The AAAtrig algorithm for rational approximation of periodic functions

TL;DR

The paper introduces AAAtrig, an extension of the AAA algorithm, for efficient rational approximation of periodic functions, with applications in solving Laplace's equation and compressing conformal maps, implemented in Chebfun.

Contribution

It develops AAAtrig, a robust and flexible algorithm for periodic function approximation using barycentric trigonometric forms, extending AAA's capabilities.

Findings

Effective approximation of periodic functions demonstrated

Reproduces tapered exponential clustering effect

Applicable to solving Laplace's equation and conformal map compression

Abstract

We present an extension of the AAA (adaptive Antoulas--Anderson) algorithm for periodic functions, called 'AAAtrig'. The algorithm uses the key steps of AAA approximation by (i) representing the approximant in (trigonometric) barycentric form and (ii) selecting the support points greedily. Accordingly, AAAtrig inherits all the favourable characteristics of AAA and is thus extremely flexible and robust, being able to consider quite general sets of sample points in the complex plane. We consider a range of applications with particular emphasis on solving Laplace's equation in periodic domains and compressing periodic conformal maps. These results reproduce the tapered exponential clustering effect observed in other recent studies. The algorithm is implemented in Chebfun.