Summary: CONTINUATION METHODS FOR THE COMPUTATION
OF ZEROS OF SZEG �
O POLYNOMIALS \Lambda
G. S. AMMAR y , D. CALVETTI z , AND L. REICHEL x
Abstract. Let fOE j g
1
j=0 be a family of monic polynomials that are orthogonal with respect to
an inner product on the unit circle. The polynomials OE j arise in time series analysis and are often
referred to as Szeg�o 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 OE n 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(n 2 ) 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 OE 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.
AMS subject classifications. 30C15, 62M10, 65E05, 65F15, 65H17, 65H20, 65L05
Key words. Szeg�o polynomial, root�finder, continuation method, linear predictor, reflection
coefficient, eigenvalue problem, unitary Hessenberg matrix, parallel computation.

Source: Ammar, Greg - Department of Mathematical Sciences, Northern Illinois University

Collections: Mathematics