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

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.