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Algorithm 988: AMGKQ: An Efficient Implementation of Adaptive Multivariate Gauss-Kronrod Quadrature for Simultaneous Integrands in Octave/MATLAB

Author:

Robert W. JohnsonAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 44, Issue 3

Article No.: 36, Pages 1 - 19

https://doi.org/10.1145/3157735

Published: 11 April 2018 Publication History

Abstract

The algorithm AMGKQ for adaptive multivariate Gauss-Kronrod quadrature over hyperrectangular regions of arbitrary dimensionality is proposed and implemented in Octave/MATLAB. It can approximate numerically any number of integrals over a common domain simultaneously. Improper integrals are addressed through singularity weakening coordinate transformations. Internal singularities are addressed through the use of breakpoints. Its accuracy performance is investigated thoroughly, and its running time is compared to other commonly available routines in two and three dimensions. Its running time can be several orders of magnitude faster than recursively called quadrature routines. Its performance is limited only by the memory structure of its operating environment. Included with the software are numerous examples of its invocation.

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References

[1]

Jarle Berntsen and Terje O. Espelid. 1991. Error estimation in automatic quadrature routines. ACM Trans. Math. Softw. 17, 2, 233--252.

Digital Library

[2]

Jarle Berntsen, Terje O. Espelid, and Alan Genz. 1991. An adaptive algorithm for the approximate calculation of multiple integrals. ACM Trans. Math. Softw. 17, 4, 437--451.

Digital Library

[3]

John Burkardt. 2009. TEST_INT: Quadrature Tests. Retrieved March 6, 2018, from http://people.sc.fsu.edu/~jburkardt/m_src/test_int/test_int.html.

[4]

John Burkardt. 2011. Quadrature Tests for 2D Finite Intervals. Retrieved March 6, 2018, from http://people.sc.fsu.edu/jburkardt/m_src/test_int_2d/test_int_2d.html.

[5]

Ronald Cools and Ann Haegemans. 2003. Algorithm 824: CUBPACK: A package for automatic cubature; framework description. ACM Trans. Math. Softw. 29, 3, 287--296. 0098-3500

Digital Library

[6]

John W. Eaton, David Bateman, and Søren Hauberg. 2009. GNU Octave Version 3.0.1 Manual: A High-Level Interactive Language for Numerical Computations. CreateSpace Independent Publishing Platform. http://www.gnu.org/software/octave/doc/interpreter ISBN 1441413006

[7]

Bengt Fornberg. 1998. Calculation of weights in finite difference formulas. SIAM Rev. 40, 685--691.

Digital Library

[8]

Walter Gander and Walter Gautschi. 2000. Adaptive quadrature revisited. BIT Numer. Math. 40, 1, 84--101.

[9]

Walter Gautschi. 1994. Algorithm 726: ORTHPOL—a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Softw. 20, 1, 21--62. 0098-3500

Digital Library

[10]

Walter Gautschi. 2004. Orthogonal Polynomials: Computation and Approximation. Oxford University Press.

[11]

Alan C. Genz. 2013. Alan Genz Software. Retrieved March 6, 2018, from http://www.math.wsu.edu/faculty/genz/software/software.html.

[12]

A. C. Genz and A. A. Malik. 1980. Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region. J. Comput. Appl. Math. 6, 4, 295--302.

[13]

Pedro Gonnet. 2010. Increasing the reliability of adaptive quadrature using explicit interpolants. ACM Trans. Math. Softw. 37, 3, Article 26, 32 pages.

Digital Library

[14]

Pavel Holoborodko. 2011. Gauss-Kronrod Quadrature Nodes and Weights. Retrieved March 6, 2018, from http://www.advanpix.com/2011/11/07/gauss-kronrod-quadrature-nodes-weights/.

[15]

David S. Johnson. 2002. A theoretician’s guide to the experimental analysis of algorithms. In Data Structures, Near Neighbor Searches, and Methodology: Fifth and Sixth DIMACS Implementation Challenges, M. H. Goldwasser, D. S. Johnson, and C. C. McGeoch (Eds.). American Mathematical Society, Providence, RI, 215--250.

[16]

Dirk P. Laurie. 1997. Calculation of Gauss-Kronrod quadrature rules. Math. Comp. 66, 219, 1133--1145.

Digital Library

[17]

Randall J. LeVeque. 2007. Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

[18]

Koen Poppe and Ronald Cools. 2013. CHEBINT: A MATLAB/Octave toolbox for fast multivariate integration and interpolation based on Chebyshev approximations over hypercubes. ACM Trans. Math. Softw. 40, 1, 2:1--2:13.

Digital Library

[19]

William Press, Saul Teukolsky, William Vetterling, and Brian Flannery. 1992. Numerical Recipes in C (2nd ed.). Cambridge University Press, Cambridge, England.

Digital Library

[20]

L. F. Shampine. 2008. Vectorized adaptive quadrature in MATLAB. J. Comput. Appl. Math. 211, 2, 131--140. 0377-0427

Digital Library

[21]

Lawrence F. Shampine. 2010. Weighted quadrature by change of variable. Neural, Parallel Sci. Comput. 18, 2, 195--206. 1061-5369 http://dl.acm.org/citation.cfm?id=1991936.1991941

Digital Library

[22]

Paul van Dooren and Luc de Ridder. 1976. An adaptive algorithm for numerical integration over an N-dimensional cube. J. Comput. Appl. Math. 2, 3, 207--217.

Cited By

Guo FLiu LYang SYin ZWu JZhang Y(2022)Coupled Resonance Mechanism of Interface Stratification of Thin Coating Structures Excited by Horizontal Shear WavesCoatings10.3390/coatings1210150912:10(1509)Online publication date: 9-Oct-2022
https://doi.org/10.3390/coatings12101509
Guo FWu J(2020)Coupling resonance mechanism of interfacial stratification of sandwich plate structures excited by SH wavesJournal of Low Frequency Noise, Vibration and Active Control10.1177/146134842097946840:3(1166-1193)Online publication date: 13-Dec-2020
https://doi.org/10.1177/1461348420979468

Index Terms

Algorithm 988: AMGKQ: An Efficient Implementation of Adaptive Multivariate Gauss-Kronrod Quadrature for Simultaneous Integrands in Octave/MATLAB
1. Mathematics of computing
  1. Mathematical analysis
    1. Quadrature

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Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 44, Issue 3

September 2018

291 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/3175005

Editor:
Daniel Kressner
EPF Lausanne, Switzerland

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Copyright © 2018 ACM.

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 the author(s) 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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Association for Computing Machinery

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Publication History

Published: 11 April 2018

Accepted: 01 October 2017

Revised: 01 September 2016

Received: 01 July 2015

Published in TOMS Volume 44, Issue 3

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Cited By

Guo FLiu LYang SYin ZWu JZhang Y(2022)Coupled Resonance Mechanism of Interface Stratification of Thin Coating Structures Excited by Horizontal Shear WavesCoatings10.3390/coatings1210150912:10(1509)Online publication date: 9-Oct-2022
https://doi.org/10.3390/coatings12101509
Guo FWu J(2020)Coupling resonance mechanism of interfacial stratification of sandwich plate structures excited by SH wavesJournal of Low Frequency Noise, Vibration and Active Control10.1177/146134842097946840:3(1166-1193)Online publication date: 13-Dec-2020
https://doi.org/10.1177/1461348420979468

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