Bounds on the extreme eigenvalues of real symmetric Toeplitz matrices

We derive upper and lower bounds on the smallest and largest eigenvalues, r espectively, of real symmetric Toeplitz matrices. The bounds are first obta ined for positive-definite matrices and then extended to the general real s ymmetric case. They are computed as the roots of rational and polynomial ap proximations to spectral, or secular, equations for the symmetric and antis ymmetric parts of the spectrum; this leads to separate bounds on the even a nd odd eigenvalues. We also present numerical results.