A fast algorithm for solving Toeplitz system of equations

A fast algorithm for the solution of Toeplitz system of equations is presented. The algorithm requires order
N (\log N)^{2}
computations where N is the number of equations. For banded Toeplitz matrices the order of computations is reduced to only
N \log N + m (\log m)^2
where 2m is the maximum number of nonzero principal subdiagonals of the Toeplitz matrix. The algorithm is in essence a fast implementation of the Trench algorithm in reverse. Thus the algorithm involves imbedding of the given matrix in a cyclic matrix and a fast HD (Half Divisor) algorithm to compute the first row of the inverse matrix. The desired solution is then obtained directly from the first row by applying Fast Fourier Transform techniques in order
N \log N
computations. Finally, the extension of the algorithm to block Toeplitz matrices is also presented.