Round-off error propagation in Durbin´s, Levinson´s, and Trench´s algorithms

The subject matter of this paper concerns the round-off error propagation in order n²algorithms for solving problems involving Toeplitz matrices. Since linear predictive techniques owe much of their appeal to the computational efficiency of Durbin´s, Levinson´s, and Trench´s algorithms, it is important to understand the accuracy of these methods. In what appears to be the first analysis of its kind, we derive bounds on the errors due to round-off and discuss their merits and tightness. In particular, it is shown that these errors enjoy certain stability properties and do not grow as quickly as may be feared. Simulations are presented to illustrate the results.