{
  "id": "2309.11864",
  "version": "v1",
  "published": "2023-09-21T08:06:50.000Z",
  "updated": "2023-09-21T08:06:50.000Z",
  "title": "A Golub-Welsch version for simultaneous Gaussian quadrature",
  "authors": [
    "Walter Van Assche"
  ],
  "comment": "19 pages, 4 tables",
  "categories": [
    "math.NA",
    "cs.NA",
    "math.CA"
  ],
  "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. 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-21T08:06:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A55",
      "65D32",
      "15A18",
      "33C45",
      "41A21",
      "42C05"
    ],
    "keywords": [
      "simultaneous gaussian quadrature",
      "golub-welsch version",
      "multiple orthogonal polynomials",
      "van assche",
      "quadrature formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}