Some fast Toeplitz least-squares algorithms
James G. Nagy, Robert J. Plemmons · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1991
We study fast preconditioned conjugate gradient (PCG) methods for solving least squares problems min (parallel)b-T(chi) (parallel)2, where T is an m X n Toeplitz matrix of rank n. Two circulant preconditioners are suggested: one, denoted by P, is based on a block partitioning of T and the other, denoted by N, is based on the displacement representation of TTT. Each is obtained without forming TTT. We prove formally that for a wide class of problems the PCG method with P converges in a small number of iterations independent of m and n, so that the computational cost of solving such Toeplitz least squares problems is O(m log n). Numerical experiments in using both P and N are reported, indicating similar good convergence properties for each preconditioner.