Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of symmetric positive deﬁnite Toeplitz matrices. Several algorithms have been proposed in the literature. Many of them compute the smallest eigenvalue in an iterative fashion, 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 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 latter computation is eﬃciently accomplished by the generalized Schur algorithm.