A generalization of Levinson algorithm for solving Toeplitz systems

A new algorithm which solves a Toeplitz system having t unknowns with computational complexity of 0(t²) is proposed. The solving algorithm is based upon iterative solutions of Toeplitz systems for sub-Toeplitz-matrices of the given Toeplitz matrix and works well even when some of the sub-Toeplitz-matrices is singular. That is, the solving algorithm is a generalized version of the Levinson algorithm in the sense that its solving process is quite similar to that of the Levinson algorithm and it overcomes the defect which the Levinson algorithm possesses. We may expect that the accuracy of the new solving algorithm is superior to those of the Brent-Gustavson-Yun algorithm and the solving algorithm based upon Euclid algorithm, when the given Toeplitz matrix is a type of matrices such as a covariance matrix, since the former algorithm uses sub-Toeplitz-matrices and the latter algorithms use sub-Hankel-matrices.