Orthogonal Polynomials in Python

Description

The py-orthpol package defines the module orthpol which can be used easily construct univariate and multivariate orthogonal polynomials in Python. The purpose of this code is to serve as a component in Python packages that could use orthogonal polynomials as basis functions for various tasks. For example:

• The polynomials can be used in least squares applications.
• The polynomials can serve as the mean in Gaussian process regression.
• etc.

The need to have an easy to use package that can generate polynomials orthogonal with respect to arbitrary weight functions is motivated by applications in the field of Uncertainty Quantification (UQ). In UQ, collections of such polynomials are known as generalized Polynomial Chaos (gPC). My end goal is to provide a tool that makes it ridiculously easy to construct these polynomials.

Where does this come from?

This package serves as a Python wrapper for the legacy Fortran code ORTHPOL. The original ORTHPOL code can be found here. The code that computes tensors products of univariate orthogonal polynomials is a transolation of Stockos C++ routines to Python.

Installation

Simply clone the repository:

git clone https://github.com/ebilionis/py-orthpol.git

Go inside the directory and run:

python setup.py install

Demos

I provide several demos that demonstrate how polynomials can be constructed both from simple weight functions as well as scipy.stats random variables. It is quite easy based on these examples to generalize to more complicated cases. All one has to do is change the weight function or the random variable. Here is a list of them:

• demos/demo1.py: Hermite polynomials using a weight function.
• demos/demo2.py: Laguerre polynomials using a weight function.
• demos/demo3.py: Chebyshev polynomials using a weight function.
• demos/demo4.py: Jacobi polynomials using a weight function.
• demos/demo5.py: Gegenbauer polynomials using a weight function.
• demos/demo6.py: Legendre polynomials using a weight function.
• demos/demo7.pv: Legendre polynomials using scipy.stats.uniform().
• demos/demo8.pv: Hermite polynomials using scipy.stats.norm().
• demos/demo9.pv: Shifted Hermite polynomials using a non-standard scipy.stats.norm().
• demos/demo10.py: Orthogonal polynomials with respect to a truncated normal.
• demos/demo11.py: 2D orthogonal polynomials using the ProductBasis class and a collection of scipy.stats random variables.

TODO

This is a list of things that need to be done:

• Implement the method orthopol.ProductBasis.d() that calculates the derivative of a product basis of polynomials with respect to x.