Advertisement

Numerische Mathematik

, Volume 19, Issue 3, pp 238–247 | Cite as

On the computation of inclusion regions for partitioned matrices

  • R. L. Johnston
  • D. D. Olesky
Article

Abstract

In this paper, we consider the problem of computing inclusion regions for the eigenvalues of a partitioned matrix. The algorithms derived are special cases of a generalization of a result of Feingold and Varga which, in turn, is a generalization to the partitioned case of the well-known Gerschgorin circle theorem.

Keywords

Mathematical Method Circle Theorem Inclusion Region Gerschgorin Circle Gerschgorin Circle Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. 1.
    Bauer, F. L.: Optimally scaled matrices. Numer. Math.5, 73–87 (1963).Google Scholar
  2. 2.
    Feingold, D. G., Varga, R. S.: Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem. Pac. J. Math.12, 1241–1250 (1962).Google Scholar
  3. 3.
    Fletcher, R.: Methods for the solution of optimization problems. Report #432, Theoretical Physics Division, U.K.A.E.A. Research Group. Atomic Energy Research Establishment, Harwell, England 1970.Google Scholar
  4. 4.
    Gerschgorin, S.: Über die Abgrenzung der Eigenwerte einer Matrix. Izv. Akad. Nauk SSSR, Ser. Mat.7, 749–754 (1931).Google Scholar
  5. 5.
    Hoffman, A. J.: On the nonsingularity of real partitioned matrices. ICC Bulletin4, 7–17 (1965).Google Scholar
  6. 6.
    Householder, A. S.: Theory of matrices in numerical analysis. New York: Ginn and Co. 1964.Google Scholar
  7. 7.
    Johnston, R. L.: Gerschgorin theorems for partitioned matrices. Lin. Alg. and Its Appns.4, 205–220 (1971).Google Scholar
  8. 8.
    Levinger, B. W., Varga, R. S.: Minimal Gerschgorin sets II Pac. J. Math.17, 199–210 (1966).Google Scholar
  9. 9.
    Olesky, D. D.: Inclusion regions for partitioned matrices. Ph. D. Thesis, University of Toronto, Toronto, Canada. Available as Technical Report #31, Dept. of Computer Science, University of Toronto, 1971.Google Scholar
  10. 10.
    Stone, B. J.: Best possible ratios of certain matrix norms. Numer. Math.4, 114–116 (1962).Google Scholar
  11. 11.
    Varga, R. S.: Matrix iterative analysis. Englewood Cliffs, New Jersey: Prenticehall, Inc. 1963.Google Scholar
  12. 12.
    Varga, R. S.: Minimal Gerschgorin sets. Pac. J. math.15, 719–729 (1965).Google Scholar
  13. 13.
    Wilkinson, J. H.: The algebraic eigenvalue problem. Oxford: Clarendon Press 1965.Google Scholar

Copyright information

© Springer-Verlag 1972

Authors and Affiliations

  • R. L. Johnston
    • 1
  • D. D. Olesky
    • 2
  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada
  2. 2.Department of MathematicsUniversity of VictoriaVictoriaCanada

Personalised recommendations