The second-order biorthogonalization procedure and its application to quadratic eigenvalue problems

Given a pair of matrices and starting vectors, we present a procedure to generate the biorthonormal basis of the second-order right and left Krylov subspaces. The application is to solve the large-scale quadratic eigenvalue problems via oblique projection technique. This method can take full advantage of the sparseness of large-scale system as well as the superior convergence behavior of Krylov subspace based methods by implicit linearization, which makes the solution acceptable in terms of both cost and time.