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An Inexact Shift-and-Invert Arnoldi Algorithm for Large Non-Hermitian Generalised Toeplitz Eigenproblems

Published online by Cambridge University Press:  28 May 2015

Ting-Ting Feng ,
Gang Wu  and
Ting-Ting Xu
Ting-Ting Feng
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, P.R. China
Gang Wu*
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, P.R. China Department of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, P.R. China
Ting-Ting Xu
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, P.R. China
*
*Corresponding author. Email addresses: tofengtingting@163.com (T.-T. Feng), gangwu76@126.com, wugangzy@gmail.com (G. Wu), xutingtingdream@163.com (T.-T. Xu)
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Abstract

The shift-and-invert Arnoldi method is a most effective approach to compute a few eigenpairs of a large non-Hermitian Toeplitz matrix pencil, where the Gohberg-Semencul formula can be used to obtain the Toeplitz inverse. However, two large non-Hermitian Toeplitz systems must be solved in the first step of this method, and the cost becomes prohibitive if the desired accuracy for this step is high — especially for some ill-conditioned problems. To overcome this difficulty, we establish a relationship between the errors in solving these systems and the residual of the Toeplitz eigenproblem. We consequently present a practical stopping criterion for their numerical solution, and propose an inexact shift-and-invert Arnoldi algorithm for the generalised Toeplitz eigenproblem. Numerical experiments illustrate our theoretical results and demonstrate the efficiency of the new algorithm.

Keywords

65F1565F10Toeplitz matrixgeneralised eigenproblemshift-and-invert Arnoldi methodGohberg-Semencul formula
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 2 , May 2015 , pp. 160 - 175
Copyright
Copyright © Global-Science Press 2015 

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