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Orthogonal similarity transformation into block‐semiseparable matrices of semiseparability rank k
Author(s) -
Van Barel M.,
Van Camp E.,
Mastronardi N.
Publication year - 2005
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.461
Subject(s) - mathematics , lanczos resampling , eigenvalues and eigenvectors , orthogonality , matrix (chemical analysis) , matrix similarity , subspace topology , block (permutation group theory) , combinatorics , rank (graph theory) , lanczos algorithm , orthonormality , krylov subspace , similarity (geometry) , algorithm , orthogonal matrix , iterative method , orthonormal basis , mathematical analysis , computer science , partial differential equation , orthogonal basis , geometry , materials science , quantum mechanics , image (mathematics) , artificial intelligence , composite material , physics
Very recently, an algorithm, which reduces any symmetric matrix into a semiseparable one of semi‐ separability rank 1 by similar orthogonality transformations, has been proposed by Vandebril, Van Barel and Mastronardi. Partial execution of this algorithm computes a semiseparable matrix whose eigenvalues are the Ritz‐values obtained by the Lanczos' process applied to the original matrix. Also a kind of nested subspace iteration is performed at each step. In this paper, we generalize the above results and propose an algorithm to reduce any symmetric matrix into a similar block‐semiseparable one of semiseparability rank k , with k ∈ ℕ, by orthogonal similarity transformations. Also in this case partial execution of the algorithm computes a block‐semiseparable matrix whose eigenvalues are the Ritz‐values obtained by the block‐Lanczos' process with k starting vectors, applied to the original matrix. Subspace iteration is performed at each step as well. Copyright © 2005 John Wiley & Sons, Ltd.