Computing Eigenvalues of Banded Symmetric Toeplitz Matrices

Abstract

A recently proposed algorithm for the computation of the eigenvalues of symmetric banded Toeplitz matrices is investigated.The basic idea of the algorithm is to embed the Toeplitz matrix in a symmetric circulant matrix of higher order. After having computed the spectral decomposition of the circulant matrix—which is trivial since the eigenvectors of circulants are known a priori and the eigenvalues can be obtained by Fourier transform—the Toeplitz eigenvalue problem is treated as a restricted eigenvalue problem. In doing this, use can be made of the theory for intermediate problems of Weinstein and Aronszajn and its recent refinements.Since the main part of the proposed algorithm consists of independent searches for zeros in disjoint real intervals, the algorithm is well suited for parallel computers. Implementations of the algorithm are discussed and some numerical results are given. The (sequential) complexity of the computation of s eigenvalues of the Toeplitz matrix T is $O(sr(n + r^2 ))$, where n is the order and rthe bandwidth of T.