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Lanczos methods for the solution of nonsymmetric systems of linear equations. (English) Zbl 0758.65026

The paper can be divided into two parts. The first one contains some theoretical results on the likelihood of breakdown, when applying the Lanczos (or biconjugate gradient) method to solve nonsymmetric systems of linear equations. These results are stated in terms of measure-zero sets of initial residues and are based on two standard facts:

1) The zero set of a polynomial has measure zero (in fact, this set is contained in a finite union of proper differentiable manifolds, as implied by the implicit function theorem).

2) Given a matrix $A$, the vectors that generate an invariant subspace of dimension smaller than the degree of the minimal polynomial of $A$ build up a measure-zero set [a more precise result is proven on page 19 of the referenced book of A. S. Householder, The theory of matrices in numerical analysis (1964; Zbl 0161.121)].

In the second part of the paper, a remedy to breakdown is proposed by means of several heuristical techniques; among them, the most promising seems to be that of restarting the application of the method. Interesting numerical tests illustrate the application of an algorithm based on those techniques.

##### MSC:
 65F10 Iterative methods for linear systems