derivkit.derivatives.fornberg module#

Implementation of Fornberg’s algorithm for numerical derivatives.

The algorithm was publised by Fornberg in: Bengt Fornberg, Calculation of Weights in Finite Difference Formulas, SIAM Review, vol. 40, No. 3, pp. 685–691, September 1998

Typical usage example:

>>> import numpy as np
>>> from derivkit.derivatives.fornberg import FornbergDerivative
>>> x0 = np.pi/4
>>> grid = x0 + np.array([-0.3, -0.25, -0.1, 0, 0.12])
>>> fornberg = FornbergDerivative(lambda x: np.tan(x), x0)
>>> bool(np.isclose(
...     fornberg.differentiate(grid=grid, order=1),
...     2.0022106298738143,
...     rtol=1e-14,
...     atol=0.0,
... ))
True

class derivkit.derivatives.fornberg.FornbergDerivative(function: Callable, x0: float64)#

Bases: object

Supplies the Fornberg derivative.

function#: the function to be differentiated.

x0#: the evaluation point for the derivative.

Initialises the class.

Parameters:

function – the function to be differentiated.
x0 – the evaluation point for the derivative.

differentiate(*, grid: array, order: int = 1) → float64#

Constructs the derivative of a given order of a function at a point.

Currently only the derivative of order order is returned. An array of orders of 0 to order can be constructed through get_weights. See section 3 of (Fornberg 1998) for more details.

Parameters:

grid – a series of points around the evaluation point.
order – the order of the derivative.

Returns:

The derivative evaluated at the given point.

get_weights(grid: array, order: int = 1) → ndarray#

Constructs the weights needed for derivatives up to a given order.

Parameters:

grid – a series of points around the evaluation point.
order – the order of the derivative.

Returns:

A 2D array of numbers representing the derivative weights. The 0th axis corresponds to the order of the derivative, with the 0th row corresponding with the function itself.