Generation of orthogonal rational functions by procedures for structured matrices

TL;DR

This paper introduces methods for generating orthogonal and biorthogonal rational functions through inverse eigenvalue problems involving structured matrices, with a focus on numerical stability and efficiency.

Contribution

It proposes two procedures for solving inverse eigenvalue problems for structured matrix pencils, extending to biorthogonal functions with more efficient recurrence relations.

Findings

The rational Arnoldi iteration is effective for orthogonal functions.

The updating procedure using unitary similarity transformations is numerically stable.

Biorthogonal functions lead to more efficient recurrence relations.

Abstract

The problem of computing recurrence coefficients of sequences of rational functions orthogonal with respect to a discrete inner product is formulated as an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two procedures are proposed to solve this inverse eigenvalue problem, via the rational Arnoldi iteration and via an updating procedure using unitary similarity transformations. The latter is shown to be numerically stable. This problem and both procedures are generalized by considering biorthogonal rational functions with respect to a bilinear form. This leads to an inverse eigenvalue problem for a pencil of tridiagonal matrices. A tridiagonal pencil implies short recurrence relations for the biorthogonal rational functions, which is more efficient than the orthogonal case. However the procedures solving this problem must rely on nonunitary operations and might not be numerically stable.