A MATLAB package computing simultaneous Gaussian quadrature rules for Multiple Orthogonal Polynomials

TL;DR

This paper introduces a MATLAB package for reliably computing simultaneous Gaussian quadrature rules for multiple orthogonal polynomials, overcoming ill-conditioning issues with a novel balancing procedure.

Contribution

The paper presents a new MATLAB package that efficiently computes simultaneous Gaussian quadrature rules using a novel balancing method to address ill-conditioning.

Findings

The package reliably computes quadrature rules in floating point arithmetic.

A new balancing procedure significantly improves eigenvalue problem conditioning.

The method is applicable to multiple orthogonal polynomials with two measures.

Abstract

The aim of this paper is to describe a Matlab package for computing the simultaneous Gaussian quadrature rules associated with a variety of multiple orthogonal polynomials. Multiple orthogonal polynomials can be considered as a generalization of classical orthogonal polynomials, satisfying orthogonality constraints with respect to $r$ different measures, with $r \ge 1$. Moreover, they satisfy $(r+2)$--term recurrence relations. In this manuscript, without loss of generality, $r$ is considered equal to $2$. The so-called simultaneous Gaussian quadrature rules associated with multiple orthogonal polynomials can be computed by solving a banded lower Hessenberg eigenvalue problem. Unfortunately, computing the eigendecomposition of such a matrix turns out to be strongly ill-conditioned and the \texttt{Matlab} function \texttt{balance.m} does not improve the condition of the eigenvalue problem. Therefore, most procedures for computing simultaneous Gaussian quadrature rules are implemented with variable precision arithmetic. Here, we propose a \texttt{Matlab} package that allows to reliably compute the simultaneous Gaussian quadrature rules in floating point arithmetic. It makes use of a variant of a new balancing procedure, recently developed by the authors of the present manuscript, that drastically reduces the condition of the Hessenberg eigenvalue problem.