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Treatment of near-breakdown in the CGS algorithm

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Abstract

Lanczos' method for solving the system of linear equationsAx=b consists in constructing a sequence of vectors (x k ) such thatr k =b−Ax k =P k (A)r 0 wherer 0=b−Ax 0.P k is an orthogonal polynomial which is computed recursively. The conjugate gradient squared algorithm (CGS) consists in takingr k =P 2k (A)r0. In the recurrence relation forP k , the coefficients are given as ratios of scalar products. When a scalar product in a denominator is zero, then a breakdown occurs in the algorithm. When such a scalar product is close to zero, then rounding errors can seriously affect the algorithm, a situation known as near-breakdown. In this paper it is shown how to avoid near-breakdown in the CGS algorithm in order to obtain a more stable method.

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Authors and Affiliations

  1. Laboratoire d'Analyse Numérique et d'Optimisation, UFR IEEA-M3, Université des Sciences et Technologies de Lille, F-59655, Villeneuve d'Ascq Cedex, France

    C. Brezinski

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

    M. Redivo-Zaglia

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  1. C. Brezinski
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  2. M. Redivo-Zaglia
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Communicated by T.F. Chan

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Brezinski, C., Redivo-Zaglia, M. Treatment of near-breakdown in the CGS algorithm. Numer Algor 7, 33–73 (1994). https://doi.org/10.1007/BF02141260

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  • DOI: https://doi.org/10.1007/BF02141260

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