Algorithms for Modifying Recurrence Relations of Orthogonal Polynomial and Rational Functions when Changing the Discrete Inner Product

TL;DR

This paper develops efficient algorithms for updating recurrence relations of orthogonal polynomials and rational functions when changing the discrete inner product, including both adding and removing nodes, with validation through numerical experiments.

Contribution

It introduces methods to efficiently downdate recurrence relations for orthogonal functions when nodes are removed, complementing previous work on adding nodes, based on eigenvalue deflation techniques.

Findings

Algorithms successfully update recurrence relations after node removal.

Numerical experiments confirm the efficiency and accuracy of the proposed methods.

The approach applies to both polynomial and rational function cases.

Abstract

Often, polynomials or rational functions, orthogonal for a particular inner product are desired. In practical numerical algorithms these polynomials are not constructed, but instead the associated recurrence relations are computed. Moreover, also typically the inner product is changed to a discrete inner product, which is the finite sum of weighted functions evaluated in specific nodes. For particular applications it is beneficial to have an efficient procedure to update the recurrence relations when adding or removing nodes from the inner product. The construction of the recurrence relations is equivalent to computing a structured matrix (polynomial) or pencil (rational) having prescribed spectral properties. Hence the solution of this problem is often referred to as solving an Inverse Eigenvalue Problem. In Van Buggenhout et al. (2022) we proposed updating techniques to add nodes to the inner product while efficiently updating the recurrences. To complete this study we present in this article manners to efficiently downdate the recurrences when removing nodes from the inner product. The link between removing nodes and the QR algorithm to deflate eigenvalues is exploited to develop efficient algorithms. We will base ourselves on the perfect shift strategy and develop algorithms, both for the polynomial case and the rational function setting. Numerical experiments validate our approach.