Block second-order Krylov subspace methods for large-scale quadratic eigenvalue problems

In this paper, we first introduce a block second-order Krylov subspace Gm1(A,B;Q1) based on a pair of square matrices A and B and an orthonormal matrix Q1. Then we present a block second-order Arnoldi procedure for generating an orthonormal basis of Gm1(A,B;Q1) and a block second-order biorthogonalization procedure for generating biorthonormal basis of Gm1(A,B;Q1) and Gm1(AT,BT;P1). By applying the projection techniques, we derive two block second-order Krylov subspace methods for solving a large-scale quadratic eigenvalue problem (QEP). These methods are applied to the QEP directly. Hence they preserve essential structures and properties of the QEP. Some theoretical results are given. Numerical experiments report the effectiveness of these methods.