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Mathematics of Computation

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On generalized averaged Gaussian formulas

Author: Miodrag M. Spalević
Journal: Math. Comp. 76 (2007), 1483-1492
MSC (2000): Primary 65D30, 65D32; Secondary 33A65.
Published electronically: March 8, 2007
Erratum: Math. Comp. 47 (1986), 767.
MathSciNet review: 2299784
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Abstract: We present a simple numerical method for constructing the optimal (generalized) averaged Gaussian quadrature formulas which are the optimal stratified extensions of Gauss quadrature formulas. These extensions exist in many cases in which real positive Kronrod formulas do not exist. For the Jacobi weight functions $w(x)\equiv w^{(\alpha ,\beta )}(x)=(1-x)^\alpha (1+x)^\beta$ ($\alpha ,\beta >-1$) we give a necessary and sufficient condition on the parameters $\alpha$ and $\beta$ such that the optimal averaged Gaussian quadrature formulas are internal.

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Additional Information

Miodrag M. Spalević
Affiliation: Department of Mathematics and Informatics, University of Kragujevac, Faculty of Science, P.O. Box 60, 34000 Kragujevac, Serbia
MR Author ID: 600543

Keywords: Averaged and anti-Gaussian quadrature formula, optimal stratified extension, three-term recurrence relation, positive quadrature formula, Gauss, Jacobi matrix, Kronrod
Received by editor(s): August 9, 2005
Received by editor(s) in revised form: May 4, 2006
Published electronically: March 8, 2007
Additional Notes: The author was supported in part by the Serbian Ministry of Science and Environmental Protection (Project #144005A: “Approximation of linear operators”).
Article copyright: © Copyright 2007 American Mathematical Society