Numdifftools
Numdifftools is a 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 Richardson 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 complex-step, central, forward or backward differences are used.
The methods provided are:
Derivative: Compute the derivatives of order 1 through 10 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.
Numdifftools also provide an easy to use interface to derivatives calculated with in _AlgoPy. Algopy stands for Algorithmic Differentiation in Python. The purpose of AlgoPy is the evaluation of higher-order derivatives in the forward and reverse mode of Algorithmic Differentiation (AD) of functions that are implemented as Python programs.
Getting Started
Visualize high order derivatives of the tanh function
>>> import numpy as np >>> import numdifftools as nd >>> import matplotlib.pyplot as plt >>> x = np.linspace(-2, 2, 100) >>> for i in range(10): ... df = Derivative(np.tanh, n=i) ... y = df(x) ... plt.plot(x, y/np.abs(y).max()) >>> plt.show()
Compute 1’st and 2’nd derivative of exp(x), at x == 1:
>>> fd = nd.Derivative(np.exp) # 1'st derivative >>> fdd = nd.Derivative(np.exp, n=2) # 2'nd derivative >>> np.allclose(fd(1), 2.7182818284590424) True >>> np.allclose(fdd(1), 2.7182818284590424) True
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 = nd.Jacobian(fun) >>> np.allclose(np.abs(Jfun([1,2,0.75])), 0) # should be numerically zero True
Compute gradient of sum(x**2):
>>> fun = lambda x: np.sum(x**2) >>> dfun = nd.Gradient(fun) >>> dfun([1,2,3]) array([ 2., 4., 6.])
Compute the same with the easy to use interface to AlgoPy:
>>> import numdifftools.nd_algopy as nda >>> import numpy as np >>> fd = nda.Derivative(np.exp) # 1'st derivative >>> fdd = nda.Derivative(np.exp, n=2) # 2'nd derivative >>> np.allclose(fd(1), 2.7182818284590424) True >>> np.allclose(fdd(1), 2.7182818284590424) True
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 = nda.Jacobian(fun, method='reverse') >>> np.allclose(np.abs(Jfun([1,2,0.75])), 0) # should be numerically zero True
Compute gradient of sum(x**2):
>>> fun = lambda x: np.sum(x**2) >>> dfun = nda.Gradient(fun) >>> dfun([1,2,3]) array([ 2., 4., 6.])
See also
scipy.misc.derivative
Documentation and code
Numdifftools works on Python 2.7+ and Python 3.0+.
Official releases available at: http://pypi.python.org/pypi/numdifftools
Official documentation available at: http://numdifftools.readthedocs.org/
Bleeding edge: https://github.com/pbrod/numdifftools.
Installation and upgrade:
with pip
$ pip install numdifftools
with conda
$ conda install -c https://conda.anaconda.org/pbrod numdifftools
with easy_install
$ easy_install numdifftools
or
$ easy_install upgrade numdifftools
to upgrade to the newest version
Unit tests
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)
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Release files for numdifftools 0.9.14
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