Implicitly Restarted Hybrid (Block) SOAR Methods For Quadratic Eigenvalue Problems

The Second Order Arnoldi method (SOAR) solves the Quadratic eigenvalue problems (QEP)with rapid convergence and less computational cost and storage. However, it is not clear how toperform a proper restart. In this paper, we introduce an improved second-order Krylov subspaceK_m(A, B; u, w)based on matrices A、 B and a pair of vectors u、 w, apply the Q-Arnoldiprocedure to generate an orthonormal basis of K_m(A, B; u, w), and present a hybrid SOAR methodfor solving QEP by using the Rayleigh-Ritz orthogonal projection technique. In order to computemultiple or cluster eigenvalues, we establish a block second-order Krylov subspace K_m(A, B; U, W)based on the matrices A、B and a pair of matrics U、W, develop a block version of the Q-Arnoldiprocedure to generate an orthonormal basis of K_m(A, B; U, W), and provide a hybrid block SOARmethod for solving QEP. In order to compute a few of extreme eigenvalues and reduce thecomputational cost, using the implicitly restarted technique, we present an implicitly restarted hybridSOAR method and its block version. To perform efficiently deflation, we apply the non-equivalencelow-rank deflation technique to the implicitly restarted hybrid SOAR method and its block version.Numerical results show that the proposed methods are effective.