A Schur-based algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite Toeplitz matrix

Abstract

Recent progress in signal processing and estimation has generated c onsiderable interest in the problem of computing the smallest eigenvalue of symmetric positive definite Toeplitz matrices. Several algorithms have been proposed in the literature. They compute the smallest eigenvalue in an iterative fashion, many of them relying on the Levinson–Durbin solution of sequences of Yule–Walker systems. Exploiting the properties of two algorithms recently developed for estimating a lower and an upper bound of the smallest singular value of upper triangular matrices, respectively, an algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite Toeplitz matrix is derived. The algorithm relies on the computation of the R factor of the QR-factorization of the Toeplitz matrix and the inverse of R. The simultaneous computation of R and R −1 is efficiently accomplished by the generalized Schur algorithm.