Continuation methods for the computation of zeros of Szegö polynomials

Abstract

Let { φ j } ∞ j = 0 be a family of monic polynomials that are orthogonal with respect to an inner product on the unit circle. The polynomials φ j arise in time series analysis and are often referred to as Szegö polynomials or Levinson polynomials. Knowledge about the location of their zeros is important for frequency analysis of time series and for filter implementation. We present fast algorithms for computing the zeros of the polynomials φ n based on the observation that the zeros are eigenvalues of a rank-one modification of a unitary upper Hessenberg matrix H n (0) of order n. The algorithms first determine the spectrum of H n (0) by one of several available schemes that require only O( n 2) arithmetic operations. The eigenvalues of the rank-one perturbation are then determined from the eigenvalues of H n (0) by a continuation method. The computation of the n zeros of φ n in this manner typically requires only O( n 2) arithmetic operations. The algorithms have a structure that lends itself well to parallel computation. The latter is of significance in real-time applications.