Volume 5, Issue 2
An Inexact Shift-and-Invert Arnoldi Algorithm for Large Non-Hermitian Generalised Toeplitz Eigenproblems

Ting-Ting Feng, Gang Wu & Ting-Ting Xu

East Asian J. Appl. Math., 5 (2015), pp. 160-175.

Published online: 2018-02

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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

Toeplitz matrix, generalised eigenproblem, shift-and-invert Arnoldi method, GohbergSemencul formula.

  • AMS Subject Headings

65F15, 65F10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-5-160, author = {}, title = {An Inexact Shift-and-Invert Arnoldi Algorithm for Large Non-Hermitian Generalised Toeplitz Eigenproblems}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {5}, number = {2}, pages = {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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.010914.130415a}, url = {http://global-sci.org/intro/article_detail/eajam/10791.html} }
TY - JOUR T1 - An Inexact Shift-and-Invert Arnoldi Algorithm for Large Non-Hermitian Generalised Toeplitz Eigenproblems JO - East Asian Journal on Applied Mathematics VL - 2 SP - 160 EP - 175 PY - 2018 DA - 2018/02 SN - 5 DO - http://doi.org/10.4208/eajam.010914.130415a UR - https://global-sci.org/intro/article_detail/eajam/10791.html KW - Toeplitz matrix, generalised eigenproblem, shift-and-invert Arnoldi method, GohbergSemencul formula. AB -

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.

Ting-Ting Feng, Gang Wu & Ting-Ting Xu. (1970). An Inexact Shift-and-Invert Arnoldi Algorithm for Large Non-Hermitian Generalised Toeplitz Eigenproblems. East Asian Journal on Applied Mathematics. 5 (2). 160-175. doi:10.4208/eajam.010914.130415a
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