Preconditioned Lanczos method for generalized Toeplitz eigenvalue problems

We employ the sine transform-based preconditioner to precondition the shifted Toeplitz matrix A(n) - rho B-n involved in the Lanczos method to compute the minimum eigenvalue of the generalized symmetric Toeplitz eigenvalue problem A(n)chi = lambda B-n chi, where A(n) and B-n are given matrices of suitable sizes. The sine transform-based preconditioner can improve the spectral distribution of the shifted Toeplitz matrix and, hence, can speed up the convergence rate of the preconditioned Lanczos method. The sine transform-based preconditioner can be implemented efficiently by the fast transform algorithm. A convergence analysis shows that the preconditioned Lanczos method converges sufficiently fast, and numerical results show that this method is highly effective for a large matrix. (c) 2008 Elsevier B.V. All rights reserved.