Bevilacqua, Roberto and Del Corso, Gianna M. (2003) Existence, Uniqueness and Algorithms for Matrix Unitary Reduction to Semiseparable Form. Technical Report del Dipartimento di Informatica . Università di Pisa, Pisa, IT.
Postscript (GZip) - Published Version Available under License Creative Commons Attribution No Derivatives. Download (283Kb) |
Abstract
This paper concerns the study of a unitary transformation from a generic symmetric matrix $A$ into a semiseparable matrix. The problem is studied both theoretically and from an algorithmic point of view. In particular, we first give a proof of the existence of such a transformation and then we discuss the uniqueness of such transformation proving an Implicit-$Q$ Theorem for semiseparable matrices. Finally, we study structural properties of the factors of the $QR$-decomposition of a semiseparable matrix. These properties allows us to design a method based on the $QR$ iterations applied to a semiseparable matrix for reducing a symmetric matrix to semiseparable form. This method has the same asymptotic cost of the reduction of a symmetric matrix to tridiagonal form. Once the transformation has been accomplished, if one is interested in computing the eigenvalues each further $QR$ iteration can be done in linear time.
Item Type: | Book |
---|---|
Uncontrolled Keywords: | Eigenvalues, QR iteration, semiseparable matrices, unitary reduction |
Subjects: | Area01 - Scienze matematiche e informatiche > INF/01 - Informatica |
Divisions: | Dipartimenti (until 2012) > DIPARTIMENTO DI INFORMATICA |
Depositing User: | dott.ssa Sandra Faita |
Date Deposited: | 17 Dec 2014 16:37 |
Last Modified: | 17 Dec 2014 16:37 |
URI: | http://eprints.adm.unipi.it/id/eprint/2099 |
Repository staff only actions
View Item |