On a Sturm Sequence of Polynomials for Unitary Hessenberg Matrices

Abstract

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.

Structured Eigenvalue Problems

“…A careful choice of shifts can lead to cubic convergence or even ensure global convergence [46,126,127]. Even better, an orthogonal (or unitary) Hessenberg matrix can be represented by O(n) so called Schur parameters [52,27]. This makes it possible to implement the QR algorithm very efficiently [53]; for an extension to unitary and orthogonal matrix pairs, see [25].…”

Efficient Implementation of the Multishift $QR$ Algorithm for the Unitary Eigenvalue Problem

“…We have already mentioned the work of Rutishauser [12] and Gragg [9]. Bunse-Gerstner and He [6] proposed a bisection method based on a Sturm sequence. Gragg and Reichel [10] and Ammar Reichel and Sorensen [2,3] * Department of Mathematics, Washington State University, Pullman, Washington 99164-3113 (e-mail: {rdavid,watkins}@math.wsu.edu).…”

Convergence of the Isometric Arnoldi Process

“…Introduction. Unitary eigenvalue problems arise in a number of different fields, for example signal processing and trigonometric approximation problems (for references, see [10]). There exist numerical methods specifically designed to solve such eigenvalue problems.…”