Analysis of the Lanczos Error Bounds and Its Application to the Explicitly Restarted Lanczos Algorithm
LExpRes is a k-step explicit restart variant of the Lanczos algorithm. In this method a periodic/selective reorthogonalization strategy is adopted in order to dampen the affects of instability incurred by a loss of orthogonality among the Lanczos vectors. Despite this however, round-off error still effects its performance particularly on the estimated Lanczos error bounds that are considerably different from the true error bound. In this paper a detailed analysis of the Lanczos error bound is presented and a new more realistic error proposed for determining a ‘good’ Ritz vector. A scheme is also proposed that uses this bound to derive start vectors for subsequent eigenpairs. The performance of the implementation of this scheme on the CRAY-T3E is critically assessed and some conclusions are drawn.
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