Tridiagonal Approach to the Algebraic Environment of Toeplitz Matrices, Part II: Zero and Eigenvalue Problems

Abstract

The central subject of this paper is the three-term recurrence formula satisfied by the symmetric (first-kind) and antisymmetric (second-kind) polynomials relative to a given sequence of reflection coefficients, the last element of which has unit modulus. The theory of these polynomials is shown to have interesting analogies with the classical theory of orthogonal polynomials on the real line. In the case of real data, the former is equivalent to a special case of the latter (by a change of variable). The main application considered here is the problem of computing the zeros of the highest-degree symmetric polynomial, which is identified as a predictor polynomial. This problem occurs not only in some modelling techniques for digital signal processing, but can also be interpreted as the eigenvalue problem for a unitary Hessenberg matrix. Attractive solution methods are derived from the “tridiagonal approach,” based on the three-term recurrence relation.