Abstract
This paper presents an algorithm for the numerical approximation of spectral projectors onto the invariant subspaces corresponding to the eigenvalues inside, on, and outside the unit circle of a symplectic matrix. The algorithm constructs iteratively three matrix sequences from which the projectors are obtained. The convergence depends essentially on the gap between the unit circle and the eigenvalues inside it. A larger gap leads to faster convergence. Theoretical and algorithmic aspects of the algorithm are developed. Numerical results are reported.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Fassbender, H.: Symplectic Methods for Symplectic Eigenproblems. Kluwer, New York (2000)
Godunov, S.K.: Modern aspects of linear algebra. Am. Math. Soc. 175 (1998)
Godunov, S.K., Sadkane, M.: Numerical determination of a canonical form of a symplectic matrix. Sib. Math. J. 42, 629–647 (2001)
Godunov, S.K., Sadkane, M.: Some new algorithms for the spectral dichotomy methods. Linear Algebra Appl. 358, 173–194 (2003)
Godunov, S.K., Sadkane, M.: Spectral analysis of symplectic matrices with application to the theory of parametric resonance. SIAM J. Matrix Anal. Appl. 28, 1083–1096 (2006)
Gohberg, I., Lancaster, P., Rodman, L.: Indefinite Linear Algebra and Applications. Birkhäuser, Basel (2005)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 2nd edn. The Johns Hopkins University Press, Baltimore (1989)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge Universtity Press, Cambridge (1990)
Lancaster, P., Rodman, L.: Algebraic Riccati Equations. Clarendon, Oxford (1995)
Malyshev, A.: On spectral trichotomy of symplectic and Hamiltonian matrices. In: O’Leary, D., et al. (eds.) Proceeding of XIII Householder Symposium on Numerical Linear Algebra, pp. 140–144, Pontresina, 17–21 June 1996
Mackey, D.S., Mackey, N., Tisseur, F.: Structured factorizations in scalar product spaces. SIAM J. Matrix Anal. Appl. 27, 821–850 (2005)
Robbé, M., Sadkane, M.: Discrete-time Lyapunov stability of large matrices. J. Comput. Appl. Math. 115, 479–494 (2000)
Salam, A.: On theoretical and numerical aspects of symplectic Gram-Schmidt-like algorithms. Numer. Algorithms 39, 437–462 (2005)
Stewart, G.W., Sun, J.-G.: Matrix Perturbation Theory. Academic, San Diego (1990)
Yakubovich, V.A., Starzhinskii, V.M.: Linear Differential Equations with Periodic Coefficients, vols. 1 & 2. Wiley, New York (1975)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dosso, M., Sadkane, M. A spectral trichotomy method for symplectic matrices. Numer Algor 52, 187–212 (2009). https://doi.org/10.1007/s11075-009-9264-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-009-9264-5