The method of Lanczos for solving systems of linear equations is implemented by various recurrence relationships. Sometimes, a division by a quantity close to zero occurs, thus producing numerical instability in the algorithm. It is shown how to treat such a near-breakdown by a look-ahead technique which, at the same time, avoids the use of the transpose of the matrix. The main idea of the algorithms given in this paperconsists of computing the coefficients of these recurrences by a look-ahead algorithm for the CGS or the BiCGSTAB and, then, to use them also in Lanczos' method. Thus, transpose-free look-ahead algorithms for treating simultaneously breakdowns and near-breakdowns in Lanczos' method and in the CGS or the BiCGSTAB will be obtained.
Transpose-free look-ahead algorithms for Lanczos' method
REDIVO ZAGLIA, MICHELA
1999
Abstract
The method of Lanczos for solving systems of linear equations is implemented by various recurrence relationships. Sometimes, a division by a quantity close to zero occurs, thus producing numerical instability in the algorithm. It is shown how to treat such a near-breakdown by a look-ahead technique which, at the same time, avoids the use of the transpose of the matrix. The main idea of the algorithms given in this paperconsists of computing the coefficients of these recurrences by a look-ahead algorithm for the CGS or the BiCGSTAB and, then, to use them also in Lanczos' method. Thus, transpose-free look-ahead algorithms for treating simultaneously breakdowns and near-breakdowns in Lanczos' method and in the CGS or the BiCGSTAB will be obtained.Pubblicazioni consigliate
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