An algorithm for solving discrete-time Wiener-Hopf equations based upon Euclid´s algorithm

An algorithm for solving a discrete-time Wiener-Hopf equation is presented based upon Euclid\´s algorithm. The discrete-time Wiener-Hopf equation is a system of linear inhomogeneous equations with a given Toeplitz matrix M, a given vector b, and an unknown vector such that . The algorithm is able to find a solution of the discrete-time Wiener-Hopf equation for any type of Toeplitz matrices except for the all-zero matrix, while the Levinson algorithm and the Trench algorithm are not available when at least one of the principal submatrices of the Toeplitz matrix is singular. The algorithm gives a solution, if one exists, even when the Toeplitz matrix is singular, while the Brent-Gustavson-Yun algorithm only states that the Toeplitz matrix is singular. The algorithm requires arithmetic operations for unknowns, in the sense that the number of multiplications or divisions is directly proportional to , like the Levinson and Trench algorithms. Furthermore, a faster algorithm is also presented based upon the half greatest common divisor algorithm, and hence it requires arithmetic operations, like the Brent-Gustavson-Yun algorithm.