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 - <p style="text-align: justify;">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.</p>