Vector algorithms for computing QR and Cholesky factors of close-to-Toeplitz matrices

We review the coupled vector recursions for the Levinson, Cybenko and LeRoux-Gueguen algorithms that Cholesky and QR factor Toeplitz matrices. We generalize the algorithms to include close-to-Toeplitz matrices, and show that these generalized algorithms may be imbedded in the same vector recursions. Then, depending on how the initial conditions are set, and how the reflection coefficients are computed, one gets either generalized Levinson recursions [1], generalized LeRoux-Gueguen recursions, or generalized Cybenko recursions. The Cybenko recursions produce an interesting way to compute the generalized reflection coefficients and also lead to generalized LeRoux-Gueguen recursions for all the cases of linear prediction.