The QR algorithm for unitary Hessenberg matrices
Journal of Computational and Applied Mathematics
An implementation of a divide and conquer algorithm for the unitary eigen problem
ACM Transactions on Mathematical Software (TOMS)
Journal of Computational and Applied Mathematics - Special issue on computational complex analysis
Some perspectives on the eigenvalue problem
SIAM Review
On a Sturm Sequence of Polynomials for Unitary Hessenberg Matrices
SIAM Journal on Matrix Analysis and Applications
A connection between quadrature formulas on the unit circle and the interval [ - 1,1]
Journal of Computational and Applied Mathematics
A Stable Divide and Conquer Algorithm for the Unitary Eigenproblem
SIAM Journal on Matrix Analysis and Applications
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the conference on orthogonal functions and related topics held in honor of Olav Njåstad
An Error Analysis of a Unitary Hessenberg QR Algorithm
SIAM Journal on Matrix Analysis and Applications
Szegő-Lobatto quadrature rules
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
CMV matrices: Five years after
Journal of Computational and Applied Mathematics
Eigenvalue computation for unitary rank structured matrices
Journal of Computational and Applied Mathematics
A matrix approach to the computation of quadrature formulas on the unit circle
Applied Numerical Mathematics
Unitary rank structured matrices
Journal of Computational and Applied Mathematics
Chasing Bulges or Rotations? A Metamorphosis of the QR-Algorithm
SIAM Journal on Matrix Analysis and Applications
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Let there be given a probability measure @m on the unit circle T of the complex plane and consider the inner product induced by @m. In this paper we consider the problem of orthogonalizing a sequence of monomials {z^r^"^k}"k, for a certain order of the r"k@?Z, by means of the Gram-Schmidt orthogonalization process. This leads to a sequence of orthonormal Laurent polynomials {@j"k}"k. We show that the matrix representation with respect to {@j"k}"k of the operator of multiplication by z is an infinite unitary or isometric matrix allowing a 'snake-shaped' matrix factorization. Here the 'snake shape' of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of S. Delvaux and M. Van Barel. We show that the shape of the snake is determined by the order in which the monomials {z^r^"^k}"k are orthogonalized, while the 'segments' of the snake are canonically determined in terms of the Schur parameters for @m. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism.