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Algorithm 32 automatic computation of integrals with singular integrand, over a finite or an infinite range

Algorithmus 31. Automatische Integration von singulären Integranden über ein endliches oder ein unendliches Intervall

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Abstract

A numerical quadrature algorithm is developed, for integrands which may exhibit some kind of singular behaviour within the finite of infinite integration range.

Using the automatical FORTRAN IV integration program, one should provide the abscissae the function is not “smooth” at.

The quadrature formula has been obtained by applying the trapezoidal rule after transformation of the integrand.

Standing severe tests which were based on the test functions of Casaletto et al. and on Kahaner's sample set, the integration scheme turned out to be of a remarkable reliability, efficiency and accuracy.

Zusammenfassung

Es wird ein Algorithmus zur numerischen Quadratur beschrieben, der für Integranden mit singulärem Verhalten in einem endlichen oder unendlichen Integrationsbereich geeignet ist.

Die Quadraturformel wurde durch Transformation des Integranden und anschließende Anwendung der Trapezregel erhalten.

Es wird ein FORTRAN IV Programm vorgestellt, das auf der Anwendung dieser Quadraturformel beruht. Beim Aufruf dieses Programms sollte der Benutzer jene Abszissenwerte vorgeben, an denen der Integrand Singularitäten aufweist.

Unter Verwendung der Testfunktionen von Casaletto et al. und von Kahaner konnte die Zuverlässigkeit, Effizienz und Genauigkeit dieses Integrationsprogramms nachgewiesen werden.

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Authors and Affiliations

  1. Applied Mathematics and Programming Division, Katholieke Universiteit Leuven, Celestijnenlaan 200 A, B-3030, Heverlee, Belgium

    Elise De Doncker & R. Piessens

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  1. Elise De Doncker
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  2. R. Piessens
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De Doncker, E., Piessens, R. Algorithm 32 automatic computation of integrals with singular integrand, over a finite or an infinite range. Computing 17, 265–279 (1976). https://doi.org/10.1007/BF02259651

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  • DOI: https://doi.org/10.1007/BF02259651

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