Extreme singular values and eigenvalues of non-Hermitian block Toeplitz matrices
type: Item
instance of: scholarly article
title: Extreme singular values and eigenvalues of non-Hermitian block Toeplitz matrices
zbMATH Open document ID: 0937.15013
DOI: 10.1016/S0377-0427(99)00104-1
author: Paolo Tilli
author: Stefano Serra Capizzano
published in: Journal of Computational and Applied Mathematics
publication date: 2000-04-06T00:00:00Z
review text: The asymptotic behaviour of singular values and eigenvalues of non-Hermitian block Toeplitz matrices \(\{T_n(f)\}\) generated by a multivariate matrix valued function \(f:Q^p\) \(\to\) \(\mathbb{C}\)\(^{h\times k}\), \(Q=(-\pi,\pi)\) is considered. It is proved that the interval of singular values is a proper cluster when \(f\) is continuous or belongs to a Klein algebra. Results for the minimal singular value of \(\{T_n(f)\}\) in the non-Hermitian case are discussed and second-order ergodic formulas and localization of eigenvalues of preconditioned matrices \(\{T^{-1}_n(g)T_n(f)\}\) are presented. The analysis of the number of outlying singular values is important to establish a priori bounds for the number of iterations we have to expect when a conjugate gradient method is used for normal equations.
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
zbMATH DE Number: 1367893
zbMATH Keywords: non-Hermitian block Toeplitz matrices
zbMATH Keywords: Klein algebra
zbMATH Keywords: singular value
zbMATH Keywords: preconditioning
zbMATH Keywords: localization of eigenvalues
zbMATH Keywords: conjugate gradient method
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1367893