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ArticleOriginal scientific text

Title

Orthogonal polynomials and the Lanczos method

Authors 1, 1, 2

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

Abstract

Lanczos method for solving a system of linear equations is well known. It is derived from a generalization of the method of moments and one of its main interests is that it provides the exact answer in at most n steps where n is the dimension of the system. Lanczos method can be implemented via several recursive algorithms known as Orthodir, Orthomin, Orthores, Biconjugate gradient,... In this paper, we show that all these procedures can be explained within the framework of formal orthogonal polynomials. This theory also provides a natural basis for curing breakdown and near-breakdown in these algorithms. The case of the conjugate gradient squared method can be treated similarly.

Keywords

ENG
projection, biconjugate gradient, orthogonal polynomials, Lanczos method

Bibliography

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Banach Center Publications
1994/29/1
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Pages:
19-33
Main language of publication
English
Published
1994
Exact and natural sciences