View in EDS

A multiple shift -step for structured rank matrices

Saved in:

Bibliographic Details
Title:	A multiple shift -step for structured rank matrices
Authors:	Vandebril, Raf¹, Van Barel, Marc¹, Mastronardi, Nicola²
Source:	Journal of Computational & Applied Mathematics. Jan2010, Vol. 233 Issue 5, p1326-1344. 19p.
Subjects:	Semiseparable matrices, Eigenvalues, Mathematical transformations, Algorithms, Nonsymmetric matrices, Numerical analysis
Abstract:	Abstract: Eigenvalue computations for structured rank matrices are the subject of many investigations nowadays. There exist methods for transforming matrices into structured rank form, -algorithms for semiseparable and semiseparable plus diagonal form, methods for reducing structured rank matrices efficiently to Hessenberg form and so forth. Eigenvalue computations for the symmetric case, involving semiseparable and semiseparable plus diagonal matrices have been thoroughly explored. A first attempt for computing the eigenvalues of nonsymmetric matrices via intermediate Hessenberg-like matrices (i.e. a matrix having all subblocks in the lower triangular part of rank at most one) was restricted to the single shift strategy. Unfortunately this leads in general to the use of complex shifts switching thereby from real to complex operations. This paper will explain a general multishift implementation for Hessenberg-like matrices (semiseparable matrices are a special case and hence also admit this approach). Besides a general multishift -step, this will also admit restriction to real computations when computing the eigenvalues of arbitrary real matrices. Details on the implementation are provided as well as numerical experiments proving the viability of the presented approach. [Copyright &y& Elsevier]
	Copyright of Journal of Computational & Applied Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database:	Engineering Source

Description
Abstract:	Abstract: Eigenvalue computations for structured rank matrices are the subject of many investigations nowadays. There exist methods for transforming matrices into structured rank form, -algorithms for semiseparable and semiseparable plus diagonal form, methods for reducing structured rank matrices efficiently to Hessenberg form and so forth. Eigenvalue computations for the symmetric case, involving semiseparable and semiseparable plus diagonal matrices have been thoroughly explored. A first attempt for computing the eigenvalues of nonsymmetric matrices via intermediate Hessenberg-like matrices (i.e. a matrix having all subblocks in the lower triangular part of rank at most one) was restricted to the single shift strategy. Unfortunately this leads in general to the use of complex shifts switching thereby from real to complex operations. This paper will explain a general multishift implementation for Hessenberg-like matrices (semiseparable matrices are a special case and hence also admit this approach). Besides a general multishift -step, this will also admit restriction to real computations when computing the eigenvalues of arbitrary real matrices. Details on the implementation are provided as well as numerical experiments proving the viability of the presented approach. [Copyright &y& Elsevier]
ISSN:	03770427
DOI:	10.1016/j.cam.2008.11.017