Algorithms for rank-de cient and ill-conditioned Toeplitz least-squares and QR factorization

In this paper we present two algorithms-one to compute the QR factorization of nearly rank-deecient Toeplitz and block Toeplitz matrices and the other to compute the solution of a severely ill-conditioned Toeplitz least-squares problem. The rst algorithm is based on adapting the generalized Schur algorithm to Cauchy-like matrices and has some rank-revealing capability. The other algorithm is based on adapting the augmented systems method to Toeplitz matrices. This algorithm does not suuer from the numerical inaccuracy of most Levinson-and Schur-based schemes proposed in the literature thus far because it avoids forming the normal equations either implicitly or explicitly. Some comparisons with more recent Toeplitz least-squares algorithms are also presented.