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# A Look-Ahead Lanczos Algorithm for Unsymmetric Matrices

Beresford N. Parlett, Derek R. Taylor and Zhishun A. Liu
Mathematics of Computation
Vol. 44, No. 169 (Jan., 1985), pp. 105-124
DOI: 10.2307/2007796
Stable URL: http://www.jstor.org/stable/2007796
Page Count: 20
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## Abstract

The two-sided Lanczos algorithm sometimes suffers from serious breakdowns. These occur when the associated moment matrix does not permit triangular factorization. We modify the algorithm slightly so that it corresponds to using a $2 \times 2$ pivot in triangular factorization whenever a $1 \times 1$ pivot would be dangerous. The likelihood of breakdown is greatly reduced. The price paid is that the tridiagonal matrix produced by the algorithm now has bumps whenever a $2 \times 2$ pivot is used. Experiments with several versions of the algorithm on a variety of matrices are described, including some large problems arising in the study of plasma instability.

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