Numerical Solution of the Eigenvalue Problem for Symmetric Rationally Generated Toeplitz matrices

A numerical method is proposed for finding all eigenvalues of symmetric Toeplitz matrices $T_n = ( t_{j - i} )_{ij = 1}^n $, where the $\{ t_j \}$ are the coefficients in a Laurent expansion of a rational function. Matrices of this kind occur, for example, as covariance matrices of ARMA processes. The technique rests on a representation of the characteristic polynomial as $\det ( \lambda I_n - T_n ) = W_n G_{0n} G_{1n} $ in which $G_{0n} ( \lambda ) = 0$ for the eigenvalues of $T_n $ associated with symmetric eigenvectors, $G_{1n} ( \lambda ) = 0$ for those associated with skew-symmetric eigenvectors, both functions are free of extreme variations, and both can be computed with cost independent of n. It is proposed that root finding techniques be used to compute the zeros of $G_{0n} $ and $G_{1n} $. Numerical experiments indicate that the method may be useful.