differint

differint

This package is used for numerically calculating fractional derivatives and integrals (differintegrals). Options for varying definitions of the differintegral are available, including the Grunwald-Letnikov (GL), the ‘improved’ Grunwald-Letnikov (GLI), the Riemann-Liouville (RL), and the Caputo (coming soon!). Through the API, you can compute differintegrals at a point or over an array of function values.

Motivation

There is little in the way of readily available, easy-to-use code for numerical fractional calculus. What is currently available are functions that are generally either smart parts of a much larger package, or only offer one numerical algorithm. The differint package offers a variety of algorithms for computing differintegrals and several auxiliary functions relating to generalized binomial coefficients.

Installation

This project requires Python 3+ and NumPy to run.

Installation from the Python Packaging index (https://pypi.python.org/pypi) is simple using pip.

.. code:: python

pip install differint

Example Usage

Taking a fractional derivative is easy with the differint package. Let’s take the 1/2 derivative of the square root function on the interval [0,1], using the Riemann-Liouville definition of the fractional derivative.

.. code:: python

import numpy as np
import differint.differint as df

def f(x):
   return x**0.5

DF = df.RL(0.5, f)
print(DF)

You can also specify the endpoints of the domain and the number of points used as follows.

.. code:: python

DF = df.RL(0.5, f, 0, 1, 128)

Tests

All tests can be run with nose from the command line. Setup will automatically install nose if it is not present on your machine.

.. code:: python

python setup.py tests

Alternatively, you can run the test script directly.

.. code:: python

cd <file_path>/differint/tests/
python test.py

API Reference

In this section we cover the usage of the various functions within the differint package.

+---------------------------------------------------+-------------------+ | Main Function | Usage | +===================================================+===================+ | GLpoint | Computes the GL | | | differintegral at | | | a point | +---------------------------------------------------+-------------------+ | GL | Computes the GL | | | differintegral | | | over an entire | | | array of function | | | values using the | | | Fast Fourier | | | Transform | +---------------------------------------------------+-------------------+ | GLI | Computes the | | | improved GL | | | differintegral | | | over an entire | | | array of function | | | values | +---------------------------------------------------+-------------------+ | RLpoint | Computes the RL | | | differintegral at | | | a point | +---------------------------------------------------+-------------------+ | RL | Computes the RL | | | differintegral | | | over an entire | | | array of function | | | values using | | | matrix methods | +---------------------------------------------------+-------------------+

+-------------------------------------------------------+--------------+ | Auxiliary Function | Usage | +=======================================================+==============+ | isInteger | Determine if | | | a number is | | | an integer | +-------------------------------------------------------+--------------+ | checkValues | Used to | | | check for | | | valid | | | algorithm | | | input types | +-------------------------------------------------------+--------------+ | GLIinterpolat | Define | | | interpolatin | | | g | | | coefficients | | | for the | | | improved GL | | | algorithm | +-------------------------------------------------------+--------------+ | functionCheck | Determines | | | if algorithm | | | function | | | input is | | | callable or | | | an array of | | | numbers | +-------------------------------------------------------+--------------+ | test_func | Testing | | | function for | | | docstring | | | examples | +-------------------------------------------------------+--------------+ | poch | Computes the | | | Pochhammer | | | symbol | +-------------------------------------------------------+--------------+ | GLcoeffs | Determines | | | the | | | convolution | | | filter | | | composed of | | | generalized | | | binomial | | | coefficients | | | used in the | | | GL algorithm | +-------------------------------------------------------+--------------+ | RLcoeffs | Calculates | | | the | | | coefficients | | | used in the | | | RLpoint and | | | RL | | | algorithms | +-------------------------------------------------------+--------------+ | RLmatrix | Determines | | | the matrix | | | used in the | | | RL algorithm | +-------------------------------------------------------+--------------+

Contribute

To contribute to this project, see the contributing guidelines_.

Credits

Baleanu, D., Diethelm, K., Scalas, E., & Trujillo, J.J. (2012). Fractional Calculus: Models and Numerical Methods. World Scientific.

Oldham, K.B. & Spanier, J. (1974). The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press Inc.

License

MIT © Matthew Adams_

.. _contributing guidelines: https://github.com/snimpids/differint/blob/master/CONTRIBUTING.md .. _Matthew Adams: 2018