The preconditioned inverse iteration for hierarchical matrices / Peter Benner, Thomas Mach
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870173855
URN
urn:nbn:de:gbv:3:2-63784
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Autorin / Autor
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Erschienen
Magdeburg : Max Planck Institute for Dynamics of Complex Technical Systems, February 11, 2011
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1 Online-Ressource (16 Seiten = 0,28 MB) : Diagramme
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Sprache
eng
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Inhaltliche Zusammenfassung
Abstract: The preconditioned inverse iteration [Ney01] is an efficient method to compute the smallest eigenpair of a symmetric positive definite matrix ℋ. Here we use this method to find the smallest eigenvalues of a hierarchical matrix [Hac99]. The storage complexity of the data-sparse ℋ-matrices is almost linear. We use ℋ-arithmetic to precondition with an approximate inverse of M or an approximate Cholesky decomposition of M. In general ℋ-arithmetic is of linear-polylogarithmic complexity, so the computation of one eigenvalue is cheap. We extend the ideas to the computation of inner eigenvalues by computing an invariant subspaces S of (M-\mu I)² by subspace preconditioned inverse iteration. The eigenvalues of the generalized matrix Rayleigh quotient \muM(S) are the wanted inner eigenvalues of M. The idea of using (M-\mu I)² instead of M is known as folded spectrum method [WanZ94]. Numerical results substantiate the convergence properties and show that the computation of the eigenvalues is superior to existing algorithms for non-sparse matrices.
Schriftenreihe
Max Planck Institute Magdeburg Preprints ; 11-01 ppn:870173030