Derivative transformation

Description

Calculates the derivative of the multidimensional function versus parameter. Uses finite differences method of 4-th order.

Arguments

1. x — parameter instance.

2. reldelta — finite difference step \(h\).

Inputs

1. derivative.y — array of size \(N\).

Outputs

1. derivative.dy — array of size \(N\).

Implementation

The second order finite difference reads as follows:

\[D_2(h) = \frac{f(x+h) - f(x-h)}{2h}.\]

The fourth order reads as follows:

\[\begin{split}\frac{dy}{dx} = D_4(h) &= \frac{1}{3} \left(4D_2\left(\frac{h}{2}\right) - D_2(h)\right) = \\ &= \frac{4}{3h} \left(f\left(x+\frac{h}{2}\right) - f\left(x-\frac{h}{2}\right)\right) - \frac{1}{6h} \left(f\left(x+h\right) - f\left(x-h\right)\right).\end{split}\]

for more information see https://en.wikipedia.org/wiki/Finite_difference_coefficient.