Computing probabilistic bounds for extreme eigenvalues of symmetric matrices with the Lanczos method

J.L.M. Dorsselaer, van, M.E. Hochstenbach, H.A. Vorst, van der

    Research output: Book/ReportReportAcademic

    Abstract

    In many applications it is important to have reliable approximations for the extreme eigenvalues of a symmetric or Hermitian matrix. A method which is often used to compute these eigenvalues is the Lanczos method. Unfortunately it is not guaranteed that the extreme Ritz values are close to the extreme eigenvalues { even when the norms of the corresponding residual vectors are small. Assuming that the starting vector has been chosen randomly, we derive probabilistic bounds for the extreme eigenvalues. Four di??erent types of bounds are obtained using Lanczos, Ritz and Chebyshev polynomials. These bounds are compared theoretically and numerically. Furthermore we show how one can determine, after each Lanczos step, an upper bound for the number of steps still needed (without performing these steps) to obtain an approximation to the largest or smallest eigenvalue within a prescribed tolerance.
    Original languageEnglish
    Place of PublicationAmsterdam
    PublisherCentrum voor Wiskunde en Informatica
    Number of pages17
    Publication statusPublished - 1999

    Publication series

    NameCWI report. MAS-R : modelling, analysis and simulation
    Volume9934
    ISSN (Print)1386-3703

    Fingerprint Dive into the research topics of 'Computing probabilistic bounds for extreme eigenvalues of symmetric matrices with the Lanczos method'. Together they form a unique fingerprint.

    Cite this