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On the monotonicity of higher-order interval iterations for bounding the inverse matrix

Über die Monotonie von intervallmäßigen Iterationsverfahren höherer Ordnung zur Einschließung der inversen Matrix

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Abstract

We are considering modifications of the higher-order interval Schulz methods for improving an initial inclusion for the inverse of a matrix. It is proved that the methods are strictly monotone in both bounds if this is the case for the first iteration step. A necessary and sufficient condition for the existence of such an initial inclusion is given as well as an easy-to-compute formula for it.

Zusammenfassung

Wir betrachten in dieser Note Modifikationen der intervallmäßigen Schulz-Verfahren höherer Ordnung zur Verbesserung einer Einschließung für die Inverse einer Matrix. Es wird ein Satz über das in beiden Schranken streng monotone Verhalten der Methoden bewiesen. Eine notwendige und hinreichende Bedingung für die Existenz solch einer Anfangseinschließung wird angegeben sowie solch eine Intervallmatrix auch berechnet.

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References

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

    Google Scholar 

  2. Herzberger, J.: On theR-order of some recurrences with applications to inclusions-methods II. Computing37, 255–259 (1986).

    Article  Google Scholar 

  3. Herzberger, J.: Remarks on the interval version of Schulz's method. Computing39, 183–186 (1987).

    Article  Google Scholar 

  4. Herzberger, J.: On the monotonicity of the interval versions of Schulz's method. II. Computing39, 371–375 (1987).

    Article  Google Scholar 

  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. Heidelberg: Springer-Verlag 1986.

    Google Scholar 

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

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

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Authors and Affiliations

  1. Fachbereich Mathematik, Universität Oldenburg, D-2900, Oldenburg, Federal Republic of Germany

    J. Herzberger

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  1. J. Herzberger
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Dedicated to Professor Dr. G. Hämmerlin on the occasion of his 60th birthday

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Herzberger, J. On the monotonicity of higher-order interval iterations for bounding the inverse matrix. Computing 40, 367–372 (1988). https://doi.org/10.1007/BF02276921

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  • DOI: https://doi.org/10.1007/BF02276921

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