ON THE METHODS FOR SOLVING YULE-WALKER EQUATIONS

We review the three well-known fast algorithms for the solution of Yule-Walker (YW) equations: the Levinson, Euclidean, and Berlekamp-Massey algorithms, and show the relation between each of them and the Pade approximation problem. This connection has already been noticed for some of them, but we intend here to offer a synthetic view of these fast algorithms. We classify the algorithms solving YW equations with reference to three criteria, namely: 1) the path they follow in the Pade table; 2) the organization of the computation: we distinguish between one-pass and two-pass algorithms; and 3) the auxiliary variables used: some algorithms use the backward predictor of same degree as intermediate variable for computing the forward predictor (or vice versa), while others use two predictors of the same type but of successive degrees. This classification shows that the set of known classical algorithms is not complete, and we propose the missing variants. With these variants of the Berlekamp-Massey and Euclid algorithms, we are able to obtain both forward and backward predictors without additional cost. Furthermore, we give a unified representation of the two-pass algorithms, in such a way that the application of the divide and conquer strategy becomes straightforward. A general doubling algorithm which represents all the associated doubling algorithms in an exhaustive way is provided.