Unitary matrices have a rich mathematicalstructure which is closely analogous to real symmetric matrices. For real symmetric matrices this structure can be exploited to develop very e cient numerical algorithms and for some of these algorithms unitary analogues are known. Here we present a unitary analogue of the bisection method for symmetric tridiagonal matrices. Recently Delsarte and Genin introduced a sequence of so-called n-symmetric polynomials which can be used to replace the classical Szeg o polynomials in several signal processing problems. These polynomials satisfy a three term recurrence relation and their roots interlace on the unit circle. Here we explain this sequence of polynomials in matrix terms. For an n n unitary Hessenberg matrix, we introduce, motivated by the Cayley transformation, a sequence of modi ed unitary submatrices. The characteristic polynomials of the modi ed unitary submatrices p k (z); k = 1; 2; : : :; n are exactly the n-symmetric polynomials up to a constant. These polynomials can be considered as a sort of Sturm sequence and can serve as a basis for a bisection method for computing the eigenvalues of the unitary Hessenberg matrix. The Sturm sequence properties allow to identify the number of roots of p n (z), the characteristic polynomial of the unitary Hessenberg matrix itself, on any arc of the unit circle by computing the sign agreements of certain related real polynomials at a given point.
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