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  • 1
    Electronic Resource
    Electronic Resource
    Estimates in quadratic formulas (1994)
    Springer
    Numerical algorithms 8 (1994), S. 241-268 
    Details
    ISSN: 1572-9265
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract LetA be a real symmetric positive definite matrix. We consider three particular questions, namely estimates for the error in linear systemsAx=b, minimizing quadratic functional minx(x T Ax−2b T x) subject to the constraint ‖x‖=α, α〈‖A −1 b‖, and estimates for the entries of the matrix inverseA −1. All of these questions can be formulated as a problem of finding an estimate or an upper and lower bound onu T F(A)u, whereF(A)=A −1 resp.F(A)=A −2,u is a real vector. This problem can be considered in terms of estimates in the Gauss-type quadrature formulas which can be effectively computed exploiting the underlying Lanczos process. Using this approach, we first recall the exact arithmetic solution of the questions formulated above and then analyze the effect of rounding errors in the quadrature calculations. It is proved that the basic relation between the accuracy of Gauss quadrature forf(λ)=λ−1 and the rate of convergence of the corresponding conjugate gradient process holds true even for finite precision computation. This allows us to explain experimental results observed in quadrature calculations and in physical chemistry and solid state physics computations which are based on continued fraction recurrences.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02142693
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    Paper (German National Licenses)
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  • 2
    Book
    Book
    Krylov subspace methods: principles and analysis / (2013)
    Oxford :Oxford Univ. Press,
    Details
    Title: Krylov subspace methods: principles and analysis /
    Author: Liesen, Jörg
    Contributer: Strakoš, Zdenek
    Publisher: Oxford :Oxford Univ. Press,
    Year of publication: 2013
    Pages: XV, 391 S. : , graph. Darst. ; , 24 cm
    Series Statement: Numerical mathematics and scientific computation
    ISBN: 978-0-19-965541-0
    Type of Medium: Book
    Language: English
    URL: http://www.gbv.de/dms/tib-ub-hannover/729075249.pdf
    URL: http://zbmath.org/?q=an:1263.65034
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  • 3
    Book
    Book
    Preconditioning and the conjugate gradient method in the context of solving PDEs /; 01 (2015)
    Philadelphia :SIAM, Soc. for Industrial and Applied Mathematics,
    Details
    Title: Preconditioning and the conjugate gradient method in the context of solving PDEs /; 01
    Author: Málek, Josef
    Contributer: Strakoš, Zdeněk
    Publisher: Philadelphia :SIAM, Soc. for Industrial and Applied Mathematics,
    Year of publication: 2015
    Series Statement: SIAM spotlights 01
    ISBN: 978-1-61197-383-9
    Type of Medium: Book
    Language: English
    URL: https://www.zbmath.org/?q=an:06423099
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