Mathematical software for Sturm-Liouville problems

Pruess, Steven; Fulton, Charles T.

doi:10.1145/155743.155791

Cited by 117 publications

(94 citation statements)

References 7 publications

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“…This is done e.g. in SLEDGE and combined with the ideas based on the Prüfer substitution to be able to home in on a particular eigenvalue (see [15]). …”

Section: Coefficient Approximationmentioning

confidence: 99%

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Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics

Ledoux

Daele

Berghe

2009

Computer Physics Communications

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Finding the eigenvalues of a Sturm-Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behaviour of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. We will discuss some techniques that yield uniform approximation over the whole eigenvalue spectrum and can take large steps even for high eigenvalues. In particular, we will focus on methods based on coefficient approximation which replace the coefficient functions of the SturmLiouville problem by simpler approximations and then solve the approximating problem. The use of (modified) Magnus or Neumann integrators allows to extend the coefficient approximation idea to higher order methods.

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“…This is done e.g. in SLEDGE and combined with the ideas based on the Prüfer substitution to be able to home in on a particular eigenvalue (see [15]). …”

Section: Coefficient Approximationmentioning

confidence: 99%

“…with t = e (x−7)/0.6 over the interval [0,15]. The eigenvalue spectrum of this Woods-Saxon problem contains 14 eigenenergies λ 0 , ..., λ 13 .…”

Section: Some Experimentsmentioning

confidence: 99%

Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics

Ledoux

Daele

Berghe

2009

Computer Physics Communications

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“…The modified Neumann schemes can be used to locate the eigenvalues by shooting. As for the Pruess method [28], some special care is needed in the shooting algorithm to avoid difficulties when the solutions show rapid exponential growth. Also we discuss a procedure to count the oscillations of the solution, as to home in on a particular eigenvalue.…”

Section: Error Dependence On E (Fixed Steps)mentioning

confidence: 99%

“…The recursion (4.1) can be unstable when many (q i − Ew i )P i > 0 because of the exponential growth. To avoid problems, we stabilize our shooting algorithm in the same way as for the second order method in [28]. We divide e hiĀi by its spectral radius at each step.…”

Section: Error Dependence On E (Fixed Steps)mentioning

confidence: 99%

“…Better results are obtained using methods based on coefficient approximation since, for such methods, the step size is not restricted by the oscillations in the solution. A method that has been successfully used to solve Sturm-Liouville problems is, e.g., the second order coefficient approximation method which was implemented in the SLEDGE package [27,28]. Pryce called this method the Pruess method in [26,25].…”

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confidence: 99%

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Solution of Sturm–Liouville Problems Using Modified Neumann Schemes

Ledoux¹,

Daele²

2010

SIAM J. Sci. Comput.

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Abstract. The main purpose of this paper is to describe the extension of the successful modified integral series methods for Schrödinger problems to more general Sturm-Liouville eigenvalue problems. We present a robust and reliable modified Neumann method which can handle a wide variety of problems. This modified Neumann method is closely related to the second-order Pruess method, but provides for higher order approximations. We show that the method can be successfully implemented in a competitive automatic general-purpose software package.

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