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Transpose-Free Look-Ahead Algorithms for Lanczos’ Method

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Applications and Computation of Orthogonal Polynomials

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 131))

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Abstract

The method of Lanczos for solving systems of linear equations is implemented by various recurrence relationships. Sometimes, a division by a quantity close to zero occurs, thus producing numerical instability in the algorithm. It is shown how to treat such a near-breakdown by a look-ahead technique which, at the same time, avoids the use of the transpose of the matrix. The main idea of the algorithms given in this paper consists of computing the coefficients of these recurrences by a look-ahead algorithm for the CGS or the BiCGSTAB and, then, to use them also in Lanczos’ method. Thus, transpose-free look-ahead algorithms for treating simultaneously breakdowns and near-breakdowns in Lanczos’ method and in the CGS or the BiCGSTAB will be obtained.

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Author information

Authors and Affiliations

  1. Laboratoire d’Analyse Numérique et d’Optimisation, Université des Sciences et Technologies de Lille, F-59655, Villeneuve d’Ascq Cedex, France

    Claude Brezinski

  2. Dipartimento di Elettronica e Informatica, Università degli Studi di Padova, Via Gradenigo 6/a, I-35131, Padova, Italy

    Michela Redivo-Zaglia

Authors
  1. Claude Brezinski
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  2. Michela Redivo-Zaglia
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Editor information

Editors and Affiliations

  1. Department of Computer Sciences, Purdue University, West Lafayette, IN, 47907-1398, USA

    Walter Gautschi

  2. Institute for Applied Mathematics, University of Hamburg, Bundesstr. 55, D-20146, Hamburg, Germany

    Gerhard Opfer

  3. Computer Science Department, Stanford University, Stanford, CA, 94305, USA

    Gene H. Golub

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© 1999 Springer Basel AG

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Brezinski, C., Redivo-Zaglia, M. (1999). Transpose-Free Look-Ahead Algorithms for Lanczos’ Method. In: Gautschi, W., Opfer, G., Golub, G.H. (eds) Applications and Computation of Orthogonal Polynomials. International Series of Numerical Mathematics, vol 131. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8685-7_2

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  • DOI: https://doi.org/10.1007/978-3-0348-8685-7_2

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9728-0

  • Online ISBN: 978-3-0348-8685-7

  • eBook Packages: Springer Book Archive

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