A Refined Iterative Block Second-Order Arnoldi Method

In this paper, we consider the numerical methods for the solution of the large-scale quadratic eigenvalue problems. First, we generalize the second-order Krylov subspace K m( A, B; u ) based on a pair of square matrices A , B and a vector u to K m( A, B; u , w) based on a pair of square matrices A, B and a pair of vectors u , w , and present the iterative second-order Arnoldi method. Then, we use a pair of rectangular matrices U and W instead of a pair of vectors u and w to define the block second-order Krylov subspace K m( A, B; U , W ). A block second-order Arnoldi process is given to generate an orthonormal basis of K m( A, B ; U , W ). By applying the Rayleigh-Ritz orthogonal projection technique, we derive an iterative block second-order Arnoldi method for solving large-scale quadratic eigenvalue problems. Finally, using the refined projection principle, we improve iterative block second-order Arnoldi method and present a refined iterative block second-order Arnoldi method. Some numerical examples are given to demonstrate the efficiency of the proposed algorithms.