Improved bisection eigenvalue method for band symmetric Toeplitz matrices

Abstract

. We apply a general bisection eigenvalue algorithm, developed for Hermitian matrices with quasisep- arable representations, to the particular case of real band symmetric Toeplitz matrices. We show that every band symmetric Toeplitz matrix T q with bandwidth q admits the representation T q = A q + H q , where the eigendata of A q are obtained explicitly and the matrix H q has nonzero entries only in two diagonal blocks of size ( q − 1) × ( q − 1) . Based on this representation, one obtains an interlacing property of the eigenvalues of the matrix T q and the known eigenvalues of the matrix A q . This allows us to essentially improve the performance of the bisection eigenvalue algorithm. We also present an algorithm to compute the corresponding eigenvectors.