Volume 45, pp. 16-32, 2016.

New filtering strategies for implicitly restarted Lanczos iteration

Alex Breuer

Abstract

Implicitly-restarted Lanczos iteration methods are among the most powerful methods for finding a small number of eigenpairs of Hermitian matrices. These methods enjoy strong convergence and have manageable memory requirements. Implicitly-restarted Lanczos methods naturally find approximations to both the largest and smallest eigenpairs of the input matrix, but stagnation of Ritz values may lead to slow convergence, especially when one only wants eigenvalues from one end of the spectrum and memory is constrained. We develop a filtering strategy that breaks Ritz value stagnation for these one-sided eigenvalue problems. We demonstrate the reduction in matrix-vector product costs and note that this new variant has marked advantages when memory is constrained and when matrix-vector products are expensive.

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Key words

Implicitly-restarted Lanczos, polynomial filtering, Hermitian eigenvalue problems, Krylov subspaces, Chebyshev filtering

AMS subject classifications

65F15, 15A18, 90C99

Links to the cited ETNA articles

[5]Vol. 7 (1998), pp. 124-140 J. Baglama, D. Calvetti, and L. Reichel: Fast Leja points
[15]Vol. 2 (1994), pp. 1-21 D. Calvetti, L. Reichel, and D. C. Sorensen: An implicitly restarted Lanczos method for large symmetric eigenvalue problems

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