Continuation methods for the computation of zeros of Szegö polynomials Original Research Article

Let {φj}∞j = 0 be a family of monic polynomials that are orthogonal with respect to an inner product on the unit circle. The polynomials φj 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 φn based on the observation that the zeros are eigenvalues of a rank-one modification of a unitary upper Hessenberg matrix Hn(0) of order n. The algorithms first determine the spectrum of Hn(0) by one of several available schemes that require only O(n2) arithmetic operations. The eigenvalues of the rank-one perturbation are then determined from the eigenvalues of Hn(0) by a continuation method. The computation of the n zeros of φn in this manner typically requires only O(n2) arithmetic operations. The algorithms have a structure that lends itself well to parallel computation. The latter is of significance in real-time applications.