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Solves automatic numerical differentiation problems in one or more variables.

Project description

Suite of tools to solve automatic numerical differentiation problems in one or more variables. All of these methods also produce error estimates on the result. A pdf file is also provided to explain the theory behind these tools.

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

import numdifftools as nd


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.


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


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


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


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


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.])

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