We consider the problem of solving least squares problems involving a matrix M of small displacement rank with respect to two matrices Z1 and Z2. We develop formulas for the generators of the matrix MHM in terms of the generators of M and show that the Cholesky factorization of the matrix MHM can be computed quickly if Z1 is close to unitary and Z2 is triangular and nilpotent. These conditions are satisfied for several classes of matrices, including Toeplitz, block Toeplitz, Hankel, and block Hankel, and for matrices whose blocks have such structure. Fast Cholesky factorization enables fast solution of least squares problems, total least squares problems, and regularized total least squares problems involving these classes of matrices.
block Toeplitz matrix, Displacement rank, errors in variables method, structured total least squares, total least squares
and O'Leary, D.
Fast Algorithms for Structured Least Squares, Journal of Research (NIST JRES), National Institute of Standards and Technology, Gaithersburg, MD, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=50670
(Accessed December 3, 2023)