Computing the smallest eigenpair of a symmetric positive definite Toeplitz matrix

An algorithm for computing the smallest eigenvalue of a symmetric positive definite Toeplitz matrix is presented. The eigenvalue is approximated from below by Newton's method applied to the characteristic polynomial of the matrix. The Newton's step is calculated by a Levinson-Durbin type recursion. Simultaneously, this recursion produces a realistic error bound of the actual approximation without additional computing effort as well as a simple and efficient way to compute the associated eigenvector.