5.1.6.1. numdifftools.nd_algopy.Derivative

class Derivative(fun, n=1, method='forward', full_output=False)[source]

Calculate n-th derivative with Algorithmic Differentiation method

Parameters:

fun (function) – function of one array fun(x, *args, **kwds)
n (int, optional) – Order of the derivative.
method (string, optional {'forward', 'reverse'}) – defines method used in the approximation

__call__: callable with the following parameters:

x: array_like: value at which function derivative is evaluated
args: tuple: Arguments for function fun.
kwds: dict: Keyword arguments for function fun.

Returns:: der – array of derivatives
Return type:: ndarray

Notes

Algorithmic differentiation is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.

References

Sebastian F. Walter and Lutz Lehmann 2013, “Algorithmic differentiation in Python with AlgoPy”, in Journal of Computational Science, vol 4, no 5, pp 334 - 344, http://www.sciencedirect.com/science/article/pii/S1877750311001013

https://en.wikipedia.org/wiki/Automatic_differentiation

Examples

# 1’st and 2’nd derivative of exp(x), at x == 1

>>> import numpy as np
>>> import numdifftools.nd_algopy as nda
>>> fd = nda.Derivative(np.exp)              # 1'st derivative
>>> np.allclose(fd(1), 2.718281828459045)
True
>>> fd5 = nda.Derivative(np.exp, n=5)         # 5'th derivative
>>> np.allclose(fd5(1), 2.718281828459045)
True

# 1’st derivative of x^3+x^4, at x = [0,1]

>>> fun = lambda x: x**3 + x**4
>>> fd3 = nda.Derivative(fun)
>>> np.allclose(fd3([0,1]), [ 0.,  7.])
True

`__init__`(fun[, n, method, full_output])
`computational_graph`(x, args, *kwds)