Abstract
This paper presents an algorithm for solving a linear Hamiltonian system arising in the study of certain ODE eigenproblems. The method follows the phase angles of an associated unitary matrix, which are essential for correct indexing of the eigenvalues of the ODE. Compared to the netlib code SL11F [11] the new method has the property that on many important problems – in particular, on matrix–vector Schrödinger equations – the cost of the integration is bounded independently of the eigenparameter λ. This allows large eigenvalues to be found much more efficiently. Numerical results show that our implementation of the new algorithm is substantially faster than the netlib code SL11F.
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References
M.H. Alexander and D.E. Manolopoulos, A stable linear reference potential algorithm for the solution of the quantum close-coupled equations in molecular scattering theory, J. Chem. Phys. 86 (1987) 2044–2050.
F.V. Atkinson, Discrete and Continuous Boundary Problems (Academic Press, New York, 1964).
F. Gilbert and G.E. Backus, Propagator matrices in elastic wave and vibration problems, Geophysics 31 (1966) 326–332.
L. Greenberg, A Prüfer method for calculating eigenvalues of self-adjoint systems of ordinary differential equations, Parts 1 and 2, Technical Report TR91–24, University of Maryland (1991).
L. Greenberg and M. Marletta, Oscillation theory and numerical solution of fourth-order Sturm– Liouville problems, IMA J. Numer. Anal. 15 (1995) 319–356.
L. Greenberg and M. Marletta, Oscillation theory and numerical solution of sixth-order Sturm– Liouville problems, preprint.
A. Iserles and S. Nørsett, On the solution of linear differential equations in Lie groups, Technical Report, DAMTP 1997–155, University of Cambridge (1997).
T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1966).
C.R. Maple and M. Marletta, Solving Hamiltonian systems arising from ODE eigenproblems, Technical Report 1997/41, University of Leicester (1997).
C.R. Maple and M. Marletta, Asymptotic behaviour of the spectral decomposition of Riccati-like variables for linear Hamiltonian systems, Technical Report 1997/44, University of Leicester (1997).
M. Marletta, Numerical solution of eigenvalue problems for Hamiltonian systems, Adv. Comput. Math. 2 (1994) 155–184.
M.A. Naimark, Linear Differential Operators (Ungar, New York, 1968).
L.F. Shampine and M.K. Gordon, Computer Solution of Ordinary Differential Equations: The Initial Problem (Freeman, New York, 1975).
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Maple, C.R., Marletta, M. Solving Hamiltonian systems arising from ODE eigenproblems. Numerical Algorithms 22, 263–284 (1999). https://doi.org/10.1023/A:1019171110743
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DOI: https://doi.org/10.1023/A:1019171110743