Orthogonal similarity transformation into block-semiseparable matrices of semiseparability rankk

Abstract

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.