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arXiv · 2309.11864

A Golub-Welsch version for simultaneous Gaussian quadrature

Walter Van Assche
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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.

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Walter Van Assche. 2023-09-21. A Golub-Welsch version for simultaneous Gaussian quadrature. https://doi.org/10.1007/s11075-024-01767-2

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