Generalized eigenvalue methods for Gaussian quadrature rules

Blekherman, Grigoriy; Kummer, Mario; Riener, Cordian; Schweighofer, Markus; Vinzant, Cynthia

doi:10.5802/ahl.62

Blekherman, Grigoriy ¹; Kummer, Mario ²; Riener, Cordian ³; Schweighofer, Markus ⁴; Vinzant, Cynthia ⁵

¹ School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, (USA)
² Technische Universität Dresden Fakultät Mathematik Institut für Geometrie, Zellescher Weg 12-14 01062 Dresden, (Germany)
³ Department of Mathematics and Statistics, Faculty of Science and Technology, UiT The Arctic University of Norway, 9037 Tromsø, (Norway)
⁴ Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, (Germany)
⁵ Department of Mathematics, North Carolina State University, Box 8205, NC State University Raleigh, NC 27695-8205, (USA)

Annales Henri Lebesgue, Volume 3 (2020), pp. 1327-1341.

Abstract
Résumé

A quadrature rule of a measure $μ$ on the real line represents a conic combination of finitely many evaluations at points, called nodes, that agrees with integration against $μ$ 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.

Une formule de quadrature pour une mesure $μ$ sur la droite réelle est une combinaison conique d’un nombre fini d’évaluations en des points, appelés nœuds, qui concorde avec l’intégration selon $μ$ pour tout polynôme jusqu’à un certain degré fixé. Dans cet article, nous introduisons un polynôme bivarié dont les racines paramètrent les nœuds des formules de quadrature minimales pour une mesure donnée. Nous donnons deux représentations déterminantales symétriques pour ce polynôme, ce qui ramène le problème de recherche des nœuds à la résolution d’un problème aux valeurs propres généralisé.

Received: 2018-06-12
Revised: 2019-10-17
Accepted: 2020-02-19
Published online: 2020-11-05

DOI: 10.5802/ahl.62

Classification: 65D32, 14H50, 14P05, 15A22
Keywords: quadrature, Gaussian quadrature, plane curves

Author's affiliations:

Blekherman, Grigoriy ¹; Kummer, Mario ²; Riener, Cordian ³; Schweighofer, Markus ⁴; Vinzant, Cynthia ⁵

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     author = {Blekherman, Grigoriy and Kummer, Mario and Riener, Cordian and Schweighofer, Markus and Vinzant, Cynthia},
     title = {Generalized eigenvalue methods for {Gaussian} quadrature rules},
     journal = {Annales Henri Lebesgue},
     pages = {1327--1341},
     publisher = {\'ENS Rennes},
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     year = {2020},
     doi = {10.5802/ahl.62},
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     url = {http://archive.numdam.org/articles/10.5802/ahl.62/}
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Blekherman, Grigoriy; Kummer, Mario; Riener, Cordian; Schweighofer, Markus; Vinzant, Cynthia. Generalized eigenvalue methods for Gaussian quadrature rules. Annales Henri Lebesgue, Volume 3 (2020), pp. 1327-1341. doi : 10.5802/ahl.62. http://archive.numdam.org/articles/10.5802/ahl.62/

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