TY - JOUR
T1 - An extended Hamiltonian QR algorithm
AU - Ferranti, Micol
AU - Iannazzo, Bruno
AU - Mach, Thomas
AU - Vandebril, Raf
N1 - Funding Information:
We would like to thank the referees for their detailed comments, which have led to a significantly improved version of the article. The research was partially supported by the Research Council KU Leuven, projects C14/16/056 Inverse-free Rational Krylov Methods: Theory and Applications, CREA-13-012 Can Unconventional Eigenvalue Algorithms Supersede the State of the Art, OT/11/055 Spectral Properties of Perturbed Normal Matrices and their Applications, and CoE EF/05/006 Optimization in Engineering (OPTEC); by the Fund for Scientific Research?Flanders (Belgium) Project G034212N Reestablishing Smoothness for Matrix Manifold Optimization via Resolution of Singularities; and by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office, Belgian Network DYSCO (Dynamical Systems, Control, and Optimization). The research of the second author was partly supported by INdAM through the GNCS Project 2016.
Publisher Copyright:
© 2017, Springer-Verlag Italia.
PY - 2017/9/11
Y1 - 2017/9/11
N2 - 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.
AB - 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.
KW - Extended Hessenberg matrices
KW - Hamiltonian eigenvalue problems
KW - QR algorithm
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U2 - 10.1007/s10092-017-0220-9
DO - 10.1007/s10092-017-0220-9
M3 - Article
AN - SCOPUS:85014691487
VL - 54
SP - 1097
JO - Calcolo
JF - Calcolo
SN - 0008-0624
IS - 3
ER -