An extended Hamiltonian QR algorithm

Micol Ferranti, Bruno Iannazzo, Thomas Mach, Raf Vandebril

Research output: Contribution to journalArticlepeer-review


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
Issue number3
Publication statusPublished - Sep 11 2017


  • Extended Hessenberg matrices
  • Hamiltonian eigenvalue problems
  • QR algorithm

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics

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