ON A STURM SEQUENCE OF POLYNOMIALS FOR UNITARY HESSENBERG MATRICES

Unitary matrices have a rich mathematical structure that is closely an alogous to real symmetric matrices. For real symmetric matrices this s tructure can be exploited to develop very efficient numerical algorith ms and for some of these algorithms unitary analogues are known. Here we present a unitary analogue of the bisection method for symmetric tr idiagonal matrices. Recently Delsarte and Genin introduced a sequence of so-called gamma(n)-symmetric polynomials that can be used to replac e the classical Szego polynomials in several signal processing problem s. These polynomials satisfy a three-term recurrence relation and thei r roots interlace on the unit circle. Here we explain this sequence of polynomials in matrix terms. For an n x n unitary Hessenberg matrix, we introduce, motivated by the Cayley transformation, a sequence of mo dified unitary submatrices. The characteristic polynomials of the modi fied unitary submatrices p(k) (z), k = 1, 2, ..., n are exactly the ga mma(n)-symmetric polynomials up to a constant. These polynomials can b e considered as a sort of Sturm sequence and can serve as a basis for a bisection method for computing the eigenvalues of the unitary Hessen berg matrix. The Sturm sequence properties allow identification of the number of roots of p(n) (z), the characteristic polynomial of the uni tary Hessenberg matrix itself, on any are of the unit circle by comput ing the sign agreements of certain related real polynomials at a given point.