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Algorithm 793: GQRAT—Gauss quadrature for rational functions

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Walter GautschiAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 25, Issue 2

Pages 213 - 239

https://doi.org/10.1145/317275.317282

Published: 01 June 1999 Publication History

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Abstract

The concern here is with Gauss-type quadrature rules that are exact for a mixture of polynomials and rational functions, the latter being selected so as to simulate poles that may be present in the integrand. The underlying theory is presented as well as methods for constructing such rational Gauss formulae. Relevant computer routines are provided and applied to a number examples, including Fermi-Dirac and Bose-Einstein integrals of interest in solid state physics.

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References

[1]

GALANT, D. 1992. Algebraic methods for modified orthogonal polynomials. Math. Comput. 59, 200 (Oct.), 541-546.

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GAUTSCHI, W. 1993a. Gauss-type quadrature rules for rational functions. In Numerical Integration IV, H. Brass and G. H mmerlin, Eds. International Series of Numerical Mathematics, vol. 112. Birkh user-Verlag, Basel, Switzerland, 111-130.

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GAUTSCHI, W. 1993b. On the computation of generalized Fermi-Dirac and Bose-Einstein integrals. Comput. Phys. Commun. 74, 233-238.

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GAUTSCHI, W. 1994. Algorithm 726: ORTHPOL--A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Softw. 20, 1 (Mar. 1994), 21-62.

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GRADSHTEYN, I. S. AND RYZHIK, I. M. 1965. Table of Integrals, Series, and Products. 4th ed. Academic Press, Inc., New York, NY.

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ROVBA, E.A. 1996. Quadrature formulae of interpolatory rational type. Dokl. Akad. Nauk Belarusi 40, 3 (May-Jun.), 42-46. In Russian.

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SAGAR, R. P. 1991. A Gaussian quadrature for the calculation of generalized Fermi-Dirac integrals. Comput. Phys. Commun. 66, 271-275.

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Cited By

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Fernandes AAtchley W(2017)Gaussian Quadrature Formulae for Arbitrary Positive MeasuresEvolutionary Bioinformatics10.1177/1176934306002000102(117693430600200)Online publication date: 9-Nov-2017
https://doi.org/10.1177/117693430600200010
Deckers KMougaida ABelhadjsalah H(2017)Algorithm 973ACM Transactions on Mathematical Software10.1145/305407743:4(1-29)Online publication date: 23-Mar-2017
https://dl.acm.org/doi/10.1145/3054077
Illn-Gonzlez JRebollido-Lorenzo J(2017)Evaluation of finite part integrals using a regularization technique that decreases instabilityJournal of Computational and Applied Mathematics10.1016/j.cam.2017.01.009319:C(210-219)Online publication date: 1-Aug-2017
https://dl.acm.org/doi/10.1016/j.cam.2017.01.009
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Index Terms

Algorithm 793: GQRAT—Gauss quadrature for rational functions
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Numerical differentiation
    2. Quadrature
2. Software and its engineering
  1. Software notations and tools
    1. General programming languages
      1. Language types

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Reviews

Reviewer: Friedemann W. Stallmann

This algorithm concerns Gaussian-type quadrature formulas that are exact for rational functions that have poles of known orders at known locations. Poles near the integration interval are treated somewhat differently to increase convergence, that is, to decrease the error. There is a detailed description of the mathematics involved, with references. Several examples are provided, such as Fermi-Dirac and Bose-Einstein integrals, which are of interest in solid state physics. The essential computations can be performed using routines from Algorithm 726, ORTHOPOL, with driver programs provided in the current algorithm.

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cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 25, Issue 2

June 1999

148 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/317275

Issue’s Table of Contents

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 01 June 1999

Published in TOMS Volume 25, Issue 2

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Fernandes AAtchley W(2017)Gaussian Quadrature Formulae for Arbitrary Positive MeasuresEvolutionary Bioinformatics10.1177/1176934306002000102(117693430600200)Online publication date: 9-Nov-2017
https://doi.org/10.1177/117693430600200010
Deckers KMougaida ABelhadjsalah H(2017)Algorithm 973ACM Transactions on Mathematical Software10.1145/305407743:4(1-29)Online publication date: 23-Mar-2017
https://dl.acm.org/doi/10.1145/3054077
Illn-Gonzlez JRebollido-Lorenzo J(2017)Evaluation of finite part integrals using a regularization technique that decreases instabilityJournal of Computational and Applied Mathematics10.1016/j.cam.2017.01.009319:C(210-219)Online publication date: 1-Aug-2017
https://dl.acm.org/doi/10.1016/j.cam.2017.01.009
(2015)Gauss rules associated with nearly singular weightsApplied Numerical Mathematics10.1016/j.apnum.2014.07.00691:C(1-10)Online publication date: 1-May-2015
https://dl.acm.org/doi/10.1016/j.apnum.2014.07.006
Deckers KBultheel A(2012)The existence and construction of rational Gauss-type quadrature rulesApplied Mathematics and Computation10.1016/j.amc.2012.04.008218:20(10299-10320)Online publication date: Jun-2012
https://doi.org/10.1016/j.amc.2012.04.008
Berriochoa Esnaola ECachafeiro López AIllán-González JMartínez-Brey E(2011)Chebyshev series method for computing weighted quadrature formulasApplied Mathematics and Computation10.1016/j.amc.2011.10.024218:8(4437-4447)Online publication date: Dec-2011
https://doi.org/10.1016/j.amc.2011.10.024
Illán González J(2009)Gaussian rational quadrature formulas for ill-scaled integrandsJournal of Computational and Applied Mathematics10.1016/j.cam.2009.02.043233:3(745-748)Online publication date: 1-Dec-2009
https://dl.acm.org/doi/10.1016/j.cam.2009.02.043
Lombardi G(2009)Design of quadrature rules for Müntz and Müntz‐logarithmic polynomials using monomial transformationInternational Journal for Numerical Methods in Engineering10.1002/nme.268480:13(1687-1717)Online publication date: 31-Jul-2009
https://doi.org/10.1002/nme.2684
Graglia RLombardi G(2008)Machine Precision Evaluation of Singular and Nearly Singular Potential Integrals by Use of Gauss Quadrature Formulas for Rational FunctionsIEEE Transactions on Antennas and Propagation10.1109/TAP.2008.91918156:4(981-998)Online publication date: Apr-2008
https://doi.org/10.1109/TAP.2008.919181
Graglia RLombardi G(2007)Exact Quadrature of Singular and Nearly Singular Potential Integrals2007 International Conference on Electromagnetics in Advanced Applications10.1109/ICEAA.2007.4387471(982-985)Online publication date: Sep-2007
https://doi.org/10.1109/ICEAA.2007.4387471
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Abstract

Supplementary Material

References

Cited By

Index Terms

Recommendations

Algorithm 867: QUADLOG—a package of routines for generating Gauss-related quadrature for two classes of logarithmic weight functions

An interpolation algorithm for orthogonal rational functions

Algorithm 855: Subroutines for the computation of Mathieu characteristic numbers and their general orders

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