Bounds for the Minimum Eigenvalue of

In a recent paper Melman 12] derived upper bounds for the smallest eigen-value of a real symmetric Toeplitz matrix in terms of the smallest roots of rational and polynomial approximations of the secular equation f() = 0, the best of which being constructed by the (1; 2)-Pad e approximation of f. In this paper we prove that this bound is the smallest eigenvalue of the projection of the given eigenvalue problem onto a Krylov space of T ?1 n of dimension 3. This interpretation of the bound suggests enhanced bounds of increasing accuracy. They can be substantially improved further by exploiting symmetry properties of the principal eigenvector of T n .