*(English)*Zbl 0748.65033

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.

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.

Pseudo-codes for MRZ and BSMRZ are given and some numerical results are presented.

##### MSC:

65F15 | Eigenvalues, eigenvectors (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##### References:

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