The Loewner framework applied to Zolotarev sign and ratio problems

TL;DR

This paper evaluates the Loewner framework for approximating functions related to Zolotarev problems, demonstrating its speed, reliability, and high accuracy compared to AAA-based methods.

Contribution

It provides a comprehensive numerical comparison showing the Loewner framework's advantages over AAA and AAA-Lawson methods for Zolotarev function approximation.

Findings

Loewner framework is faster and more reliable.

High-degree Loewner approximants are often more accurate.

Loewner method's runtime remains constant with increasing degree.

Abstract

In this work, we propose a numerical study concerning the approximation of functions associated with the 3rd and 4th Zolotarev problems. We compare various methods, in particular the Loewner framework, the standard AAA algorithm, and recently-proposed extensions of AAA (namely, the sign and Lawson variants). We show that the Loewner framework is fast and reliable, and provides approximants with a high level of accuracy. When the approximants are of a higher degree, Loewner approximants are often more accurate than near-optimal ones computed with AAA-Lawson. Last but not least, the Loewner framework is a direct method for which the running time is typically lower than that of the iterative AAA-Lawson variants. Moreover, for the latter, the running time increases substantially with the degree of the approximant, whereas for the Loewner method, it remains constant. These claims are supported by an extensive numerical treatment.