Iterative refinement for ill-conditioned linear systems

Shinichi Oishi, Takeshi Ogita, Siegfried M. Rump

    Research output: Contribution to journalArticle

    3 Citations (Scopus)

    Abstract

    This paper treats a linear equation Aυ = b, where A ∈ F n×n and b ∈ Fn. Here, F is a set of floating point numbers. Let u be the unit round-off of the working precision and κ(A) = ∥A∥∞∥A-1∥∞ be the condition number of the problem. In this paper, ill-conditioned problems with 1 < uκ(A) < ∞ are considered and an iterative refinement algorithm for the problems is proposed. In this paper, the forward and backward stability will be shown for this iterative refinement algorithm.

    Original languageEnglish
    Pages (from-to)465-476
    Number of pages12
    JournalJapan Journal of Industrial and Applied Mathematics
    Volume26
    Issue number2-3
    Publication statusPublished - 2009 Oct

    Fingerprint

    Iterative Refinement
    Linear systems
    Linear Systems
    Linear equations
    Floating point
    Condition number
    Linear equation
    Unit

    Keywords

    • Ill-conditioned linear systems
    • Iterative refinement
    • Verified numerical computation

    ASJC Scopus subject areas

    • Applied Mathematics
    • Engineering(all)

    Cite this

    Iterative refinement for ill-conditioned linear systems. / Oishi, Shinichi; Ogita, Takeshi; Rump, Siegfried M.

    In: Japan Journal of Industrial and Applied Mathematics, Vol. 26, No. 2-3, 10.2009, p. 465-476.

    Research output: Contribution to journalArticle

    Oishi, Shinichi ; Ogita, Takeshi ; Rump, Siegfried M. / Iterative refinement for ill-conditioned linear systems. In: Japan Journal of Industrial and Applied Mathematics. 2009 ; Vol. 26, No. 2-3. pp. 465-476.
    @article{fbf6417fd0e74f569a6df147e045b290,
    title = "Iterative refinement for ill-conditioned linear systems",
    abstract = "This paper treats a linear equation Aυ = b, where A ∈ F n×n and b ∈ Fn. Here, F is a set of floating point numbers. Let u be the unit round-off of the working precision and κ(A) = ∥A∥∞∥A-1∥∞ be the condition number of the problem. In this paper, ill-conditioned problems with 1 < uκ(A) < ∞ are considered and an iterative refinement algorithm for the problems is proposed. In this paper, the forward and backward stability will be shown for this iterative refinement algorithm.",
    keywords = "Ill-conditioned linear systems, Iterative refinement, Verified numerical computation",
    author = "Shinichi Oishi and Takeshi Ogita and Rump, {Siegfried M.}",
    year = "2009",
    month = "10",
    language = "English",
    volume = "26",
    pages = "465--476",
    journal = "Japan Journal of Industrial and Applied Mathematics",
    issn = "0916-7005",
    publisher = "Springer Japan",
    number = "2-3",

    }

    TY - JOUR

    T1 - Iterative refinement for ill-conditioned linear systems

    AU - Oishi, Shinichi

    AU - Ogita, Takeshi

    AU - Rump, Siegfried M.

    PY - 2009/10

    Y1 - 2009/10

    N2 - This paper treats a linear equation Aυ = b, where A ∈ F n×n and b ∈ Fn. Here, F is a set of floating point numbers. Let u be the unit round-off of the working precision and κ(A) = ∥A∥∞∥A-1∥∞ be the condition number of the problem. In this paper, ill-conditioned problems with 1 < uκ(A) < ∞ are considered and an iterative refinement algorithm for the problems is proposed. In this paper, the forward and backward stability will be shown for this iterative refinement algorithm.

    AB - This paper treats a linear equation Aυ = b, where A ∈ F n×n and b ∈ Fn. Here, F is a set of floating point numbers. Let u be the unit round-off of the working precision and κ(A) = ∥A∥∞∥A-1∥∞ be the condition number of the problem. In this paper, ill-conditioned problems with 1 < uκ(A) < ∞ are considered and an iterative refinement algorithm for the problems is proposed. In this paper, the forward and backward stability will be shown for this iterative refinement algorithm.

    KW - Ill-conditioned linear systems

    KW - Iterative refinement

    KW - Verified numerical computation

    UR - http://www.scopus.com/inward/record.url?scp=77149123329&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=77149123329&partnerID=8YFLogxK

    M3 - Article

    AN - SCOPUS:77149123329

    VL - 26

    SP - 465

    EP - 476

    JO - Japan Journal of Industrial and Applied Mathematics

    JF - Japan Journal of Industrial and Applied Mathematics

    SN - 0916-7005

    IS - 2-3

    ER -