Integer Levinson algorithms for Toeplitz and certain Toeplitz-like matrices

The paper presents an integer Levinson algorithm for certain Toeplitz-like (quasi-Toeplitz) matrices. The integer preserving (IP) property means that for a Toeplitz matrix with (complex or real) integers, the algorithm is completed over integers without encountering quotients. The algorithm also produces triangular factorization of the inverse matrix with integer matrices. The derivation begins with an intermediate algorithm that is IP simply because it is division-free but it produces integers whose size increases at a severe rate. Next, the main algorithm is obtained by identifying and recursively dividing out common integers that the division-free algorithm is shown to produce systematically. The result is an efficient integer algorithm with integers of least size. This way of derivation also provides a constructive proof for the IP property of the algorithm. The integer Levinson algorithms for a non-symmetric (real or complex) Toeplitz is deduced from the more general main result from where the integer algorithm for the Hermitian Toeplitz case follows readily.