A survey of some estimates of eigenvalues and condition numbers for certain preconditioned matrices

Owe Axelsson, H. Lu

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    Abstract

    Eigenvalue and condition number estimates for preconditioned iteration matrices provide the information required to estimate the rate of convergence of iterative methods, such as preconditioned conjugate gradient methods. In recent years various estimates have been derived for (perturbed) modified (block) incomplete factorizations. We survey and extend some of these and derive new estimates. In particular we derive upper and lower estimates of individual eigenvalues and of condition number. This includes a discussion that the condition number of preconditioned second order elliptic difference matrices is O(h-1). Some of the methods are applied to compute certain parameters involved in the computation of the preconditioner.
    Original languageEnglish
    Pages (from-to)241-264
    Number of pages24
    JournalJournal of computational and applied mathematics
    Volume80
    Issue number80
    DOIs
    Publication statusPublished - 1997

    Keywords

    • Preconditioning method
    • Perturbation
    • Eigenvalue bound
    • Condition number estimate

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