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The Stabilization of Weights in the Lanczos and Conjugate Gradient Method

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Abstract

The Lanczos and the Conjugate Gradient method both compute implicitly a sequence of Gauss quadrature approximations to a certain Riemann-Stieltjes integral. In finite precision computations the corresponding weight function will be slightly perturbed. The purpose of this paper is to solve a conjecture posed by Anne Greenbaum and Zdeněk Strakoš on the stabilization of weights for the Gauss quadrature approximations, i.e. in particular we prove that for a tight well separated cluster of Ritz values (nodes) an upper bound for the change in the sum of the corresponding weights can be developed that depends mainly on the ratio of the cluster diameter and the gap in the spectrum.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    Wolfgang Wülling

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  1. Wolfgang Wülling
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Correspondence to Wolfgang Wülling.

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AMS subject classification (2000)

65F10, 65F15, 65F50

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Wülling, W. The Stabilization of Weights in the Lanczos and Conjugate Gradient Method. Bit Numer Math 45, 395–414 (2005). https://doi.org/10.1007/s10543-005-0017-0

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  • DOI: https://doi.org/10.1007/s10543-005-0017-0

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