PolyRat: Polynomial and Rational Function Library

PolyRat is a library for polynomial and rational approximation. Formally we can think of polynomials as a sum of powers of :

A rational function is a ratio of two polynomial functions

The goal of this library is to construct polynomial and rational approximations given a collection of point data consisting of pairs of inputs and outputs that minimizes (for example)

The ultimate goal of this library is to provide algorithms to construct these approximations in a variety of norms with a variety of constraints on the approximants.

The polynomial approximation problem is relatively straightfoward as it is a convex problem for any p-norm with p≥1. However, there is still a need to be careful in the construction of the polynomial basis for high-degree polynomials to avoid ill-conditioning. Here we provide access to a number of polynomial bases:

• tensor-product polynomials based on Numpy (e.g., Monomial, Legendre, etc.);
• Vandermonde with Arnoldi polynomial basis;
• barycentric Lagrange polynomial bases.

The rational approximation problem is still an open research problem. This library provides a variety of algorithms for constructing rational approximations including:

• Adaptive Anderson Antoulas
• Sanathanan Koerner iteration
• Stabilized Sanathanan Koerner iteration
• Vector Fitting

Installation

> pip install --upgrade polyrat

Documentation

Full documentation is hosted on Read the Docs.

Usage

Using PolyRat follows the general pattern of scikit-learn. For example, to construct a rational approximation of the tangent function

import numpy as np
import polyrat

x = np.linspace(-1,1, 1000).reshape(-1,1)  # Input data 🚨 must be 2-dimensional
y = np.tan(2*np.pi*x.flatten())            # Output data

num_degree, denom_degree = 10, 10          # numerator and denominator degrees 
rat = polyrat.StabilizedSKRationalApproximation(num_degree, denom_degree)
rat.fit(x, y)

After constructing this approximation, we can then evaluate the resulting approximation by calling the class-instance

y_approx = rat(x)		# Evaluate the rational approximation on X

Comparing this to training data, we note that this degree-(10,10) approximation is highly accurate

Reproducibility

This repository contains the code to reproduce the figures in the associated papers

• Multivariate Rational Approximation Using a Stabilized Sanathanan-Koerner Iteration in Reproducibility/Stabilized_SK

Related Projects

• baryrat: Pure python implementation of the AAA algorithm
• Block-AAA: Matlab implementation of a matrix-valued AAA variant
• RationalApproximations: Julia implementation AAA variants
• RatRemez Rational Remez algorithm (Silviu-Ioan Filip)
• BarycentricDC Barycentric Differential Correction (see SISC paper)