Skip to main content
Log in

Some infra-max bounds for the spectral radii of splittings ofH-matrices

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

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.

References

  1. Arms, R. J., andL. D. Gates: Iterative methods of solving linear equations, comparisons between point and block iteration. Report, U.S. Naval Proving Ground. Dahlgren Virginia: 1956.

  2. —— andB. Zondek: A method of block iteration. J. Soc. Indust. Appl. Math.4, 220–229 (1956).

    Google Scholar 

  3. Collatz, L.: Einschließungssatz für die Charakteristischen Zahlen von Matrizen Math. Z.48, 221–226 (1942).

    Google Scholar 

  4. Fiedler, M., u.V. Ptak: Über die Konvergenz des verallgemeinerten Seidelschen Verfahrens zur Lösung von Systemen linearer Gleichungen. Math. Nachr.15, 31–38 (1956).

    Google Scholar 

  5. ——: Some inequalities for the spectrum of a matrix. Mat. Fyz. Casopis. Slovensk. Akad. Vied.10, 148–166 (1960).

    Google Scholar 

  6. ——: Generalised norms of matrices and the location of the spectrum. Czechoslovak Math. J.12, 558–571 (1962).

    Google Scholar 

  7. Forsythe, G. E., andW. Wasow: Finite-difference methods for partial differential equations. New York: Wiley 1960.

    Google Scholar 

  8. Frobenius, G.: Über Matrizen aus nicht negative Elementen. Sitzungsberichte der Akademie der Wissenschaften zu Berlin 1912, S. 456–477.

  9. Gerschgorin, S.: Über die Abgrenzung der Eigenwerte einer Matrix. Izv. Akad. Nauk. S.S.S.R.7, 749–754 (1931).

    Google Scholar 

  10. Householder, A. S.: The approximate solution of matrix problems. J.A.C.M.5, 209–226 (1958).

    Google Scholar 

  11. Kahan, W. S.: Gauss-Seidel method of solving systems of linear equations. Doctoral Thesis. Univ. of Toronto, 1958.

  12. Lynn, M. S.: Contributions to the theory of the numerical solution of sets of linear equations. Doctoral dissertation. University of California, Los Angeles, 1962.

    Google Scholar 

  13. Oldenburger, R.: Infinite powers of matrices and characteristic roots. Duke Math. J.6, 357–361 (1940).

    Google Scholar 

  14. Ostrowski, A.: Über die Determinanten mit überwiegender Hauptdiagonale. Comm. Math. Helv.10, 69–96 (1937).

    Google Scholar 

  15. —: Determinanten mit überwiegender Hauptdiagonale und die absolute Konvergenz von linearen Iterationsprozessen. Comm. Math. Helv.30, 175–210 (1956).

    Google Scholar 

  16. —: On some metrical properties of operator matrices and matrices partitioned into blocks. J. Math. Anal. Appl.2, 161–209 (1961).

    Google Scholar 

  17. —: Iterative solution of linear systems of functional equations. J. Math. Anal. Appl.2, 351–369 (1961).

    Google Scholar 

  18. Perron, O.: Zur Theorie der Matrizen. Math. Ann.64, 259–263 (1907).

    Google Scholar 

  19. Price, G. B.: Bounds for determinants with dominant principal diagonal. Proc. Amer. Math. Soc.2, 497–504 (1951).

    Google Scholar 

  20. Stein, P., andR. L. Rosenberg: On the solution of linear simultaneous equations by iteration. J. Lond. Math. Soc.23, 111–118 (1948).

    Google Scholar 

  21. Taussky, O.: A recurring theorem on determinants. Amer. Math. Monthly56, 672–676 (1949).

    Google Scholar 

  22. Varga, R. S.: Matrix Iterative Analysis. New Jersey: Prentice-Hall 1962.

    Google Scholar 

  23. Varga, R. S.: Factorisation and normalised iterative methods. Boundary Problems in Differential Equations. Ed. R. Langer, Univ. of Wisconsin, 1960.

  24. Varga, R. S.: Iterative Numerical Analysis. Computation and Data Processing Center Univ. of Pittsburgh, 1959.

  25. Varga, R. S., andD. C. Feingold: Block diagonally dominant matrices and generalisations of the Gerschgorin circle theorem. Pac. J. Math. (to appear).

  26. Wielandt, H.: Unzerlegbare, nicht negative Matrizen. Math. Z.52, 642–648 (1950).

    Google Scholar 

  27. Young, D.: Iterative methods for solving partial differential equations of the elliptic type. Trans. Amer. Math. Soc.76, 92–111 (1954).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

The author extends his appreciation to ProfessorP. K. Henrici for his guidance in the preparation of the dissertation upon which this paper is largely based. He also wishes to acknowledge with gratitude the constructive criticisms of Dr.D. W. Martin, ProfessorR. S. Varga and Dr.J. H. Wilkinson. The work described above was carried out partly at the University of California, Los Angeles (where it was sponsored by the Office of Naval Research, US Navy) and partly within the research programme of the National Physical Laboratory. The paper is published by permission of the Director of the Laboratory.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lynn, M.S. Some infra-max bounds for the spectral radii of splittings ofH-matrices. Numer. Math. 5, 152–174 (1963). https://doi.org/10.1007/BF01385887

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01385887

Keywords

Navigation