QR-based Parallel Set-Valued Approximation with Rational Functions

Abstract

In this article a fast and parallelizable algorithm for rational approximation is presented. The method, called (P)QR-AAA, is a (parallel) set-valued variant of the AAA algorithm for scalar functions. It builds on the set-valued AAA framework introduced by Lietaert, Meerbergen, P{\'e}rez and Vandereycken, accelerating it by using an approximate orthogonal basis obtained from a truncated QR decomposition. We demonstrate both theoretically and numerically this method's accuracy and efficiency. We show how it can be parallelized while maintaining the desired accuracy, with minimal communication cost.