IMPROVED BISECTION EIGENVALUE METHOD FOR BAND SYMMETRIC TOEPLITZ MATRICES^∗

We apply a general bisection eigenvalue algorithm, developed for Hermitian matrices with quasiseparable representations, to the particular case of real band symmetric Toeplitz matrices. We show that every band symmetric Toeplitz matrix Tq with bandwidth q admits the representation Tq = Aq + Hq, where the eigendata of Aq are obtained explicitly and the matrix Hq 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 Tq and the known eigenvalues of the matrix Aq. This allows us to essentially improve the performance of the bisection eigenvalue algorithm. We also present an algorithm to compute the corresponding eigenvectors.