Abstract
This paper discusses the numerical solution of eigenvalue problems for Hamiltonian systems of ordinary differential equations. Two new codes are presented which incorporate the algorithms described here; to the best of the author’s knowledge, these are the first codes capable of solving numerically such general eigenvalue problems. One of these implements a new new method of solving a differential equation whose solution is a unitary matrix. Both codes are fully documented and are written inPfort-verifiedFortran 77, and will be available in netlib/aicm/sl11f and netlib/aicm/sl12f.
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References
M.H. Alexander and D.E. Manolopoulos, A stable linear reference potential algorithm for 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 Value Problems (Academic Press, 1964).
L. Dieci, R.D. Russell and E.S. Van Vleck, Unitary integrators and applications to continuous orthonormalization techniques, Preprint (1992).
C. Fulton and S. Pruess, Mathematical software for Sturm-Liouville problems, NSF Final Report for Grants DMS88-13113 and DMS88-00839, Computational Mathematics Division (1991).
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).
W. Magnus, On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math. 7(1954)649–673.
M. Marletta, Theory and implementation of algorithms for Sturm-Liouville computations, Ph.D. Thesis, Royal Military College of Science (1991).
M. Marletta, Computation of eigenvalues of regular and singular vector Sturm-Liouville systems, Numer. Algor. 4(1993)65–99.
M. Marletta and J.D. Pryce, Automatic solution of Sturm-Liouville problems using the Pruess method. J. Comp. Appl. Math. 39(1992)57–78.
S. Pruess, Estimating the eigenvalues of Sturm-Liouville problems by approximating the differential equation, SIAM J. Numer. Anal. 10(1973)55–68.
J.D. Pryce, Error control of phase-function shooting methods for Sturm-Liouville problems, IMA J. Numer. Anal. 6(1986)103–123.
W.T. Reid, A continuity property of principal solutions of linear Hamiltonian differential systems, Scripta Mathematica 29(1973)337–350.
W.T. Reid,Sturmian Theory for Ordinary Differential Equations, Applied Mathematical Sciences 31 (Springer, 1980).
J.M. Sanz-Serna, Runge-Kutta schemes for Hamiltonian systems, BIT 28(1988)877–883.
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Marletta, M. Numerical solution of eigenvalue problems for Hamiltonian systems. Adv Comput Math 2, 155–184 (1994). https://doi.org/10.1007/BF02521106
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DOI: https://doi.org/10.1007/BF02521106