Fast verification of solutions of matrix equations

Shinichi Oishi, Siegfried M. Rump

    Research output: Contribution to journalArticle

    35 Citations (Scopus)

    Abstract

    In this paper, we are concerned with a matrix equation Ax = b where A is an n × n real matrix and x and b are n-vectors. Assume that an approximate solution x is given together with an approximate LU decomposition. We will present fast algorithms for proving nonsingularity of A and for calculating rigorous error bounds for ∥A-1 b - x̃∥. The emphasis is on rigour of the bounds. The purpose of this paper is to propose different algorithms, the fastest with 2/3n3 flops computational cost for the verification step, the same as for the LU decomposition. The presented algorithms exclusively use library routines for LU decomposition and for all other matrix and vector operations.

    Original languageEnglish
    Pages (from-to)755-773
    Number of pages19
    JournalNumerische Mathematik
    Volume90
    Issue number4
    DOIs
    Publication statusPublished - 2002 Feb

    Fingerprint

    LU decomposition
    Matrix Equation
    Decomposition
    Nonsingularity
    Error Bounds
    Fast Algorithm
    Computational Cost
    Approximate Solution
    Costs

    ASJC Scopus subject areas

    • Mathematics(all)
    • Applied Mathematics
    • Computational Mathematics

    Cite this

    Fast verification of solutions of matrix equations. / Oishi, Shinichi; Rump, Siegfried M.

    In: Numerische Mathematik, Vol. 90, No. 4, 02.2002, p. 755-773.

    Research output: Contribution to journalArticle

    Oishi, Shinichi ; Rump, Siegfried M. / Fast verification of solutions of matrix equations. In: Numerische Mathematik. 2002 ; Vol. 90, No. 4. pp. 755-773.
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