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.