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
A fitting algorithm for real coefficient polynomial rooting
Journal of Computational and Applied Mathematics
An algorithm for computing the eigenvalues of block companion matrices
Numerical Algorithms
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In this paper an implicit (double) shifted QR-method for computing the eigenvalues of companion and fellow matrices will be presented. Companion and fellow matrices are Hessenberg matrices, that can be decomposed into the sum of a unitary and a rank 1 matrix. The Hessenberg, the unitary as well as the rank 1 structures are preserved under a step of the QR-method. This makes these matrices suitable for the design of a fast QR-method. Several techniques already exist for performing a QR-step. The implementation of these methods is highly dependent on the representation used. Unfortunately for most of the methods compression is needed since one is not able to maintain all three, unitary, Hessenberg and rank 1 structures. In this manuscript an implicit algorithm will be designed for performing a step of the QR-method on the companion or fellow matrix based on a new representation consisting of Givens transformations. Moreover, no compression is needed as the specific representation of the involved matrices is maintained. Finally, also a double shift version of the implicit method is presented.