Bounds on the Extreme Eigenvalues of Real Symmetric Toeplitz Matrices

We derive upper and lower bounds on the smallest and largest eigenvalues, respectively, of real symmetric Toeplitz matrices. The bounds are rst obtained for positive-deenite matrices and then extended to the general real symmetric case. Our bounds are computed as the roots of rational and polynomial approximations to spectral, or secular, equations. The decomposition of the spectrum into even and odd parts is exploited to obtain separate bounds on the even and odd eigenvalues. This leads to signiicantly improved bounds, as illustrated by extensive numerical results.