Transpose-Free Look-Ahead Algorithms for Lanczos’ Method
Claude Brezinski, Michela Redivo‐Zaglia · Birkhäuser Basel eBooks · 1999
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 paper consists 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. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.