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Mathematics > Numerical Analysis

arXiv:1805.12047 (math)
[Submitted on 30 May 2018]

Title:Generalized eigenvalue methods for Gaussian quadrature rules

Authors:Grigoriy Blekherman, Mario Kummer, Cordian Riener, Markus Schweighofer, Cynthia Vinzant
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Abstract:A quadrature rule of a measure $\mu$ on the real line represents a convex combination of finitely many evaluations at points, called nodes, that agrees with integration against $\mu$ for all polynomials up to some fixed degree. In this paper, we present a bivariate polynomial whose roots parametrize the nodes of minimal quadrature rules for measures on the real line. We give two symmetric determinantal formulas for this polynomial, which translate the problem of finding the nodes to solving a generalized eigenvalue problem.
Comments: 13 pages, 2 figures
Subjects: Numerical Analysis (math.NA); Algebraic Geometry (math.AG)
MSC classes: Primary 65D32, Secondary 14H50, 14P05, 15A22
Cite as: arXiv:1805.12047 [math.NA]
  (or arXiv:1805.12047v1 [math.NA] for this version)
  https://doi.org/10.48550/arXiv.1805.12047
Related DOI: https://doi.org/10.5802/ahl.62

Submission history

From: Cynthia Vinzant [view email]
[v1] Wed, 30 May 2018 15:58:01 UTC (42 KB)
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