Convergence of Rump's method for inverting arbitrarily ill-conditioned matrices

Shinichi Oishi, Kunio Tanabe, Takeshi Ogita, Siegfried M. Rump

    Research output: Contribution to journalArticle

    19 Citations (Scopus)

    Abstract

    In this paper, the problem of inverting regular matrices with arbitrarily large condition number is treated in double precision defined by IEEE 754 floating point standard. In about 1984, Rump derived a method for inverting arbitrarily ill-conditioned matrices. The method requires the possibility to calculate a dot product in higher precision. Rump's method is of theoretical interest. Rump made it clear that inverting an arbitrarily ill-conditioned matrix in single or double precision does not produce meaningless numbers, but contains a lot of information in it. Rump's method uses such inverses as preconditioners. Numerical experiments exhibit that Rump's method converges rapidly for various matrices with large condition numbers. Why Rump's method is so efficient for inverting arbitrarily ill-conditioned matrices is a little mysterious. Thus, to prove its convergence is an interesting problem in numerical error analysis. In this article, a convergence theorem is presented for a variant of Rump's method.

    Original languageEnglish
    Pages (from-to)533-544
    Number of pages12
    JournalJournal of Computational and Applied Mathematics
    Volume205
    Issue number1
    DOIs
    Publication statusPublished - 2007 Aug 1

    Fingerprint

    Condition number
    Error analysis
    Floating point
    Error Analysis
    Scalar, inner or dot product
    Convergence Theorem
    Preconditioner
    Numerical Analysis
    Numerical Experiment
    Converge
    Calculate
    Experiments
    Standards

    Keywords

    • Accurate dot product
    • Ill-conditioned matrix
    • Matrix inversion
    • Precondition

    ASJC Scopus subject areas

    • Applied Mathematics
    • Computational Mathematics
    • Numerical Analysis

    Cite this

    Convergence of Rump's method for inverting arbitrarily ill-conditioned matrices. / Oishi, Shinichi; Tanabe, Kunio; Ogita, Takeshi; Rump, Siegfried M.

    In: Journal of Computational and Applied Mathematics, Vol. 205, No. 1, 01.08.2007, p. 533-544.

    Research output: Contribution to journalArticle

    Oishi, Shinichi ; Tanabe, Kunio ; Ogita, Takeshi ; Rump, Siegfried M. / Convergence of Rump's method for inverting arbitrarily ill-conditioned matrices. In: Journal of Computational and Applied Mathematics. 2007 ; Vol. 205, No. 1. pp. 533-544.
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