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arXiv · 1805.12047

Generalized eigenvalue methods for Gaussian quadrature rules

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.

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Grigoriy Blekherman, Mario Kummer, Cordian Riener, Markus Schweighofer, Cynthia Vinzant. 2018-05-30. Generalized eigenvalue methods for Gaussian quadrature rules. https://doi.org/10.5802/ahl.62

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