Numerical computation of the coefficients in exponential fitting
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Numerical computation of the coefficients in exponential fitting L. Gr. Ixaru1,2 Received: 24 June 2020 / Accepted: 6 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We show that a direct numerical computation of the coefficients of any method based on the exponential fitting is possible. This makes unnecessary the knowledge of long sets of analytical expressions for the coefficients, as usually presented in the literature. Consequently, the task of any potential user for writing his/her own code becomes much simpler. The approach is illustrated on the case of the Numerov method for the Schr¨odinger equation, on a version for which the analytic expressions of coefficients are not known. Keywords Numerov method · Schr¨odinger equation · Regularization · Coffey-Evans potential
1 Introduction Exponential fitting (ef for short) is a mathematical procedure for generating numerical methods for various operations (solution of differential or integral equations, numerical differentiation, quadrature, least-square approximation, interpolation, etc.) on functions with a pronounced oscillatory or hyperbolic variation. The typical condition for generating the coefficients of such methods is that the method must be exact for a set of functions including exponential functions. The literature is very vast, see, e.g., [1–17] and references therein, but in the large majority of papers the efforts were directed on the analytic determination of the coefficients. However, the resulting expressions are often so long and cumbersome that their use becomes a difficult task for any potential reader who wants to apply such
L. Gr. Ixaru
[email protected] 1
“Horia Hulubei” National Institute of Physics and Nuclear Engineering, Department of Theoretical Physics, P.O. Box MG-6, Bucharest, Romania
2
Academy of Romanian Scientists, 54 Splaiul Independent¸ei, 050094, Bucharest, Romania
Numerical Algorithms
methods. Attempts of reducing the number of the expressions do also exist, e.g., in [18], but for a potential user the problem continues to remain quite complicated. As a matter of fact, the need for analytic expressions was important especially in the early stages of the field when properties of the new ef versions, for example the stability properties of solvers of differential equations, were better seen on analytic expressions. At present, however, the main interest consists in applications such that by now only the numerical determination of the coefficients is sufficient. In this paper, we consider the problem from this perspective. The need of analytic expressions, which in many papers fill pages and pages of formulas, becomes redundant. Only the generating system of equations has to be mentioned, and everything after that is done numerically by using a dedicated subroutine. We illustrate this on the case of the Numerov method for the Schr¨odinger equation.
2 Numerov method This is to solve the Schr¨odinger equation y = (V (x) − E)y,
x ∈ [a, b],
(1)
where V (x), called the po
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