A Golub-Welsch version for simultaneous Gaussian quadrature

09/21/2023

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by   Walter Van Assche, et al.

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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 μ_1,…,μ_r in the spirit of Gaussian quadrature. This was first suggested by Borges in 1994. 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.