Abstract
A unitary symplectic similarity transformation for a special class of Hamiltonian matrices to extended Hamiltonian Hessenberg form is presented. Whereas the classical Hessenberg form links to Krylov subspaces, the extended Hessenberg form links to extended Krylov subspaces. The presented algorithm generalizes thus the classic reduction to Hamiltonian Hessenberg form and offers more freedom in the choice of Hamiltonian condensed forms, to be used within an extended Hamiltonian QR algorithm. Theoretical results identifying the structure of the extended Hamiltonian Hessenberg form and proofs of uniqueness of the reduction process are included. Numerical experiments confirm the validity of the approach.
Similar content being viewed by others
Notes
The name extended refers to the link with extended Krylov subspaces, see [12] for details.
If the matrix exhibits particular structure, core transformations could equal the identity and are therefore not shown in the pyramidal structure.
References
Ammar G., Mehrmann V.: A geometric perspective on condensed forms for Hamiltonian matrices. In: Computation and control, II (Bozeman, MT, 1990), vol. 11 of Progr. Systems Control Theory, pp. 1–11. Birkhäuser, Boston (1991)
Ammar, G., Mehrmann, V.: On Hamiltonian and symplectic Hessenberg forms. Linear Algebra Appl. 149, 55–72 (1991)
Beckermann, B., Güttel, S., Vandebril, R.: On the convergence of rational Ritz-values. SIAM J. Matrix Anal. A. 31, 1740–1774 (2010)
Bini, D.A., Iannazzo, B., Meini, B.: Numerical Solution of Algebraic Riccati Equations, Fundamentals of Algorithms. SIAM (2012)
Byers, R.: A Hamiltonian \({QR}\)-algorithm. SIAM J. Sci. Stat. Comp. 7, 212–229 (986)
Ferranti, M., Iannazzo, B., Mach, T., Vandebril, R.: An extended Hamiltonian QR algorithm (2016) (in preparation)
Ferranti, M., Mach, T., Vandebril, R.: Extended Hamiltonian Hessenberg matrices arise in projection based model order reduction. Proc. Appl. Math. Mech. 15, 583–584 (2015)
Francis, J.G.F.: The QR Transformation a unitary analogue to the LR transformation-Part 1. Comput. J. 4, 265–271 (1961)
Francis, J.G.F.: The QR transformation-part 2. Comput. J. 4, 332–345 (1962)
Kuijlaars, A.B.J.: Which eigenvalues are found by the Lanczos method? SIAM J. Matrix Anal. A. 22, 306–321 (2000)
Kuijlaars, A.B.J.: Convergence analysis of Krylov subspace iterations with methods from potential theory. SIAM Rev. 48, 3–40 (2006)
Mach, T., Pranić, M.S., Vandebril, R.: Computing approximate extended Krylov subspaces without explicit inversion. Electron. T. Numer. Anal. 40, 414–435 (2013)
Paige, C., Van Loan, C.: A Schur decomposition for Hamiltonian matrices. Linear Algebra Appl. 41, 11–32 (1981)
Raines, A.C.I.I.I., Watkins, D.S.: A class of Hamiltonian-symplectic methods for solving the algebraic Riccati equation. Linear Algebra Appl. 205(206), 1045–1060 (1994)
Vandebril, R.: Chasing bulges or rotations? A metamorphosis of the QR-algorithm. SIAM J. Matrix Anal. A. 32, 217–247 (2011)
Watkins, D.S.: Bulge exchanges in algorithms of \({QR}\) type. SIAM J. Matrix Anal. A. 19, 1074–1096 (1998)
Watkins, D.S.: Francis’s algorithm. Am. Math. Mon. 118, 387–403 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research was partially supported by the Research Council KU Leuven, Projects 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 Bruno Iannazzo was partly supported by INdAM through GNCS Project 2015.
Rights and permissions
About this article
Cite this article
Ferranti, M., Iannazzo, B., Mach, T. et al. An extended Hessenberg form for Hamiltonian matrices. Calcolo 54, 423–453 (2017). https://doi.org/10.1007/s10092-016-0192-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10092-016-0192-1