Preconditioned Lanczos method for generalized Toeplitz eigenvalue problems

Abstract

a b s t r a c tWe employ the sine transform-based preconditioner to precondition the shifted Toeplitz matrix A n − ρB n involved in the Lanczos method to compute the minimum eigenvalue of the generalized symmetric Toeplitz eigenvalue problem A n x = λB n x, 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 transformbased 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.

An inexact Krylov subspace method for large generalized Hankel eigenproblems

“…The test matrices used in this example are derived from the references [33,34]. We generate the Hankel matrix A using the even function θ 2 defined on [0, π], which was introduced in Example 1.…”