A normalization scheme for the non-symmetric s-step Lanczos algorithm.

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Abstract

The Lanczos algorithm is among the most frequently used techniques for computing a few dominant eigenvalues of a large sparse nonsymmetric matrix. When variants of this algorithm are implemented on distributedmemory computers, the synchronization time spent in computing dot products is increasingly limiting the parallel scalability. The goal of sstep algorithms is to reduce the harmful influence of dot products on the parallel performance by grouping several of these operations for joint execution; thus, plummeting synchronization time when using a large number of processes. This paper extends the nonsymmetric sstep Lanczos method introduced by Kim and Chronopoulos (J. Comput. Appl. Math., 42(3), 357374, 1992) by a novel normalization scheme. Compared to the unnormalized algorithm, the normalized variant improves numerical stability and reduces the possibility of breakdowns.