  In this article a fast and parallelizable algorithm for rational
approximation is presented. The method, called (P)QR-AAA, is a set valued
variant of the Adaptive Antoulas Anderson (AAA) algorithm. It builds on the
Set-Valued AAA framework from [16], accelerating it by using an approximate
orthogonal basis obtained from a truncated QR decomposition. We demonstrate
both theoretically and numerically this method's robustness. We show how it can
be parallelized while maintaining the desired accuracy, with minimal
communication cost.
