Fast algorithms for QR and Cholesky factors of Toeplitz operators

The Levinson recursions, the Cybenko recursions and the Le Roux-Gueguen recursions provide alternative ways of computing reflection coefficients for stationary time series. In this paper we show that when a Toeplitz correlation matrix is a product of two Toeplitz data matrices, as in the correlation method of linear prediction, then the Levinson recursions may be used to derive the Cybenko recursions, and the Cybenko recursions may be used to derive the Le Roux-Gueguen recursions. We explore the close relation between QR and Cholesky algorithms in the Toeplitz case and we compare their respective numerical properties when run in finite precision arithmetic.