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We investigate the generalized second-order Arnoldi (GSOAR) method, ageneralization of the SOAR method proposed by Bai and Su [{\em SIAM J. MatrixAnal. Appl.}, 26 (2005): 640--659.], and the Refined GSOAR (RGSOAR) method forthe quadratic eigenvalue problem (QEP). The two methods use the GSOAR procedureto generate an orthonormal basis of a given generalized second-order Krylovsubspace, and with such basis they project the QEP onto the subspace andcompute the Ritz pairs and the refined Ritz pairs, respectively. We developimplicitly restarted GSOAR and RGSOAR algorithms, in which we propose certainexact and refined shifts for respective use within the two algorithms.Numerical experiments on real-world problems illustrate the efficiency of therestarted algorithms and the superiority of the restarted RGSOAR to therestarted GSOAR. The experiments also demonstrate that both IGSOAR and IRGSOARgenerally perform much better than the implicitly restarted Arnoldi methodapplied to the corresponding linearization problems, in terms of the accuracyand the computational efficiency.

Implicitly Restarted Generalized Second-order Arnoldi Type Algorithms for the Quadratic Eigenvalue Problem