Constructing Orthogonal Rational Function Vectors with an application in Rational Approximation

TL;DR

This paper introduces two algorithms for constructing orthonormal rational function vectors using a pencil-based formulation, enabling effective rational approximation of functions like a0b0 with clustered poles.

Contribution

It extends the inverse eigenvalue problem framework to rational vectors of arbitrary length and develops algorithms based on similarity transformations and Krylov methods.

Findings

Successfully approximated a0b0 on [0,1] with optimal convergence

Demonstrated robustness for functions with clustered poles near singularities

Extended pencil-based formulation to arbitrary-length rational vectors

Abstract

We present two algorithms for constructing orthonormal bases of rational function vectors with respect to a discrete inner product, and discuss how to use them for a rational approximation problem. Building on the pencil-based formulation of the inverse generalized eigenvalue problem by Van Buggenhout et al. (2022), we extend it to rational vectors of arbitrary length $k$, where the recurrence relations are represented by a pair of $k$-Hessenberg matrices, i.e., matrices with possibly $k$ nonzero subdiagonals. An updating algorithm based on similarity transformations using rotations and a Krylov-type algorithm related to the rational Arnoldi method are derived. The performance is demonstrated on the rational approximation of $\sqrt{z}$ on $[0,1]$, where the optimal lightning + polynomial convergence rate of Herremans, Huybrechs, and Trefethen (2023) is successfully recovered. This illustrates the robustness of the proposed methods for handling exponentially clustered poles near singularities.