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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

The structure of matrices in rational Gauss quadrature


Authors: Carl Jagels and Lothar Reichel
Journal: Math. Comp. 82 (2013), 2035-2060
MSC (2010): Primary 65F25, 65F60, 65D32
Published electronically: April 9, 2013
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Abstract: This paper is concerned with the approximation of matrix functionals defined by a large, sparse or structured, symmetric definite matrix. These functionals are Stieltjes integrals with a measure supported on a compact real interval. Rational Gauss quadrature rules that are designed to exactly integrate Laurent polynomials with a fixed pole in the vicinity of the support of the measure may yield better approximations of these functionals than standard Gauss quadrature rules with the same number of nodes. Therefore it can be attractive to approximate matrix functionals by these rational Gauss rules. We describe the structure of the matrices associated with these quadrature rules, derive remainder terms, and discuss computational aspects. Also discussed are rational Gauss-Radau rules and the applicability of pairs of rational Gauss and Gauss-Radau rules to computing lower and upper bounds for certain matrix functionals.


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Additional Information

Carl Jagels
Affiliation: Department of Mathematics and Computer Science, Hanover College, Hanover, Indiana 47243
Email: jagels@hanover.edu

Lothar Reichel
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
Email: reichel@math.kent.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2013-02695-6
PII: S 0025-5718(2013)02695-6
Keywords: Extended Krylov subspace, orthogonal Laurent polynomial, recursion relation, matrix functional, rational Gauss quadrature
Received by editor(s): August 15, 2011
Received by editor(s) in revised form: February 27, 2012
Published electronically: April 9, 2013
Additional Notes: This research was supported in part by a grant from the Hanover College Faculty Development Committee
This research was supported in part by NSF grant DMS-1115385
Article copyright: © Copyright 2013 American Mathematical Society