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Algorithm 45. Automatic computation of improper integrals over a bounded or unbounded planar region

Algorithmus 45. Automatische Berechnung uneigentlicher Integrale über beschränkte oder unbeschränkte Bereiche

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Abstract

An automatic quadrature algorithm especially designed for double integration of functions with some form of singular behaviour on the boundary of the integration region is described, and its FORTRAN code is presented.

The algorithm is based on the use of the product trapezoidal rule, after a non-linear transformation of the integrand in both variables renders a new integrand function whose derivatives vanish on the (transformed) boundary.

Numerical results demonstrate the ability of the algorithm to obtain high accuracies in dealing automatically with pathological singularities of non-specific types.

Zusammenfassung

Es wird ein Algorithmus zur automatischen zweidimensionalen Quadratur beschrieben, der speziell für Integranden mit singulärem Verhalten am Rand des Integrationsgebietes geeignet ist. Eine Implementierung dieses Algorithmus in Standard FORTRAN wird vorgestellt.

Der Algorithmus beruht auf einer nichtlinearen Transformation des Integranden in beiden Veränderlichen, die so gewählt, wird, daß der transformierte Integrand und alle seine Ableitungen am Rand des (transformierten) Integrationsgebiets verschwinden. Auf den transformierten Integranden wird die Trapezregel in der Produktform angewendet.

An Hand von numerischen Testresultaten wird die Effizienz des vorgestellten Algorithmus bei der automatischen Ermittlung von sehr genauen Näherungswerten für Integrale mit unspezifischen, pathologischen Singularitäten demonstriert.

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References

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Authors and Affiliations

  1. Computer Science Department, La Trobe University, 3083, Bundoora, Victoria, Australia

    I. Robinson

  2. Applied Mathematics and Programming Division, University of Leuven, Celestijnenlaan 200 A, B-3030, Heverlee, Belgium

    Elise de Doncker

Authors
  1. I. Robinson
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  2. Elise de Doncker
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Robinson, I., de Doncker, E. Algorithm 45. Automatic computation of improper integrals over a bounded or unbounded planar region. Computing 27, 253–284 (1981). https://doi.org/10.1007/BF02237983

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  • DOI: https://doi.org/10.1007/BF02237983

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