Utility Functions

This folder contains mathematical classes and functions.

Derivative tools

Functions for calculating derivatives with different methods:

• spline_1d_partial_derivative.hpp : calculate derivatives using a 1d spline representation.
• spline_2d_partial_derivative.hpp : calculate derivatives using a 2d spline representation.
• central_fdm_partial_derivatives.hpp : calculate derivatives using Finite Difference Method.

Spline derivatives

The spline_builder_2d_cache.hpp file contains a class to be used with a 2d spline representation to compute partial derivatives. The cache class allows for calling the 2d splines builder only once per iteration, even if partial derivatives are evaluated in two directions.

Finite difference derivatives

The method is as follow: Take a function $f$. We want to approximate the value of $f'(x_1)$ knowing the value of $f(x_1)$, $f(x_2)$ and $f(x_3)$ for $x_3>x_2>x_1$. Denote $\alpha :=|x_2-x_1|$ and $\beta :=|x_3-x_2|$. We call $Df(x_1)$ the approximate value of $f'(x_1)$. We want $|f'(x_1)-Df(x_1)|=o\left(\max(\alpha,\beta\right)^2)$. To this end, write

$$Df(x_1)=c_1f(x_1)+c_2f(x_2)+c_3f(x_3),$$

and expand $f$ to the second order.

$$\begin{aligned} Df(x_1) & = c_1f(x_1)+c_2\left(f(x_1)+\alpha f'(x_1)+\frac{1}{2}\alpha^2f''(x_1)+o(\alpha^2)\right) \\\ & +c_3\left(f(x_1)+(\alpha+\beta) f'(x_1)+\frac{1}{2}(\alpha+\beta)^2f''(x_1)+o((\alpha+\beta)^2)\right). \end{aligned}$$

A term-by-term comparison gives us the following system:

$$\left\{ \begin{aligned} c_1+c_2+c_3 & =0 \\\ \alpha c_2+(\alpha+\beta)c_3 & =1 \\\ \alpha^2 c_2+(\alpha+\beta)^2c_3 & =0. \end{aligned} \right.$$

The solution is therefore given by

$$c_1=-\frac{2\alpha+\beta}{\alpha^2+\alpha\beta}\qquad c_2=\frac{1}{\alpha}+\frac{1}{\beta}\qquad c_3=-\frac{\alpha}{\beta(\alpha+\beta)}.$$

The same computation can be made for the backward and the centred FDM scheme, and lead to these three formulas:

$$\begin{aligned} Df(x_1) & = -\frac{2\alpha+\beta}{\alpha(\alpha+\beta)}f(x_1)+\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)f(x_2)-\frac{\alpha}{\beta(\alpha+\beta)}f(x_3) \\\ Df(x_2) & = -\frac{\beta}{\alpha(\alpha+\beta)}f(x_1)+\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)f(x_2)+\frac{\alpha}{\alpha(\alpha+\beta)}f(x_3) \\\ Df(x_3) & = \frac{2\alpha+\beta}{\alpha(\alpha+\beta)}f(x_3)-\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)f(x_2)+\frac{\alpha}{\beta(\alpha+\beta)}f(x_1). \end{aligned}$$

One can check than in the uniform case ($\alpha=\beta$) we recover the well known coefficient $-1/2$ and $1/2$ for the centred case and $-3/2$, $2$ and $-1/2$ for the decentred case.

Utility tools

The l_norm_tools.hpp file contains functions computing the infinity norm. For now, it computes the infinity norm of

• a double: $\Vert x \Vert_{\infty} = x$;
• a coordinate: $\Vert x \Vert_{\infty} = \max_{i} (|x_i|)$.