A Restarted Generalized Second-order Biorthogonalization Procedure For The Quadratic Eigenvalue Problems

The second-order Krylov subspace (?)m(A, B; u1, u2) based on a pair of square matrices A and B and a pair of vectors u1, u2is first reviewed. Then we extend this subspace by introducing the generalized second-order right and left Krylov subspaces which are denoted as (?)m(A, B;u1, u2) and (?)m(AT, BT; v1, v2), respectively. Here (?)m(A, B; u1, u2) is a subspace spanned by a sequence of vectors defined via a second-order linear homogenous recurrence relation with coefficient matrices A and B and two initial vectors u1and u2, which is a generalization of the second-order Krylov subspace (?)m(A, B; u) with an initial vector u.After giving the definition of the generalized second-order right and left Krylov subspaces, we investigate the differences and relations between the subspaces and the standard Krylov subspace. Then a modified second-order biorthogonalization proce-dure is given, which is based on the second-order biorthogonalization method(SOB)[18] to generate the biorthogonal bases of the Krylov subspaces. With less storage and computation requirements, an improved version of the procedure is presented as well.With the oblique projection technique [12,17] and restarted initial vectors [13], a restarted generalized second-order biorthogonal method(RMSOB(m)) is proposed for solving the large-scale quadratic eigenvalue problems. The advantages of the method are as follows:1. By solving the quadratic eigenvalue problem(QEP) directly, it preserves essential structures and properties (such as symmetric and so on) of the QEP.2. The right and left eigenvectors of the QEP can be obtained simultaneously.3. For the SOB like method, the increasing of iteration can lead to a big storage and computation burden, as well as missing of the biorthogonality. The restarted version of our method can avoid these weaknesses.Theoretical analysis and numerical experiments are displayed to illustrate the effectiveness of the proposed methods.