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

TL;DR

This paper introduces implicitly restarted generalized second-order Arnoldi algorithms for quadratic eigenvalue problems, demonstrating their efficiency and superiority over existing methods through numerical experiments.

Contribution

It develops new implicitly restarted GSOAR and RGSOAR algorithms with specific shifts, improving solution accuracy and computational efficiency for QEPs.

Findings

IRGSOAR outperforms GSOAR in accuracy and efficiency.

Restarted algorithms are more efficient than non-restarted versions.

Both methods outperform Arnoldi linearization in tests.

Abstract

We investigate the generalized second-order Arnoldi (GSOAR) method, a generalization of the SOAR method proposed by Bai and Su [{\em SIAM J. Matrix Anal. Appl.}, 26 (2005): 640--659.], and the Refined GSOAR (RGSOAR) method for the quadratic eigenvalue problem (QEP). The two methods use the GSOAR procedure to generate an orthonormal basis of a given generalized second-order Krylov subspace, and with such basis they project the QEP onto the subspace and compute the Ritz pairs and the refined Ritz pairs, respectively. We develop implicitly restarted GSOAR and RGSOAR algorithms, in which we propose certain exact and refined shifts for respective use within the two algorithms. Numerical experiments on real-world problems illustrate the efficiency of the restarted algorithms and the superiority of the restarted RGSOAR to the restarted GSOAR. The experiments also demonstrate that both IGSOAR and IRGSOAR generally perform much better than the implicitly restarted Arnoldi method applied to the corresponding linearization problems, in terms of the accuracy and the computational efficiency.