Krylov and core transformation algorithms for an inverse eigenvalue problem to compute recurrences of multiple orthogonal polynomials

TL;DR

This paper introduces algorithms based on Krylov and core transformation techniques to solve an inverse eigenvalue problem for computing recurrence coefficients of multiple orthogonal polynomials, with analysis of their accuracy and stability.

Contribution

The paper presents novel algorithms linking inverse eigenvalue problems with multiple orthogonal polynomials, utilizing Krylov subspaces and Gaussian eliminations, and analyzes their numerical stability.

Findings

Algorithms effectively compute recurrence coefficients for multiple orthogonal polynomials.

Numerical experiments demonstrate the stability and accuracy of the proposed methods.

Applications include Kravchuk and Hahn polynomials with varying conditioning.

Abstract

In this paper, we develop algorithms for computing the recurrence coefficients corresponding to multiple orthogonal polynomials on the step-line. We reformulate the problem as an inverse eigenvalue problem, which can be solved using numerical linear algebra techniques. We consider two approaches: the first is based on the link with block Krylov subspaces and results in a biorthogonal Lanczos process with multiple starting vectors; the second consists of applying a sequence of Gaussian eliminations on a diagonal matrix to construct the banded Hessenberg matrix containing the recurrence coefficients. We analyze the accuracy and stability of the algorithms with numerical experiments on the ill-conditioned inverse eigenvalue problemshave related to Kravchuk and Hahn polynomials, as well as on other better conditioned examples.