Abstract
The Lanczos algorithm is a well known procedure to compute few eigenvalues of large symmetric matrices. We slightly modify this algorithm in order to obtain the eigenvalues of Hamiltonian matrices H = JS with S symmetric and positive definite. These matrices represent a significant subclass of Hamiltonian matrices since their eigenvalues lie on the imaginary axis. An implicitly restarted procedure is also considered in order to speed-up the convergence of the algorithm.
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References
Ammar, G., Mehrmann, V.: On Hamiltonian and symplectic Hessenberg forms. Linear Algebra Appl. 149 (1991), 55–72
Amodio, P., Iavernaro, F., Trigiante, D.: Conservative perturbations of positive definite Hamiltonian matrices, Numer. Linear Algebra Appl., (2003), in press
Benner, P.: Symplectic balancing of Hamiltonian matrices, SIAM J. Sci. Comput., 22(5) (2001), 1885–1904
Benner, P., Faßbender, H.: An implicit restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem, Linear Algebra Appl., 263 (1997), 75–111
Brugnano, L., Trigiante, D.: Solving ODEs by Linear Multistep Initial and Boundary Value Methods. Gordon & Breach, Amsterdam, 1998
Bunse-Gerstner, A.: Matrix factorization for symplectic QR-like methods, Linear Algebra Appl., 83 (1986), 49–77
Byers, R.: A Hamiltonian QR-algorithm, SIAM J. Sci. Stat. Comput., 7 (1986), 212–229
Calvetti, D., Reichel, L., Sorensen, D.C.: An implicitly restarted Lanczos method for large symmetric eigenvalue problems, ETNA, Electron. Trans. Numer. Anal., 2 (1994), 1–21
Lancaster, P., Rodman, L.: The algebraic Riccati equation. Oxford University Press, Oxford, 1995
Lin, W., Mehrmann, V., Xu, H.: Canonical forms for Hamiltonian and symplectic matrices and pencils, Linear Algebra Appl. 302-303 (1999), 469–533
Rosen, I., Wang, C.: A multilevel technique for the approximate solution of operator Lyapunov and algebraic Riccati equations. SIAM J. Matrix Anal. Appl. 32 (1992), 514–541
Sorensen, D.C.: Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl. 13 (1992), 357–385
Van Loan, C.: A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix, Linear Algebra Appl., 16 (1984), 233–251
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© 2003 Springer-Verlag Berlin Heidelberg
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Amodio, P. (2003). A Symplectic Lanczos-Type Algorithm to Compute the Eigenvalues of Positive Definite Hamiltonian Matrices. In: Sloot, P.M.A., Abramson, D., Bogdanov, A.V., Gorbachev, Y.E., Dongarra, J.J., Zomaya, A.Y. (eds) Computational Science — ICCS 2003. ICCS 2003. Lecture Notes in Computer Science, vol 2658. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44862-4_16
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DOI: https://doi.org/10.1007/3-540-44862-4_16
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