Lanczos maximal algorithm for unsymmetric eigenvalue problems (Q753415)

The look-ahead Lanczos (LAL) algorithm has been suggested by \textit{B. N. Parlett}, \textit{D. R. Taylor} and \textit{Z. A. Liu} [Math. Comput. 44, 105- 124 (1985; Zbl 0564.65022)] to deal with some breakdowns of the standard Lanczos algorithm for the solution of the eigenvalue problems. The author presents a modification of the LAL algorithm which maximizes the lower bound of \(p_ i^ Tq_ i\), where \(p_ i\) and \(q_ i\) are vectors obtained by some normalization of those generated by the Lanczos algorithm. Then it is shown that the algorithm minimizes an error estimate based on a result by \textit{W. Kahan}, \textit{B. N. Parlett}, and \textit{E. Jiang} [SIAM J. Numer. Anal. 19, 470-484 (1982; Zbl 0483.65024)]. The relation of the algorithm to Arnoldi's method is explained. The paper is supplied with results of numerical experiments.