 
Summary: Computing the Poles of Autoregressive Models
from the Reflection Coefficients 1
Gregory S. Ammar 2 Daniela Calvetti 3 Lothar Reichel 4
Abstract
A new approach to the computation of the poles of a stable autoregressive system
from the reflection coefficients is proposed. Equivalently, we compute the zeros of Szego
polynomials from the associated Schur parameters. The numerical method utilizes an
efficient algorithm for computing the (unimodular) zeros of a unitary Hessenberg matrix;
this step can be regarded as the computation of the poles of an associated lossless system.
These eigenvalues are then used as starting points for a continuation procedure for finding
the zeros of the desired polynomial. The procedure is efficient and parallelizable, and
may therefore be suitable for realtime applications.
1. Introduction
Autoregressive models are of fundamental importance in time series analysis and discrete
time control. For example, in linear prediction, one is often given the autocorrelation
matrix of a wide sense stationary realvalued signal fx j g 1
j=\Gamma1
. This matrix is a real
symmetric positive definite Toeplitz matrix
M n+1 = [¯ j \Gammak ] n
