Skip to main content
SpringerLink
Log in

Fast numerical solution of fredholm integral equations with stationary kernels

  • Part II Numerical Mathematics
  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

A fast recursive matrix method for the numerical solution of Fredholm integral equations with stationary kernels is derived. IfN denotes the number of nodal points, the complexity of the algorithm isO(N 2), which should be compared toO(N 3) for conventional algorithms for solving such problems. The method is related to fast algorithms for inverting Toeplitz matrices.

Applications to equations of the first and second kind as well as miscellaneous problems are discussed and illustrated with numerical examples. These show that the theoretical improvement in efficiency is indeed obtained, and that no problems with numerical stability or accuracy are encountered.

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. H. Akaike:Block Toeplitz inversion, SIAM J. Appl. Math., Vol. 24, pp. 234–241, March 1973.

    Google Scholar 

  2. H. C. Andrews and B. R. Hunt:Digital Image Restoration, Prentice Hall, 1977.

  3. K. E. Atkinson:A survey of numerical methods for the solution of Fredholm integral equations of the second kind, SIAM, 1976.

  4. C. T. H. Baker:The Numerical Treatment of Integral Equations, Oxford University Press, 1977.

  5. A. B. Bakushinskii:A numerical method for solving Fredholm integral equations of the first kind, Zhurnal Vychisl. Mat. i Mat. Fiz. (1975), pp. 226–233.

  6. R. Bitmead and B. Andersson:Asymptotically fast solution of Toeplitz and related systems of linear equations, Linear algebra and its applications, Vol. 34, pp. 103–116, 1980.

    Google Scholar 

  7. G. Dahlquist and Å. Björck,Numerical Methods, Prentice Hall, Inc., 1974.

  8. Å. Björck:Numerical algorithms for linear least squares problems, Mathematics and Computation 2/78, Lecture notes presented at Department of Numerical Mathematics. The Universiyt of Trondheim, Norway.

  9. Å. Björck and L. Eldén:Methods in numerical algebra for ill-posed problems, Report LiTH-MAI-R-33-1979, Department of Mathematics, Linköping University.

  10. J. A. Cochran:The Analysis of Linear Integral Equations, McGraw-Hill, 1977.

  11. L. M. Delves and J. Walsh:Numerical Solution of Integral Equations, Clarendon Press, Oxford, 1974.

    Google Scholar 

  12. L. Eldén:Algorithms for the regularization of ill-conditioned least squares problems, BIT (1977), No. 2, pp. 134–145.

  13. B. Friedlander, M. Morf, T. Kailath and L. Ljung:New inversion formulas for matrices classified in terms of their distance from Toeplitz matrices, Linear algebra and its applications, Vol. 27 (October 1979), pp. 31–60.

    Google Scholar 

  14. M. Jaswon and G. Symm:Integral Equation Methods in Potential Theory and Elastostatics, Academic Press, London, 1977.

    Google Scholar 

  15. T. Kailath and L. Ljung:A scattering theory framework for fast least-squares algorithms, P. R. Krishnaiah, ed., Multivariable Analysis IV, North Holland Publishing Company (1977), pp. 387–406.

    Google Scholar 

  16. T. Kailath, L. Ljung and M. Morf:Generalized Krein-Levinson equations for efficient calculation of Fredholm resolvents of nondisplacement kernels. In Surveys in Mathematical Analysis; Essays dedicated to M. G. Krein, N. Y. Academic Press, 1978.

  17. N. Levinson:The Wiener RMS (Root-Mean-Square) error criterion in filter design and prediction, J. Math. Phys., 25 (1947), pp. 261–278.

    Google Scholar 

  18. B. Lindberg:A simple interpolation algorithm for improvement of the numerical solution of a differential equaton, SIAM J. Numer Anal. Vol. 9, No. 4, 1972.

  19. A. Lindquist:On Fredholm integral equations, Toeplitz equations and Kalman-Bucy filtering, Appl. Math. & Optimization, Vol. 1, pp. 355–373, 1975.

    Google Scholar 

  20. S. Ljung:Fast numerical solution of Laplace's equation and the biharmonic equation for circular boundaries, Internal report LiTH-ISY-I-0220.

  21. S. Ljung:Fast numerical algorithms to solve two-dimensional Fredholm integral equation of the first kind with a stationary kernel, Internal report LiTH-ISY-I-0345.

  22. D. L. Phillips:A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. for Comput. Machinery, 9, pp. 84–97, 1962.

    Google Scholar 

  23. W. F. Trench:An algorithm for the inversion of finite Toeplitz matrices, J. SIAM Appl. Math., Vol. 12, pp. 515–522, September 1964.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Linköping University, S-581 83, Linköping, Sweden

    Stefan Ljung & Lennart Ljung

Authors
  1. Stefan Ljung
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lennart Ljung
    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

Ljung, S., Ljung, L. Fast numerical solution of fredholm integral equations with stationary kernels. BIT 22, 54–72 (1982). https://doi.org/10.1007/BF01934395

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01934395

Keywords

Search

Navigation