NUMDIFFTOOLS

Suite of tools written in Python to solve automatic numerical differentiation problems in one or more variables. Finite differences are used in an adaptive manner, coupled with a Romberg extrapolation methodology to provide a maximally accurate result. The user can configure many options like; changing the order of the method or the extrapolation, even allowing the user to specify whether central, forward or backward differences are used. The methods provided are:

Derivative: Compute the derivatives of order 1 through 4 on any scalar function.

Gradient: Compute the gradient vector of a scalar function of one or more variables.

Jacobian: Compute the Jacobian matrix of a vector valued function of one or more variables.

Hessian: Compute the Hessian matrix of all 2nd partial derivatives of a scalar function of one or more variables.

Hessdiag: Compute only the diagonal elements of the Hessian matrix

All of these methods also produce error estimates on the result. The documentation for these tools are given at http://numdifftools.readthedocs.org.

To test if the toolbox is working paste the following in an interactive python session:

import numdifftools as nd
nd.test(coverage=True, doctests=True)

>>> import numpy as np
>>> import numdifftools as nd
>>> fd = nd.Derivative(np.exp)              # 1'st derivative
>>> fdd = nd.Derivative(np.exp, n=2)  # 2'nd derivative
>>> fd(1)
array([ 2.71828183])

>>> xdata = np.reshape(np.arange(0,1,0.1),(-1,1))
>>> ydata = 1+2*np.exp(0.75*xdata)
>>> fun = lambda c: (c[0]+c[1]*np.exp(c[2]*xdata) - ydata)**2
>>> Jfun = nd.Jacobian(fun)
>>> np.abs(Jfun([1,2,0.75])) < 1e-14 # should be numerically zero
array([[ True,  True,  True],
       [ True,  True,  True],
       [ True,  True,  True],
       [ True,  True,  True],
       [ True,  True,  True],
       [ True,  True,  True],
       [ True,  True,  True],
       [ True,  True,  True],
       [ True,  True,  True],
       [ True,  True,  True]], dtype=bool)

>>> fun = lambda x: np.sum(x**2)
>>> dfun = nd.Gradient(fun)
>>> dfun([1,2,3])
array([ 2.,  4.,  6.])