On pole-swapping algorithms for the eigenvalue problem
Numerical Analysis
2022-05-02 v3 Numerical Analysis
Abstract
Pole-swapping algorithms, which are generalizations of the QZ algorithm for the generalized eigenvalue problem, are studied. A new modular (and therefore more flexible) convergence theory that applies to all pole-swapping algorithms is developed. A key component of all such algorithms is a procedure that swaps two adjacent eigenvalues in a triangular pencil. An improved swapping routine is developed, and its superiority over existing methods is demonstrated by a backward error analysis and numerical tests. The modularity of the new convergence theory and the generality of the pole-swapping approach shed new light on bi-directional chasing algorithms, optimally packed shifts, and bulge pencils, and allow the design of novel algorithms.
Cite
@article{arxiv.1906.08672,
title = {On pole-swapping algorithms for the eigenvalue problem},
author = {Daan Camps and Thomas Mach and Raf Vandebril and David S. Watkins},
journal= {arXiv preprint arXiv:1906.08672},
year = {2022}
}