Derivative:

A flexible tool for the computation of derivatives of order 1 through 4 on any scalar function. 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 of the options, changing the order of the method or the extrapolation, even allowing the user to specify whether central, forward or backward differences are used.

Gradient

Computes the gradient vector of a scalar function of one or more variables at any location.

Jacobian

Computes the Jacobian matrix of a vector (or array) valued function of one or more variables.

Hessian

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

Hessdiag

The diagonal elements of the Hessian matrix are the pure second order partial derivatives.

Examples

Compute 1'st and 2'nd derivative of exp(x), at x == 1:

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

Nonlinear least squares:

>>> 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 = Jacobian(fun)
>>> Jfun([1,2,0.75]) # should be numerically zero
array([[  0.00000000e+00,   0.00000000e+00,   0.00000000e+00],
       [  0.00000000e+00,   0.00000000e+00,  -1.30229526e-17],
       [  0.00000000e+00,  -2.12916532e-17,   6.35877095e-17],
       [  0.00000000e+00,   0.00000000e+00,   6.95367972e-19],
       [  0.00000000e+00,   0.00000000e+00,  -2.13524915e-17],
       [  0.00000000e+00,  -3.08563327e-16,   7.43577440e-16],
       [  0.00000000e+00,   1.16128292e-15,   1.71041646e-15],
       [  0.00000000e+00,   0.00000000e+00,  -5.51592310e-16],
       [  0.00000000e+00,  -4.51138245e-19,   1.90866225e-15],
       [ -2.40861944e-19,  -1.82530534e-15,  -4.02819694e-15]])

Compute gradient of sum(x**2):

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

See also

scipy.misc.derivative