A Refined Second-order Arnoldi (RSOAR) Method for the Quadratic Eigenvalue Problem and Implicitly Restarted Algorithms

To implicitly restart the second-order Arnoldi (SOAR) method proposed by Bai and Su for the quadratic eigenvalue problem (QEP), it appears that the SOAR procedure must be replaced by a modified SOAR (MSOAR) one. However, implicit restarts fails to work provided that deflation takes place in the MSOAR procedure. In this paper, we first propose a Refined MSOAR (abbreviated as RSOAR) method that is based on the refined projection principle. We derive upper bounds for residual norms of the approximate eigenpairs obtained by the MSOAR and RSOAR methods. Based on them, we propose a reliable tolerance criterion for numerical breakdown that makes the MSOAR and RSOAR methods converge to a prescribed accuracy. This criterion also serves to decide numerical deflation. We consider the central issue of selecting the shifts involved when implicitly restarting the MSOAR and RSOAR algorithms. We propose the exact and refined shifts for the two algorithms, respectively, and present an effective approach to treat the deflation issue in implicit restarts, so that the implicit restarting scheme works unconditionally. Numerical examples illustrate the efficiency of the restarted algorithms and the superiority of the restarted RSOAR to the restarted MSOAR.