A Golub-Welsch version for simultaneous Gaussian quadrature

TL;DR

This paper extends the Golub-Welsch method to compute simultaneous Gaussian quadrature nodes and weights using eigenvalues of a banded Hessenberg matrix, specifically for multiple orthogonal polynomials, with detailed examples for two measures.

Contribution

It introduces a novel extension of the Golub-Welsch approach for simultaneous Gaussian quadrature based on multiple orthogonal polynomials, providing a practical computational method.

Findings

Method computes quadrature nodes and weights via eigenvalues of a banded Hessenberg matrix.

Detailed description and examples for the case of two measures.

The approach generalizes classical Gaussian quadrature to multiple measures.

Abstract

The zeros of type II multiple orthogonal polynomials can be used for quadrature formulas that approximate $r$ integrals of the same function $f$ with respect to $r$ measures $\mu_1,\ldots,\mu_r$ in the spirit of Gaussian quadrature. This was first suggested by Borges in 1994, even though he does not mention multiple orthogonality. We give a method to compute the quadrature nodes and the quadrature weights which extends the Golub-Welsch approach using the eigenvalues and left and right eigenvectors of a banded Hessenberg matrix. This method was already described by Coussement and Van Assche in 2005 but it seems to have gone unnoticed. We describe the result in detail for $r=2$ and give some examples.