 100644 --- a/notebook/introduction/01-ode_introduction.ipynb +++ b/notebook/introduction/01-ode_introduction.ipynb @@ -3,7 +3,7 @@ { "cell_type": "markdown", "source": [ - "## Basic Introduction Via Ordinary Differential Equations\n\nThis notebook will get you started with DifferentialEquations.jl by introducing you to the functionality for solving ordinary differential equations (ODEs). The corresponding documentation page is the [ODE tutorial](https://docs.juliadiffeq.org/dev/tutorials/ode_example/). While some of the syntax may be different for other types of equations, the same general principles hold in each case. Our goal is to give a gentle and thorough introduction that highlights these principles in a way that will help you generalize what you have learned.\n\n### Background\n\nIf you are new to the study of differential equations, it can be helpful to do a quick background read on [the definition of ordinary differential equations](https://en.wikipedia.org/wiki/Ordinary_differential_equation). We define an ordinary differential equation as an equation which describes the way that a variable $u$ changes, that is\n\n$$u' = f(u,p,t)$$\n\nwhere $p$ are the parameters of the model, $t$ is the time variable, and $f$ is the nonlinear model of how $u$ changes. The initial value problem also includes the information about the starting value:\n\n$$u(t_0) = u_0$$\n\nTogether, if you know the starting value and you know how the value will change with time, then you know what the value will be at any time point in the future. This is the intuitive definition of a differential equation.\n\n### First Model: Exponential Growth\n\nOur first model will be the canonical exponential growth model. This model says that the rate of change is proportional to the current value, and is this:\n\n$$u' = au$$\n\nwhere we have a starting value $u(0)=u_0$. Let's say we put 1 dollar into Bitcoin which is increasing at a rate of $98\\%$ per year. Then calling now $t=0$ and measuring time in years, our model is:\n\n$$u' = 0.98u$$\n\nand $u(0) = 1.0$. We encode this into Julia by noticing that, in this setup, we match the general form when" + "## Basic Introduction Via Ordinary Differential Equations\n\nThis notebook will get you started with DifferentialEquations.jl by introducing you to the functionality for solving ordinary differential equations (ODEs). The corresponding documentation page is the [ODE tutorial](https://docs.sciml.ai/dev/tutorials/ode_example/). While some of the syntax may be different for other types of equations, the same general principles hold in each case. Our goal is to give a gentle and thorough introduction that highlights these principles in a way that will help you generalize what you have learned.\n\n### Background\n\nIf you are new to the study of differential equations, it can be helpful to do a quick background read on [the definition of ordinary differential equations](https://en.wikipedia.org/wiki/Ordinary_differential_equation). We define an ordinary differential equation as an equation which describes the way that a variable $u$ changes, that is\n\n$$u' = f(u,p,t)$$\n\nwhere $p$ are the parameters of the model, $t$ is the time variable, and $f$ is the nonlinear model of how $u$ changes. The initial value problem also includes the information about the starting value:\n\n$$u(t_0) = u_0$$\n\nTogether, if you know the starting value and you know how the value will change with time, then you know what the value will be at any time point in the future. This is the intuitive definition of a differential equation.\n\n### First Model: Exponential Growth\n\nOur first model will be the canonical exponential growth model. This model says that the rate of change is proportional to the current value, and is this:\n\n$$u' = au$$\n\nwhere we have a starting value $u(0)=u_0$. Let's say we put 1 dollar into Bitcoin which is increasing at a rate of $98\\%$ per year. Then calling now $t=0$ and measuring time in years, our model is:\n\n$$u' = 0.98u$$\n\nand $u(0) = 1.0$. We encode this into Julia by noticing that, in this setup, we match the general form when" ], "metadata": {} }, @@ -51,7 +51,7 @@ { "cell_type": "markdown", "source": [ - "and that's it: we have succesfully solved our first ODE!\n\n#### Analyzing the Solution\n\nOf course, the solution type is not interesting in and of itself. We want to understand the solution! The documentation page which explains in detail the functions for analyzing the solution is the [Solution Handling](https://docs.juliadiffeq.org/dev/basics/solution/) page. Here we will describe some of the basics. You can plot the solution using the plot recipe provided by [Plots.jl](http://docs.juliaplots.org/dev/):" + "and that's it: we have succesfully solved our first ODE!\n\n#### Analyzing the Solution\n\nOf course, the solution type is not interesting in and of itself. We want to understand the solution! The documentation page which explains in detail the functions for analyzing the solution is the [Solution Handling](https://docs.sciml.ai/dev/basics/solution/) page. Here we will describe some of the basics. You can plot the solution using the plot recipe provided by [Plots.jl](http://docs.juliaplots.org/dev/):" ], "metadata": {} }, @@ -179,7 +179,7 @@ { "cell_type": "markdown", "source": [ - "#### Controlling the Solver\n\nDifferentialEquations.jl has a common set of solver controls among its algorithms which can be found [at the Common Solver Options](https://docs.juliadiffeq.org/dev/basics/common_solver_opts/) page. We will detail some of the most widely used options.\n\nThe most useful options are the tolerances `abstol` and `reltol`. These tell the internal adaptive time stepping engine how precise of a solution you want. Generally, `reltol` is the relative accuracy while `abstol` is the accuracy when `u` is near zero. These tolerances are local tolerances and thus are not global guarantees. However, a good rule of thumb is that the total solution accuracy is 1-2 digits less than the relative tolerances. Thus for the defaults `abstol=1e-6` and `reltol=1e-3`, you can expect a global accuracy of about 1-2 digits. If we want to get around 6 digits of accuracy, we can use the commands:" + "#### Controlling the Solver\n\nDifferentialEquations.jl has a common set of solver controls among its algorithms which can be found [at the Common Solver Options](https://docs.sciml.ai/dev/basics/common_solver_opts/) page. We will detail some of the most widely used options.\n\nThe most useful options are the tolerances `abstol` and `reltol`. These tell the internal adaptive time stepping engine how precise of a solution you want. Generally, `reltol` is the relative accuracy while `abstol` is the accuracy when `u` is near zero. These tolerances are local tolerances and thus are not global guarantees. However, a good rule of thumb is that the total solution accuracy is 1-2 digits less than the relative tolerances. Thus for the defaults `abstol=1e-6` and `reltol=1e-3`, you can expect a global accuracy of about 1-2 digits. If we want to get around 6 digits of accuracy, we can use the commands:" ], "metadata": {} }, @@ -291,7 +291,7 @@ { "cell_type": "markdown", "source": [ - "Note that similarly on the other side there is `save_end=false`.\n\nMore advanced saving behaviors, such as saving functionals of the solution, are handled via the `SavingCallback` in the [Callback Library](https://docs.juliadiffeq.org/dev/features/callback_library/#saving_callback-1) which will be addressed later in the tutorial.\n\n#### Choosing Solver Algorithms\n\nThere is no best algorithm for numerically solving a differential equation. When you call `solve(prob)`, DifferentialEquations.jl makes a guess at a good algorithm for your problem, given the properties that you ask for (the tolerances, the saving information, etc.). However, in many cases you may want more direct control. A later notebook will help introduce the various *algorithms* in DifferentialEquations.jl, but for now let's introduce the *syntax*.\n\nThe most crucial determining factor in choosing a numerical method is the stiffness of the model. Stiffness is roughly characterized by a Jacobian `f` with large eigenvalues. That's quite mathematical, and we can think of it more intuitively: if you have big numbers in `f` (like parameters of order `1e5`), then it's probably stiff. Or, as the creator of the MATLAB ODE Suite, Lawrence Shampine, likes to define it, if the standard algorithms are slow, then it's stiff. We will go into more depth about diagnosing stiffness in a later tutorial, but for now note that if you believe your model may be stiff, you can hint this to the algorithm chooser via `alg_hints = [:stiff]`." + "Note that similarly on the other side there is `save_end=false`.\n\nMore advanced saving behaviors, such as saving functionals of the solution, are handled via the `SavingCallback` in the [Callback Library](https://docs.sciml.ai/dev/features/callback_library/#saving_callback-1) which will be addressed later in the tutorial.\n\n#### Choosing Solver Algorithms\n\nThere is no best algorithm for numerically solving a differential equation. When you call `solve(prob)`, DifferentialEquations.jl makes a guess at a good algorithm for your problem, given the properties that you ask for (the tolerances, the saving information, etc.). However, in many cases you may want more direct control. A later notebook will help introduce the various *algorithms* in DifferentialEquations.jl, but for now let's introduce the *syntax*.\n\nThe most crucial determining factor in choosing a numerical method is the stiffness of the model. Stiffness is roughly characterized by a Jacobian `f` with large eigenvalues. That's quite mathematical, and we can think of it more intuitively: if you have big numbers in `f` (like parameters of order `1e5`), then it's probably stiff. Or, as the creator of the MATLAB ODE Suite, Lawrence Shampine, likes to define it, if the standard algorithms are slow, then it's stiff. We will go into more depth about diagnosing stiffness in a later tutorial, but for now note that if you believe your model may be stiff, you can hint this to the algorithm chooser via `alg_hints = [:stiff]`." ], "metadata": {} }, diff --git a/pdf/introduction/01-ode_introduction.pdf b/pdf/introduction/01-ode_introduction.pdf index 60043494722f2931d8bb2315303c426210ff75d9..11080e93b2a4a296d90da6445516732f084483b0 100644 GIT binary patch delta 63842 zcmY&;Q*fS-6K;|QjT_sx?WD17+l`$!c4OPNoi=u3+xlYj{N~KL_@9esX0LZ#DhTU@_phjAYGpRub*jHZfgl`d-z1`nu6GZX=;^OXRO3jJ3JwOtucmj9>C2H% zQ-B6ZfT7@}tXx6?CIIk=_5C@%A~_VQY2kWaTA(__stwJ%)0yG*pIsBxj?Qxy>g6)L zJI>rbZ1O9`bU4K6d-}=tHGJijRt+BjL0%*}3Bj%8C5{0W|K1Zn zvJYVf9SIIFh3t3-2K0x@JZt2NzaT{9tD@{VH@?i*$%~oO01izlzyU|G#CO1)6YN2% zSkwRu4Vaq+zNjf1R0`)S!BUg7TXc%*Af3o)`z@taWT?rQ)5M1-KgJ}s5iK63joLxe z{SGZt>jOFVgNogZ;lh6VD>^UQ2;FCaAUwLw$M%UF!!k!|N(L&e6njos zTNZCN&!1nwAlo%PKUN304^P6W64?WMX7=((rNaGmKZT$;o^Gq-PsM-EN$RI|a|C1G 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