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 language | English |
|---|---|
| Pages (from-to) | 1097 |
| Number of pages | 1120 |
| Journal | Calcolo |
| Volume | 54 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - Sep 11 2017 |
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Keywords
- Extended Hessenberg matrices
- Hamiltonian eigenvalue problems
- QR algorithm
ASJC Scopus subject areas
- Algebra and Number Theory
- Computational Mathematics
Cite this
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 journal › Article
}
TY - JOUR
T1 - An extended Hamiltonian QR algorithm
AU - Ferranti, Micol
AU - Iannazzo, Bruno
AU - Mach, Thomas
AU - Vandebril, Raf
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
UR - http://www.scopus.com/inward/record.url?scp=85014691487&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85014691487&partnerID=8YFLogxK
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 -