TY - JOUR
T1 - The QR iteration method for Hermitian quasiseparable matrices of an arbitrary order
AU - Eidelman, Yuli
AU - Gohberg, Israel
AU - Olshevsky, Vadim
N1 - Funding Information:
∗ Corresponding author. E-mail addresses: [email protected] (Y. Eidelman), [email protected] (I. Gohberg), [email protected] (V. Olshevsky). 1 The draft of this paper was completed during the visit of Y. Eidelman at U Conn in September 2003. 2 Supported by the NSF grant 0242518.
PY - 2005/7/15
Y1 - 2005/7/15
N2 - The QR iteration method for tridiagonal matrices is in the heart of one classical method to solve the general eigenvalue problem. In this paper we consider the more general class of quasiseparable matrices that includes not only tridiagonal but also companion, comrade, unitary Hessenberg and semiseparble matrices. A fast QR iteration method exploiting the Hermitian quasiseparable structure (and thus generalizing the classical tridiagonal scheme) is presented. The algorithm is based on an earlier work [Y. Eidelman and I. Gohberg, A modification of the Dewilde-van der Veen method for inversion of finite structured matrices, Linear Algebra Appl. 343-344 (2002) 419-450], and it applies to the general case of Hermitian quasiseparable matrices of an arbitrary order. The algorithm operates on generators (i.e., a linear set of parameters defining the quasiseparable matrix), and the storage and the cost of one iteration are only linear. The results of some numerical experiments are presented. An application of this method to solve the general eigenvalue problem via quasiseparable matrices will be analyzed separately elsewhere.
AB - The QR iteration method for tridiagonal matrices is in the heart of one classical method to solve the general eigenvalue problem. In this paper we consider the more general class of quasiseparable matrices that includes not only tridiagonal but also companion, comrade, unitary Hessenberg and semiseparble matrices. A fast QR iteration method exploiting the Hermitian quasiseparable structure (and thus generalizing the classical tridiagonal scheme) is presented. The algorithm is based on an earlier work [Y. Eidelman and I. Gohberg, A modification of the Dewilde-van der Veen method for inversion of finite structured matrices, Linear Algebra Appl. 343-344 (2002) 419-450], and it applies to the general case of Hermitian quasiseparable matrices of an arbitrary order. The algorithm operates on generators (i.e., a linear set of parameters defining the quasiseparable matrix), and the storage and the cost of one iteration are only linear. The results of some numerical experiments are presented. An application of this method to solve the general eigenvalue problem via quasiseparable matrices will be analyzed separately elsewhere.
KW - Eigenvalue problem
KW - QR iteration
KW - Quasiseparable matrices
KW - Semiseparable matrices
KW - Tridiagonal matrices
UR - http://www.scopus.com/inward/record.url?scp=20344378938&partnerID=8YFLogxK
U2 - 10.1016/j.laa.2005.02.037
DO - 10.1016/j.laa.2005.02.037
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AN - SCOPUS:20344378938
SN - 0024-3795
VL - 404
SP - 305
EP - 324
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
IS - 1-3
ER -