Numerical computation of the coefficients in exponential fitting

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 Schrödinger equation, on a version for which the analytic expressions of coefficients are not known.

Appendix

Eta functions η_− 1(Z),η₀(Z),η₁(Z),... are defined as follows [1, 19, 21]:

The first two are

$$ \eta_{-1}(Z)=\left \{ \begin{array}{ll} \cos(|Z|^{1/2})& \text{if} Z\leq 0\\ \cosh(Z^{1/2})& \text{if} Z>0 \end{array}\right. ,~~~~\eta_{0}(Z)=\left \{ \begin{array}{ll} \frac{\sin(|Z|^{1/2})}{|Z|^{1/2}} & \text{if} Z< 0\\ 1& \text{if} Z=0\\ \frac{\sinh(Z^{1/2})}{Z^{1/2}}& \text{if} Z>0 \end{array}\right. . $$

(18)

In some papers, function η_− 1(Z) is denoted ξ(Z).

Functions η_m(Z) with m > 0 are generated by recurrence

$$ \eta_m(Z)=\frac{\eta_{m-2}(Z)-(2m-1)\eta_{m-1}(Z)}{Z},~~~~m=1,2,3,... $$

(19)

if Z≠ 0, and by following values at Z = 0 :

$$ \eta_m(0)=\frac{1}{(2m+1)!!},~~~~m=1,2,3,... $$

(20)

Some useful properties are:

Series expansion:

$$ \eta_m(Z)=2^m\sum\limits_{q=0}^{\infty}\frac{(q+m)!}{q!(2q+2m+1)!}Z^q,~~~m=0,1,2,... $$

(21)

Asymptotic behavior at large |Z| :

$$ \eta_m(Z)\approx \left \{ \begin{array}{ll} \frac{\eta_{-1}(Z)}{Z^{(m+1)/2}}& \text{for~ odd}~ m\\ \frac{\eta_0(Z)}{Z^{m/2}} & \text{for~ even}~ m. \end{array}\right. $$

(22)

Differentiation properties:

$$ \eta^{\prime}_m(Z)=\frac{1}{2}\eta_{m+1}(Z),~~~m=-1,0,1,2,... $$

(23)

The latter is important in subroutine REGSOLV for solving (3) because there the expressions of the derivatives of functions F_n(Z),n = 1,2,⋯ ,N and B(Z) are also requested.

The expressions needed in system (4) for these functions and their derivatives are:

$$ F_{1}(Z)=- 1 \text{and} F_{1}^{(m)}(Z)=0 \text{for} m\geq 1, $$

$$ \begin{array}{@{}rcl@{}} &&F_{2}(Z)=2Z\eta_{-1}(Z), F^{(1)}_{2}(Z)= 2\eta_{-1}(Z)+Z\eta_{0}(Z) \text{and}\\ &&F^{(m)}_{2}(Z) = 2^{1-m} [\eta_{m-3}(Z)+3\eta_{m-2}(Z)] \text{for} m\geq 2, \end{array} $$

$$ F_{3}(Z)=Z, F^{(1)}_{3}(Z)=1 \text{and} F^{(m)}_{3}(Z)=0 \text{for} m\geq 2, $$

$$ B(Z)=2\eta_{-1}(Z) \text{and} B^{(m)}(Z)=2^{1-m}\eta_{m-1}(Z) \text{for} m\geq 1. $$

Codes for the computation of the eta functions are available: subroutines GEBASE and GEBASEV (fortran 95, double precision arithmetic) in the compact disk attached to book [1]; see also formConv (mathematica) in [24].

For solving (4), we have used derivatives up to m_max = 30 with eta functions computed by GEBASEV.