Preconditioned Lanczos method for generalized Toeplitz eigenvalue problems

We employ the sine transform-based preconditioner to precondition the shifted Toeplitz matrix An−ρBnAn−ρBn involved in the Lanczos method to compute the minimum eigenvalue of the generalized symmetric Toeplitz eigenvalue problem Anx=λBnxAnx=λBnx, where AnAn and BnBn 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.