An Inexact Shift-and-Invert Arnoldi Algorithm for Large Non-Hermitian Generalised Toeplitz Eigenproblems
Year: 2015
East Asian Journal on Applied Mathematics, Vol. 5 (2015), Iss. 2 : pp. 160–175
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/eajam.010914.130415a
East Asian Journal on Applied Mathematics, Vol. 5 (2015), Iss. 2 : pp. 160–175
Published online: 2015-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 16
Keywords: Toeplitz matrix generalised eigenproblem shift-and-invert Arnoldi method GohbergSemencul formula.
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