An extended Hamiltonian QR algorithm

Micol Ferranti, Bruno Iannazzo, Thomas Mach, Raf Vandebril

Research output: Contribution to journalArticle

Abstract

An extended QR algorithm specifically tailored for Hamiltonian matrices is presented. The algorithm generalizes the customary Hamiltonian QR algorithm with additional freedom in choosing between various possible extended Hamiltonian Hessenberg forms. We introduced in Ferranti et al. (Calcolo, 2015. doi:10.1007/s10092-016-0192-1) an algorithm to transform certain Hamiltonian matrices to such forms. Whereas the convergence of the classical QR algorithm is related to classical Krylov subspaces, convergence in the extended case links to extended Krylov subspaces, resulting in a greater flexibility, and possible enhanced convergence behavior. Details on the implementation, covering the bidirectional chasing and the bulge exchange based on rotations are presented. The numerical experiments reveal that the convergence depends on the selected extended forms and illustrate the validity of the approach.

Original languageEnglish
Pages (from-to)1097
Number of pages1120
JournalCalcolo
Volume54
Issue number3
DOIs
Publication statusPublished - Sep 11 2017

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QR Algorithm
Hamiltonians
Hamiltonian Matrix
Krylov Subspace
Covering
Flexibility
Numerical Experiment
Transform
Generalise
Form
Experiments

Keywords

  • Extended Hessenberg matrices
  • Hamiltonian eigenvalue problems
  • QR algorithm

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics

Cite this

Ferranti, M., Iannazzo, B., Mach, T., & Vandebril, R. (2017). An extended Hamiltonian QR algorithm. Calcolo, 54(3), 1097. https://doi.org/10.1007/s10092-017-0220-9

An extended Hamiltonian QR algorithm. / Ferranti, Micol; Iannazzo, Bruno; Mach, Thomas; Vandebril, Raf.

In: Calcolo, Vol. 54, No. 3, 11.09.2017, p. 1097.

Research output: Contribution to journalArticle

Ferranti, M, Iannazzo, B, Mach, T & Vandebril, R 2017, 'An extended Hamiltonian QR algorithm', Calcolo, vol. 54, no. 3, pp. 1097. https://doi.org/10.1007/s10092-017-0220-9
Ferranti M, Iannazzo B, Mach T, Vandebril R. An extended Hamiltonian QR algorithm. Calcolo. 2017 Sep 11;54(3):1097. https://doi.org/10.1007/s10092-017-0220-9
Ferranti, Micol ; Iannazzo, Bruno ; Mach, Thomas ; Vandebril, Raf. / An extended Hamiltonian QR algorithm. In: Calcolo. 2017 ; Vol. 54, No. 3. pp. 1097.
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