# Introduction to 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, multicomplex, central, forward or backward differences are used. The methods provided are:

Derivative:
  Computates the derivative of order 1 through 10 on any scalar function.
Gradient:
  Computes the gradient vector of a scalar function of one or more variables.
Jacobian:
  Computes the Jacobian matrix of a vector 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:
  Computes 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 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.

Documentation is at: http://numdifftools.readthedocs.org/

Code and issue tracker is at https://github.com/pbrod/numdifftools.

Latest stable release is at http://pypi.python.org/pypi/Numdifftools.

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)

## Getting Started

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, 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