The QR algorithm for unitary Hessenberg matrices
Journal of Computational and Applied Mathematics
On the spectral decomposition of Hermitian matrices modified by low rank perturbations
SIAM Journal on Matrix Analysis and Applications
An implementation of a divide and conquer algorithm for the unitary eigen problem
ACM Transactions on Mathematical Software (TOMS)
Discrete linearized least-squares rational approximation on the unit circle
ICCAM'92 Proceedings of the fifth international conference on Computational and applied mathematics
Vector Orthogonal Polynomials and Least Squares Approximation
SIAM Journal on Matrix Analysis and Applications
On a Sturm Sequence of Polynomials for Unitary Hessenberg Matrices
SIAM Journal on Matrix Analysis and Applications
The symmetric eigenvalue problem
The symmetric eigenvalue problem
Convergence of the unitary QR algorithm with a unimodular Wilkinson shift
Mathematics of Computation
Convergence of the shifted QR algorithm for unitary Hessenberg matrices
Mathematics of Computation
A Stable Divide and Conquer Algorithm for the Unitary Eigenproblem
SIAM Journal on Matrix Analysis and Applications
An Error Analysis of a Unitary Hessenberg QR Algorithm
SIAM Journal on Matrix Analysis and Applications
Structures preserved by matrix inversion
SIAM Journal on Matrix Analysis and Applications
Efficient Implementation of the Multishift $QR$ Algorithm for the Unitary Eigenvalue Problem
SIAM Journal on Matrix Analysis and Applications
Rank structures preserved by the QR-algorithm: The singular case
Journal of Computational and Applied Mathematics
Unitary rank structured matrices
Journal of Computational and Applied Mathematics
Trigonometric orthogonal systems and quadrature formulae
Computers & Mathematics with Applications
Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations
Journal of Approximation Theory
An algorithm for computing the eigenvalues of block companion matrices
Numerical Algorithms
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In this paper we describe how to compute the eigenvalues of a unitary rank structured matrix in two steps. First we perform a reduction of the given matrix into Hessenberg form, next we compute the eigenvalues of this resulting Hessenberg matrix via an implicit QR-algorithm. Along the way, we explain how the knowledge of a certain 'shift' correction term to the structure can be used to speed up the QR-algorithm for unitary Hessenberg matrices, and how this observation was implicitly used in a paper due to William B. Gragg. We also treat an analogue of this observation in the Hermitian tridiagonal case.