Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A method for obtaining bounds on eigenvalues and eigenfunctions by solving non-homogeneous integral equations

Eine Methode zur Berechnung der Grenzen von Eigenwerten und Eigenfunktionen durch Lösen inhomogener Integralgleichungen

  • 55 Accesses

  • 9 Citations


A new technique is presented for obtaining upper and lower bounds on eigenvalues and eigenfunctions for linear integral equations. The method is unique in that the bounds are obtained by solving non-homogeneous equations. In order to solve the non-homogeneous equations, non-linear sequence-to-sequence transformations are used to accelerate convergence of the Neumann series inside the radius of convergence and are used to “sum” the Neumann series outside the radius of convergence. Since the reciprocals of the eigenvalues appear as poles in the solution of the non-homogeneous equation, a very sensitive bounding criterion can be given. The method applies to quite general kernels, and has been successfully applied to symmetric and non-symmetric kernels. In addition, thek th eigenfunction may be obtained without a knowledge of the first (k−1) eigenvalues or eigenfunctions.


Es wird eine neue Methode zur Berechnung oberer und unterer Grenzen der Eigenwerte und Eigenfunktionen linearer Integralgleichungen dargestellt. Dadurch, daß die Lösungen inhomogener Gleichungen diese Grenzwerte bestimmen, ist die Methode neuartig. Zur Lösung der inhomogenen Gleichungen werden nichtlineare Reihentransformationen verwendet um die Konvergenz der Neumann-Reihen innerhalb des Konvergenzradius zu beschleunigen und werden ferner zur Summierung der Neumann-Reihen außerhalb des Konvergenzradius benützt. Da die reziproken Eigenwerte Wurzelsingularitäten der inhomogenen Gleichung sind, kann eine sehr empfindliche Begrenzungsbedingung angegeben werden. Die Methode bezieht sich auf ganz allgemeine Kerne und konnte im Falle symmetrischer sowohl wie unsymmetrischer Kerne angewandt werden. Ferner kann diek-te Eigenfunktion ohne Kenntnis der ersten (k−1) Eigenwerte oder Eigenfunktionen berechnet werden.

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


  1. [1]

    Brakhage, H.: A special method of successive approximations for Fredholm integral equations. Duke Math. J.15, 197–206 (1948).

  2. [2]

    Burgmeier, J. W., andM. R. Scott: Convergence of Picard's method for |λ|>|λ1|. Amer. Math. Mon.77, No. 8, 865–867 (1970).

  3. [3]

    Burgmeier, J. W.: On the reduction of a class of Fredholm integral equations to equivalent matrix problems. J. Math. Anal. Appl.31, 529–544 (1970).

  4. [4]

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

  5. [5]

    Householder, A. S.: The Numerical Treatment of a Single Non-linear Equation. New York: McGraw-Hill. 1970.

  6. [6]

    Roark, A. L., andG. M. Wing: A method for computing the eigenvalues of certain integral equations. Numer. Math.7, 159–170 (1965).

  7. [7]

    Shanks, D.: Non-linear transformations of divergent and slowly convergent sequences. J. Math. and Phys.34, 1–42 (1955).

  8. [8]

    Slepian, D., andE. Sonnenblick: Eigenvalues associated with prolate spheriodal wave functions of zero order. Bell System Tech. J.44, 1745–1759 (1965).

  9. [9]

    Tricomi, F. G.: Integral Equations. New York: Interscience. 1957.

  10. [10]

    Wing, G. M.: On a method for obtaining bounds on the eigenvalues of certain integral equations. J. Math. Anal. Appl.11, 160–174 (1965).

  11. [11]

    Wielandt, H.: Error bounds for eigenvalues of symmetric integral equations. Proc. Symp. Appl. Math. VI, pp. 261–282. New York-Toronto-London: 1956.

  12. [12]

    Wynn, P.: On a device for computing thee m (Sn) transformation. MTAC10, 91–96 (1956).

  13. [13]

    Wynn, P.: Singular rules for certain non-linear algorithms. B. I. T.3, 175–195 (1963).

Download references

Author information

Additional information

With 3 Figures

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Burgmeier, J.W., Scott, M.R. A method for obtaining bounds on eigenvalues and eigenfunctions by solving non-homogeneous integral equations. Computing 10, 9–22 (1972).

Download citation


  • Integral Equation
  • Lower Bound
  • Computational Mathematic
  • Neumann Series
  • Linear Integral Equation