An algorithm for automatically selecting a suitable verification method for linear systems

Katsuhisa Ozaki, Takeshi Ogita, Shinichi Oishi

    Research output: Contribution to journalArticle

    1 Citation (Scopus)

    Abstract

    Several methods have been proposed to calculate a rigorous error bound of an approximate solution of a linear system by floating-point arithmetic. These methods are called 'verification methods'. Applicable range of these methods are different. It depends mainly on the condition number and the dimension of the coefficient matrix whether such methods succeed to work or not. In general, however, the condition number is not known in advance. If the dimension or the condition number is large to some extent, then Oishi-Rump's method, which is known as the fastest verification method for this purpose, may fail. There are more robust verification methods whose computational cost is larger than the Oishi-Rump's one. It is not so efficient to apply such robust methods to well-conditioned problems. The aim of this paper is to choose a suitable verification method whose computational cost is minimum to succeed. First in this paper, four fast verification methods for linear systems are briefly reviewed. Next, a compromise method between Oishi-Rump's and Ogita-Oishi's one is developed. Then, an algorithm which automatically and efficiently chooses an appropriate verification method from five verification methods is proposed. The proposed algorithm does as much work as necessary to calculate error bounds of approximate solutions of linear systems. Finally, numerical results are presented.

    Original languageEnglish
    Pages (from-to)363-382
    Number of pages20
    JournalNumerical Algorithms
    Volume56
    Issue number3
    DOIs
    Publication statusPublished - 2011 Mar

    Fingerprint

    Linear systems
    Linear Systems
    Computational methods
    Condition number
    Digital arithmetic
    Error Bounds
    Computational Cost
    Approximate Solution
    Costs
    Choose
    Calculate
    Floating-point Arithmetic
    Robust Methods
    Numerical Results
    Necessary

    Keywords

    • Linear systems
    • Verified computation

    ASJC Scopus subject areas

    • Applied Mathematics

    Cite this

    An algorithm for automatically selecting a suitable verification method for linear systems. / Ozaki, Katsuhisa; Ogita, Takeshi; Oishi, Shinichi.

    In: Numerical Algorithms, Vol. 56, No. 3, 03.2011, p. 363-382.

    Research output: Contribution to journalArticle

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