RationalFunctionApproximation.jl: Rational Approximation On Discrete and Continuous Domains

TL;DR

This paper introduces RationalFunctionApproximation.jl, a Julia package that provides fast, arbitrary-precision rational approximation methods for functions on various domains, enabling highly accurate representations especially for functions with poles or branch cuts.

Contribution

It offers the fastest implementations of AAA and greedy Thiele algorithms, including arbitrary-precision capabilities, and integrates with ComplexRegions for versatile domain approximations.

Findings

Achieves root-exponential convergence for complex functions.

Provides the fastest known implementations of key rational approximation algorithms.

Enables accurate function representations over diverse complex domains.

Abstract

Unlike polynomials, rational functions can represent functions having poles or branch cuts with root-exponential convergence and no Runge phenomenon. Recent developments of the AAA and greedy Thiele algorithms have sparked renewed interest in computational rational approximation. The \textsf{RationalFunctionApproximation} package supplies the fastest known implementations of these methods and the only arbitrary-precision ones. Combined with the \textsf{ComplexRegions} package, it can produce compact and accurate representations of a huge variety of functions over intervals, circles, or other domains in the complex plane.