# numdifftools Release 0.9.41

Solves automatic numerical differentiation problems in one or more variables.

BSD-3-Clause
Install
``` pip install numdifftools==0.9.41 ```

# numdifftools

The numdifftools library 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.
• directionaldiff: Compute directional derivative of a function of n variables
• 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 = nd.Derivative(np.tanh, n=i) ... y = df(x) ... h = plt.plot(x, y/np.abs(y).max())

>>> plt.show() # doctest: +SKIP

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``````

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

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``````

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

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

Bleeding edge: https://github.com/pbrod/numdifftools.

## Installation

If you have pip installed, then simply type:

\$ pip install numdifftools

to get the lastest stable version. Using pip also has the advantage that all requirements are automatically installed.

## Unit tests

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

``````import numdifftools as nd
nd.test('--doctest-modules', '--disable-warnings')``````

## Acknowledgement

The numdifftools package for Python was written by Per A. Brodtkorb based on the adaptive numerical differentiation toolbox written in Matlab by John D'Errico [DErrico06].

Later the package was extended with some of the functionality found in the statsmodels.tools.numdiff module written by Josef Perktold [JPerktold14] which is based on [Rid09]. The implementation of bicomplex numbers is based on the matlab implementation described in the project report of [Ver14] which is based on [GLD12]. For completeness the [For98] method for computing the weights and points in general finite difference formulas as well as the [For81] method for cumputing the taylor coefficients of complex analytic function using FFT, was added.

## References

DErrico06

For81

Fornberg, B. (1981). "Numerical Differentiation of Analytic Functions", ACM Transactions on Mathematical Software (TOMS), 7(4), 512-526. http://doi.org/10.1145/355972.355979

For98

B. Fornberg (1998) "Calculation of weights_and_points in finite difference formulas", SIAM Review 40, pp. 685-691.

GLD12

Gregory Lantoine, R.P. Russell, and T. Dargent (2012) "Using multicomplex variables for automatic computation of high-order derivatives", ACM Transactions on Mathematical Software, Vol. 38, No. 3, Article 16, April 2012, 21 pages, http://doi.acm.org/10.1145/2168773.2168774

JML66

Lyness, J. M., Moler, C. B. (1966). "Vandermonde Systems and Numerical Differentiation", Numerische Mathematik.

JML69

Lyness, J. M., Moler, C. B. (1969). "Generalized Romberg Methods for Integrals of Derivatives", Numerische Mathematik.

JPerktold14

Perktold, J (2014), numdiff package http://statsmodels.sourceforge.net/0.6.0/_modules/statsmodels/tools/numdiff.html

KLLK05

K.-L. Lai, J.L. Crassidis, Y. Cheng, J. Kim (2005), "New complex step derivative approximations with application to second-order kalman filtering", AIAA Guidance, Navigation and Control Conference, San Francisco, California, August 2005, AIAA-2005-5944.

Lan10

Gregory Lantoine (2010), "A methodology for robust optimization of low-thrust trajectories in multi-body environments", Phd thesis, Georgia Institute of Technology

MELEV12

M.E. Luna-Elizarraras, M. Shapiro, D.C. Struppa1, A. Vajiac (2012), "Bicomplex Numbers and Their Elementary Functions", CUBO A Mathematical Journal, Vol. 14, No 2, (61-80). June 2012.

NAG

NAG Library. NAG Fortran Library Document: D04AAF

Rid09

Ridout, M.S. (2009) "Statistical applications of the complex-step method of numerical differentiation", The American Statistician, 63, 66-74

Ver14

Adriaen Verheyleweghen, (2014) "Computation of higher-order derivatives using the multi-complex step method", Project report, NTNU