Modified Gauss–Laguerre Exponential Fitting Based Formulae

Abstract

Modified Gauss–Laguerre exponentially fitted quadrature rules are introduced for the computation of integrals of oscillatory functions over the whole positive semiaxis. Their weights and nodes depend on the frequency of oscillation in the integrand, thus increasing the accuracy of classical Gauss–Laguerre formulae. The asymptotic order is discussed, and an algorithm for determining weights and nodes for a general number N of nodes is provided, resulting an improvement of the existing quadrature formulae. Numerical illustrations are also presented.

Appendix 1

The set of functions $\eta _m(Z),\,m=-1,0,1,2,\ldots$, has been originally introduced in [26] in the context of CP methods for the Schrödinger equation. The functions $\eta _{m}(Z)$ with $m=-1,0$ are first defined by some formulae, namely:

$$\begin{aligned} \eta _{-1}(Z) = \left\{ \begin{array} [c]{ll} \displaystyle \cos (|Z|^{1/2}) &{} \quad \mathrm {if\ } Z \le 0\,\\ \displaystyle \cosh (Z^{1/2}) &{} \quad \mathrm {if\ } Z > 0\, \end{array} \right. , \,\eta _{0}(Z) = \left\{ \begin{array} [c]{ll} \displaystyle \sin (|Z|^{1/2})/|Z|^{1/2} &{} \quad \mathrm {if\ } Z < 0\,\\ \displaystyle 1 &{} \quad \mathrm {if\ } Z=0\,\\ \displaystyle \sinh (Z^{1/2})/Z^{1/2} &{} \quad \mathrm {if\ } Z > 0 \end{array} \right. \end{aligned}$$

(7.1)

and those with $m>0$ are further generated by recurrence

$$\begin{aligned} {\eta }_{m}(Z)=\dfrac{1}{Z}[{\eta }_{m-2}(Z)-(2m-1){\eta }_{m-1}(Z)],\quad m=\ 1,\ 2,\ 3,\ldots \end{aligned}$$

(7.2)

if $\ Z\ne 0,$ and by following values at $Z=0$:

$$\begin{aligned} \eta _{m}(0)=\dfrac{1}{(2m+1)!!}, \quad m=\ 1,\ 2,\ 3,\ldots \end{aligned}$$

(7.3)

The differentiation of these functions is of direct concern for this paper. The rule is

$$\begin{aligned} \eta ^{\prime }_{m}(Z) = {\frac{ 1 }{2}} \eta _{m+1}(Z)\,,\; m=-1,\,0,\,1,\,2,\,3,\ldots \end{aligned}$$

(7.4)

For more details on these functions see [9, 26, 30].