Treatment of near-breakdown in the CGS algorithm |
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| Authors: | C. Brezinski M. Redivo-Zaglia |
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| Affiliation: | (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;(2) Dipartimento di Elettronica e Informatica, Università degli Studi di Padova, via Gradenigo 6/a, I-35131 Padova, Italy |
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| Abstract: | Lanczos' method for solving the system of linear equationsAx=b consists in constructing a sequence of vectors (xk) such thatrk=b–Axk=Pk(A)r0 wherer0=b–Ax0.Pk is an orthogonal polynomial which is computed recursively. The conjugate gradient squared algorithm (CGS) consists in takingrk=Pk2(A)r0. In the recurrence relation forPk, 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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| Keywords: | Conjugate gradient Lanczos' method systems of linear equations orthogonal polynomials |
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