Skip to main content
SpringerLink
Log in

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

  • Algorithm
  • Published:
Computing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. de Doncker, E., Piessens, R.: Algorithm 32. Automatic computation of integrals with singular integrand, over a finite or infinite range. Computing17, 265–279 (1976).

    Google Scholar 

  2. Haegemans, A.: Algorithm 34. An algorithm for the automatic integration over a triangle. Computing19, 179–187 (1977).

    Google Scholar 

  3. Iri, M., Moriguti, S., Takasawa, Y.: On a numerical integration formula. Kolyuroku Research Institute for Mathematical Sciences, Kyoto University, Nr. 91,1970, 82. (In Japanese.)

    Google Scholar 

  4. Lyness, J. N., Kaganove, J. J.: Comments on the nature of automatic quadrature routines. Comp. J.20, 170–177 (1977).

    Google Scholar 

  5. Takahasi, H., Mori, M.: Error estimation in the numerical integration of analytic functions. Report of the Computer Centre (University of Tokyo)3, 41–108 (1970).

    Google Scholar 

  6. van Dooren, P., de Ridder, L.: Algorithm 6. An adaptive algorithm for numerical integration over ann-dimensional cube. J. of Comp. and Appl. Math.2, 207–217 (1976).

    Google Scholar 

Download references

Author information

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elise de Doncker
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Issue Date:

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

Keywords

Search

Navigation