Structured matrix methods for polynomial root-finding
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Unitary rank structured matrices
Journal of Computational and Applied Mathematics
WSEAS Transactions on Mathematics
A multiple shift QR-step for structured rank matrices
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
Real and complex polynomial root-finding with eigen-solving and preprocessing
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
New progress in real and complex polynomial root-finding
Computers & Mathematics with Applications
Chasing Bulges or Rotations? A Metamorphosis of the QR-Algorithm
SIAM Journal on Matrix Analysis and Applications
Efficient polynomial root-refiners: A survey and new record efficiency estimates
Computers & Mathematics with Applications
An algorithm for computing the eigenvalues of block companion matrices
Numerical Algorithms
The rational approximations of the unitary groups
Quantum Information Processing
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Let $\mathcal H_n\subset \mathbb C^{n\times n}$ be the class of $n\times n$ Hessenberg matrices $A$ which are rank-one modifications of a unitary matrix, that is, $A=H +\B u\B w^H$, where $H$ is unitary and $\mathbf{u}, \mathbf{w}\in \mathbb C^n$. The class $\mathcal H_n$ includes three well-known subclasses: unitary Hessenberg matrices, companion (Frobenius) matrices, and fellow matrices. The paper presents some novel fast adaptations of the shifted QR algorithm for computing the eigenvalues of a matrix $A\in \mathcal H_n$ where each step can be performed with $O(n)$ flops and $O(n)$ memory space. Numerical experiments confirm the effectiveness and the robustness of these methods.