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Generalized eigenvalue methods for Gaussian quadrature rules. (Méthodes de valeurs propres généralisées pour les formules de quadrature de Gauss.) (English. French summary) Zbl 1458.65023
Summary: A quadrature rule of a measure \(\mu\) on the real line represents a conic 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.
MSC:
65D32 Numerical quadrature and cubature formulas
14H50 Plane and space curves
15A22 Matrix pencils
Software:
JDQR; JDQZ
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