Numerical behaviour of Toeplitz solutions

The numerical behavior of the order-recursive algorithms for the solution of a number of typical Toeplitz systems is studied. Use is made of a novel methodology that gives a direct interpretation of the positive definiteness that leads to the explicit computation of the minimum absolute bounds of the 1-step and l-step-ahead linear predictors. Ill-conditioning is discussed, and an algorithmic scheme with improved numerical behavior is given for the l-step-ahead LP case. It is proved that the Levinson algorithm, when it is used for the solution of the least-squares finite-impulse-response filtering problem, manifests a fundamentally different numerical behavior than when it is used for the l-step-ahead problem