Skip to main content
SpringerLink
Log in

On the monotonicity of the interval versions of Schulz's method II

Über die Monotonie der intervallmäßigen Schulz-Verfahren II

  • Short Communication
  • Published:
Computing Aims and scope Submit manuscript

Abstract

In this note we are considering the interval versions of Schulz's method for bounding the inverse of a matrix. We prove a necessary and sufficient criterion for the existence of an interval inclusion such that the iterates are strictly monotonic in both bounds. In particular, such an initial interval-matrix is constructed.

Zusammenfassung

Für die intervallmäßigen Schulz-Verfahren zur Einschließung der Inversen einer Matrix wird eine notwendige und hinreichende Bedingung dafür angegeben, daß eine Ausgangseinschließung derart existiert, so daß die Iterierten in beiden Schranken streng monotone Folgen bilden. Ferner wird eine berechenbare Startintervallmatrix mit dieser Eigenschaft angegeben.

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

References

  1. Alefeld, G., Herzberger, J.: Introduction to Interval Computations. New York: Academic Press 1983.

    Google Scholar 

  2. Fan, K.: Topological proof of certain theorems on matrices with nonnegative elements. Monatsh. Math.62, 219–237 (1958).

    Google Scholar 

  3. Herzberger, J.: On the monotonicity of the interval versions of Schulz's method. Computing38, 71–74 (1987).

    Google Scholar 

  4. Petković, Lj., Petković, M.: On the monotonicity of the higher-order Schulz's method. Z. angew. Math. Mech. (to appear).

  5. Rump, S. M.: New results on verified inclusions. In: Accurate Scientific Computations (Miranker, W. L., Toupin R. A., eds.), pp. 31–69. Lecture Notes in Computer Science 235, Springer-Verlag 1986.

  6. Schmidt, J. W.: Monotone Eingrenzung von inversen Elementen durch ein quadratisch konvergentes Verfahren ohne Durchschnittsbildung. Z. Angew. Math. Mech.60, 202–204 (1980).

    Google Scholar 

  7. Varga, R. S.: Matrix Iterative Analysis. Englewood Cliffs, N. J.: Prentice-Hall 1962.

    Google Scholar 

  8. Alefeld, G., Herzberger, J.: Matrizeninvertierung mit Fehlererfassung. Elektron. Datenverarbeitung9, 410–416 (1970).

    Google Scholar 

  9. Kulisch, U., Miranker, W. L.: Computer Arithmetic in Theory and Practice. New York: Academic Press 1981.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Universität Oldenburg, D-2900, Oldenburg, Germany

    Jürgen Herzberger

Authors
  1. Jürgen Herzberger
    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

Herzberger, J. On the monotonicity of the interval versions of Schulz's method II. Computing 39, 371–375 (1987). https://doi.org/10.1007/BF02239979

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02239979

AMS Subject Classification

Key words

Search

Navigation