The first five years of the AAA algorithm

TL;DR

This paper reviews the five-year development and impact of the AAA algorithm, highlighting its speed, robustness, and diverse applications in rational approximation and PDE solutions.

Contribution

It provides a comprehensive overview of the AAA algorithm's evolution, extensions, and significance in computational mathematics over five years.

Findings

AAA algorithm is faster and more robust than previous methods.

It has been successfully applied to solve PDEs in irregular domains.

The algorithm has been extended and remains influential in the field.

Abstract

The AAA algorithm, introduced in 2018, computes best or near-best rational approximations to functions or data on subsets of the real line or the complex plane. It is much faster and more robust than previous algorithms for such problems and has been used in many applications since its appearance, including the numerical solution of Laplace, Poisson, and biharmonic PDE problems in irregular domains. AAA has also been extended in new directions and seems likely to be a tool of lasting importance in the future.