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

 

Gaussian quadrature for sums: A rapidly convergent summation scheme


Author: H. Monien
Journal: Math. Comp. 79 (2010), 857-869
MSC (2000): Primary 40A25; Secondary 33C90, 65D32, 33F05
Published electronically: July 21, 2009
MathSciNet review: 2600547
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Abstract: Gaussian quadrature is a well-known technique for numerical integration. Recently Gaussian quadrature with respect to discrete measures corresponding to finite sums has found some new interest. In this paper we apply these ideas to infinite sums in general and give an explicit construction for the weights and abscissae of Gaussian formulas. The abscissae of the Gaussian summation have a very interesting asymptotic distribution function with a kink singularity. We apply the Gaussian summation technique to two problems which have been discussed in the literature. We find that the Gaussian summation has a very rapid convergence rate for the Hardy-Littlewood sum for a large range of parameters.


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Additional Information

H. Monien
Affiliation: Bethe Center for Theoretical Physics, Universität Bonn, Nussallee 12, 53115 Bonn, Germany
Email: monien@th.physik.uni-bonn.de

DOI: http://dx.doi.org/10.1090/S0025-5718-09-02289-3
PII: S 0025-5718(09)02289-3
Received by editor(s): December 19, 2006
Received by editor(s) in revised form: October 16, 2008, and April 3, 2009
Published electronically: July 21, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.