The AAA algorithm for rational approximation

TL;DR

The paper presents the AAA algorithm, a simple, efficient, and parameter-free method for rational approximation that outperforms existing techniques on various domains, with broad applicability and competitive results.

Contribution

Introduction of the AAA algorithm, a new adaptive, barycentric-based rational approximation method that is easy to implement and effective across diverse problems.

Findings

Outperforms existing methods on complex domains

Requires no user input parameters

Implemented in 40 lines of Matlab

Abstract

We introduce a new algorithm for approximation by rational functions on a real or complex set of points, implementable in 40 lines of Matlab and requiring no user input parameters. Even on a disk or interval the algorithm may outperform existing methods, and on more complicated domains it is especially competitive. The core ideas are (1) representation of the rational approximant in barycentric form with interpolation at certain support points and (2) greedy selection of the support points to avoid exponential instabilities. The name AAA stands for "adaptive Antoulas--Anderson" in honor of the authors who introduced a scheme based on (1). We present the core algorithm with a Matlab code and nine applications and describe variants targeted at problems of different kinds. Comparisons are made with vector fitting, RKFIT, and other existing methods for rational approximation.