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Zbl 0748.65033
Brezinski, C.; Redivo Zaglia, M.; Sadok, H.
Avoiding breakdown and near-breakdown in Lanczos type algorithms. (English)
[J] Numer. Algorithms 1, No.3, 261-284 (1991). ISSN 1017-1398; ISSN 1572-9265/e

The paper deals with methods which the authors have developed to avoid breakdown and near-breakdown due to division by a scalar product whose value is zero or is different from zero but small in Lanczos type algorithms for solving linear systems.\par In particular, the bulk of the paper concentrates on a method called by the authors the method of recursive zoom (MRZ) and its variants: SMRZ, BMRZ, GMRZ and BSMRZ, where S, B, G stand for symmetric, balancing and general, respectively. It is shown that a breakdown can be avoided by considering only the existing orthogonal polynomials in the Lanczos type algorithms. The methods described in the paper are able to detect if such a polynomial does not exist in order to jump over it.\par Pseudo-codes for MRZ and BSMRZ are given and some numerical results are presented.
[F.Luban (Bucureşti)]

MSC 2000:: *65F15 Eigenvalues (numerical linear algebra)
65F10 Iterative methods for linear systems
65F25 Orthogonalization (numerical linear algebra)

Keywords: Lanczos method; biconjugate gradient; projection; orthogonal polynomials; methods of moments; method of recursive zoom

Cited in: Zbl 0769.65014

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