Some fast Toeplitz least-squares algorithms

James G. Nagy; Robert J. Plemmons

doi:10.1117/12.49810

1 December 1991 Some fast Toeplitz least-squares algorithms

James G. Nagy, Robert J. Plemmons

Proceedings Volume 1566, Advanced Signal Processing Algorithms, Architectures, and Implementations II; (1991) https://doi.org/10.1117/12.49810
Event: San Diego, '91, 1991, San Diego, CA, United States

Abstract

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 T^TT. Each is obtained without forming T^TT. 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.

Citation Download Citation

James G. Nagy and Robert J. Plemmons "Some fast Toeplitz least-squares algorithms", Proc. SPIE 1566, Advanced Signal Processing Algorithms, Architectures, and Implementations II, (1 December 1991); https://doi.org/10.1117/12.49810

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