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On computing rational Gauss-Chebyshev quadrature formulas


Authors: Joris Van Deun, Adhemar Bultheel and Pablo González Vera
Journal: Math. Comp. 75 (2006), 307-326
MSC (2000): Primary 42C05, 65D32
DOI: https://doi.org/10.1090/S0025-5718-05-01774-6
Published electronically: October 4, 2005
MathSciNet review: 2176401
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Abstract: We provide an algorithm to compute the nodes and weights for Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary real poles outside $ [-1,1]$. Contrary to existing rational quadrature formulas, the computational effort is very low, even for extremely high degrees, and under certain conditions on the poles it can be shown that the complexity is of order $ O(n)$. This method is based on the derivation of explicit expressions for Chebyshev orthogonal rational functions, which are (thus far) the only examples of explicitly known orthogonal rational functions on $ [-1,1]$ with arbitrary real poles outside this interval.


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Additional Information

Joris Van Deun
Affiliation: Department of Computer Science, K.U.Leuven, B-3001 Heverlee, Belgium
Email: joris.vandeun@cs.kuleuven.ac.be

Adhemar Bultheel
Affiliation: Department of Computer Science, K.U.Leuven, B-3001 Heverlee, Belgium
Email: adhemar.bultheel@cs.kuleuven.ac.be

Pablo González Vera
Affiliation: Depto. Análisis Matemático, Univ. La Laguna, 38206 La Laguna, Tenerife, Canary Islands, Spain
Email: pglez@ull.es

DOI: https://doi.org/10.1090/S0025-5718-05-01774-6
Keywords: quadrature formulas, orthogonal rational functions
Received by editor(s): August 5, 2004
Published electronically: October 4, 2005
Additional Notes: The work of the first author was partially supported by the Fund for Scientic Research (FWO), projects “CORFU: Constructive study of orthogonal functions”, grant #G.0184.02 and, “RAM: Rational modelling: optimal conditioning and stable algorithms”, grant #G.0423.05, and by the Belgian Programme on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office. The scientific responsibility rests with the author.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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