Applied and computational complex analysis. Vol. 3: discrete Fourier analysis—Cauchy integrals—construction of conformal maps---univalent functions
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)
On a Sturm Sequence of Polynomials for Unitary Hessenberg Matrices
SIAM Journal on Matrix Analysis and Applications
ACM Transactions on Mathematical Software (TOMS)
On computing givens rotations reliably and efficiently
ACM Transactions on Mathematical Software (TOMS)
The restarted QR-algorithm for eigenvalue computation of structured matrices
Journal of Computational and Applied Mathematics
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
Orthogonal Rational Functions and Structured Matrices
SIAM Journal on Matrix Analysis and Applications
Original article: Further properties of random orthogonal matrix simulation
Mathematics and Computers in Simulation
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In this paper, we present a novel method for solving the unitary Hessenberg eigenvalue problem. In the first phase, an algorithm is designed to transform the unitary matrix into a diagonal-plus-semiseparable form. Then we rely on our earlier adaptation of the QR algorithm to solve the dpss eigenvalue problem in a fast and robust way. Exploiting the structure of the problem enables us to yield a quadratic time using a linear memory space. Nonetheless the algorithm remains robust and converges as fast as the customary QR algorithm. Numerical experiments confirm the effectiveness and the robustness of our approach.