5.1.6.2. numdifftools.nd_algopy.Gradient

class Gradient(fun, n=1, method='forward', full_output=False)

Calculate Gradient with Algorithmic Differentiation method

Parameters

fun: function

function of one array fun(x, *args, **kwds)

method: string, optional {‘forward’, ‘reverse’}

defines method used in the approximation

Returns

grad: array

gradient

See also

Derivative

Jacobian

Hessdiag

Hessian

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

>>> import numpy as np
>>> import numdifftools.nd_algopy as nda
>>> fun = lambda x: np.sum(x**2)
>>> df = nda.Gradient(fun, method='reverse')
>>> np.allclose(df([1,2,3]), [ 2.,  4.,  6.])
True

#At [x,y] = [1,1], compute the numerical gradient #of the function sin(x-y) + y*exp(x)

>>> sin = np.sin; exp = np.exp
>>> z = lambda xy: sin(xy[0]-xy[1]) + xy[1]*exp(xy[0])
>>> dz = nda.Gradient(z)
>>> grad2 = dz([1, 1])
>>> np.allclose(grad2, [ 3.71828183,  1.71828183])
True

#At the global minimizer (1,1) of the Rosenbrock function, #compute the gradient. It should be essentially zero.

>>> rosen = lambda x : (1-x[0])**2 + 105.*(x[1]-x[0]**2)**2
>>> rd = nda.Gradient(rosen)
>>> grad3 = rd([1,1])
>>> np.allclose(grad3, [ 0.,  0.])
True

Methods

__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.

__init__(fun, n=1, method='forward', full_output=False)

Methods

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

Attributes

fun