Skip to main content
Log in

On the calculation of the largest eigenvalue of an integral equation

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

Two methods are given for obtaining computable bounds for the largest eigenvalue of a linear integral equation with a continuous, symmetric, and nonnegative kernel. Several numerical examples are presented.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Atkinson, K. E.: Extensions of the Nyström method for the numerical solution of linear integral equations of the second kind. Ph. D. thesis. Madison: University of Wisconsin 1966.

    Google Scholar 

  2. Baker, C. T. H., L. Fox, D. F. Mayers, andK. Wright: Numerical solution of Fredholm integral equations of the first kind. Computer J.7, 141 (1964).

    Article  Google Scholar 

  3. Brakhage, H.: Zur Fehlerabschätzung für die numerische Eigenwertbestimmung bei Integralgleichungen. Numer. Math.3, 174 (1961).

    Article  Google Scholar 

  4. Bückner, H.: Die praktische Behandlung von Integralgleichungen. Berlin-Göttingen-Heidelberg: Springer 1952.

    Google Scholar 

  5. Collatz, L.: Schrittweise Näherungen bei Integralgleichungen und Eigenwertschranken. Math. Z.46, 692 (1940).

    Article  Google Scholar 

  6. —— Numerische Behandlung von Differentialgleichungen. Berlin-Göttingen-Heidelberg: Springer 1955.

    Google Scholar 

  7. Courant, R., andD. Hilbert: Methods of mathematical physics, vol. I. New York: Interscience 1953.

    Google Scholar 

  8. Cryer, C. W.: On the calculation of the largest eigenvalue of an integral equation. Tech. Report No 704, Math. Res. Ctr., U. S. Army, University of Wisconsin 1966.

  9. Fettis, H. E.: Note on the matrix equation Aχ= λB χ. Computer J.8, 279 (1965).

    Article  Google Scholar 

  10. Gantmacher, F. R.: The theory of matrices, vol. I. New York: Chelsea 1959

    Google Scholar 

  11. Goodwin, E. T.: Electronic states at the surfaces of crystals, II. Proc. Camb. Phil. Soc.35, 221 (1939).

    Google Scholar 

  12. Jahnke, E., u.F. Emde: Funktionentafeln mit Formeln und Kurven, second edition. Leipzig: Teubner 1933.

    Google Scholar 

  13. Jentzsch, R.: Über Integralgleichungen mit positivem Kern. J. Reine Angew. Math.141, 235 (1912).

    Google Scholar 

  14. Krasnoselskii, M. A.: Positive solutions of operator equations. Groningen: Noordhoff 1964.

    Google Scholar 

  15. Krein, M. G., andM. A. Rutman: Linear operators leaving invariant a cone in a Banach space. A. M. S. Transl. No. 26. New York: American Mathematical Society 1950. Uspekhi Matem. Nauk (N. S.)3, 3 (1948).

    Google Scholar 

  16. Martin, R. S., andJ. H. Wilkinson: Symmetric decomposition of positive definite band matrices. Numerische Mathematik7, 355 (1965).

    Article  Google Scholar 

  17. Mikhlin, S. G.: Integral equations. New York: MacMillan 1957.

    Google Scholar 

  18. ——. Variational methods in mathematical physics. New York: MacMillan 1964.

    Google Scholar 

  19. Rall, L. B.: Numerical integration and the solution of integral equations by the use of Riemann sums. SIAM Rev.7, 55 (1965).

    Article  Google Scholar 

  20. Wielandt, H.: Error bounds for eigenvalues of symmetric integral equations. Proc. Amer. Math. Soc. Symposia Appl. Math.6, 261 (1956).

    Google Scholar 

  21. Wilkinson, J. H.: The algebraic eigenvalue problem. Oxford: Clarendon 1965.

    Google Scholar 

  22. Wing, G. M.: On a method for obtaining bounds on the eigenvalues of certain integral equations. J. Math. Anal. Appl.11, 160 (1965).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No.: DA-31-9 24-ARO-D-462.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cryer, C.W. On the calculation of the largest eigenvalue of an integral equation. Numer. Math. 10, 165–176 (1967). https://doi.org/10.1007/BF02174151

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02174151

Keywords

Navigation