Integral Generation

Traditional ERIs require the evaluation of the integral
$$\begin{align}I &= \left(\frac{\zeta \eta}{\pi^{2}}\right)^{3/2} \iint e^{-\zeta ... ...^{2}} \int_{0}^{\infty} u \sin(u) e^{-u^{2}/4\mathrm{T}}\Im(u/R)\,du \end{align}$$
where $\Im(k)$ is the Fourier transform of the operator. In this case
$$\begin{align}\Im(k) = \frac{4 \pi}{ k^{2}} \end{align}$$
which, proceeding in the same way as section 5.2, leads to
$$\begin{align}I = R^{-1} \erf \left(R/\sqrt{\frac{1}{\zeta}+\frac{1}{\eta}}\right). \end{align}$$
Note that this is the same result as equation (5.14), as the error function and G₀(T) are closely related:
$$\begin{align}G_{0}(\mathrm{T}) = \frac{\erf(\sqrt{\mathrm{T}})}{\sqrt{2\mathrm{T}}}. \end{align}$$
However the $\erfc(\omega r)/r$ operator requires the evaluation of
$$\begin{align}I_{CASE} &= \left(\frac{\zeta \eta}{\pi^{2}}\right)^{3/2} \iint e^{... ...^{2}} \int_{0}^{\infty} u \sin(u) e^{-u^{2}/4\mathrm{T}}\Im(u/R)\,du \end{align}$$
with
$$\begin{align}\Im(k) = \frac{4 \pi}{ k^{2}} \left( 1 - e^{-k^{2}/4\omega^{2}} \right) \end{align}$$
which produces, in effect, two integrals, one of which is the same as before
$$\begin{align}I_{CASE} &= \frac{1}{2\pi^{2}R^{2}} \left(\int_{0}^{\infty} u \sin(... ...frac{1}{\zeta}+\frac{1}{\eta}+\frac{1}{\omega^{2}}}\right) \right\}. \end{align}$$
Rearranging the above formula into a more computationally useful form gives
$$\begin{align}[0]^{(m)}_{\mathrm{CASE}} = U \left\{ \left( 2\upsilon^{2} \right) ... ...lon_{\omega}^{2} \right) ^{m+1/2} G_m(\mathrm{T}_{\omega}) \right\}, \end{align}$$
which is very similar to the traditional [0]^(m) formula of
$$\begin{align}[0]^{(m)} = U \left( 2 \upsilon^{2} \right)^{m+1/2} G_{m}(\mathrm{T}) \end{align}$$
except for the addition of two new variables:
$$\begin{align}\mathrm{T}_{\omega} &= \upsilon^{2}_{\omega} R^{2} \\ \upsilon_{\o... ...ft[ \frac{1}{\zeta} + \frac{1}{\eta} + \frac{1}{\omega^{2}} \right]. \end{align}$$

Thus, computation of a CASE integral is very similar to a traditional integral. All the traditional work of before must still be done, and in addition, $\upsilon_{\omega}$ and $\mathrm{T}_{\omega}$ must be formed and the interpolation $G_{m}(\mathrm{T}_{\omega})$ must be calculated. Luckily, T and $\mathrm{T}_{\omega}$ are always greater than zero and, as $\omega$ is real,
$$\begin{align}\mathrm{T}_{\omega} \le \mathrm{T}, \end{align}$$
This is useful as it means that the interpolation table built for G_m(T) does not need to be extended to include $G_{m}(\mathrm{T}_{\omega})$. With all of these features in mind it should not be surprising if the computation of CASE integrals takes double the amount of time of the traditional integrals. This is time that needs to be made up by the removal of insignificant integrals.

There is a difference between CASE and traditional methods, however, when the values of T and $\mathrm{T}_{\omega}$ become large. If T is large enough to use the classical multipole expression mentioned in Chapter 5 (and $\mathrm{T}_{\omega}$ is not large enough), ie:
$$\begin{align}\mathrm{T}_{\mathrm{crit}} < \mathrm{T} \end{align}$$
a computational saving can be made for one of the two parts of the CASE integral. However, if
$$\begin{align}\mathrm{T}_{\mathrm{crit}} < \mathrm{T}_{\omega} \quad ( < \mathrm{T} ) \end{align}$$
then, applying the G_m(T) expansion for large T
$$\begin{align}G_{m}(\mathrm{T}) \approx \frac{(2m-1)!!}{(2\mathrm{T})^{m+1/2}} \end{align}$$
to both parts of the integral yields
$$\begin{align}[0]^{(m)}_{\mathrm{CASE}} &\approx U \frac{(2m-1)!!}{2^{m+1/2}} \le... ...{\left(\upsilon^{2}_{\omega}R^{2}\right)^{m+1/2}} \right\} \\ &= 0. \end{align}$$
Therefore the size of $\mathrm{T}_{\omega}$ is a convenient way to check the significance of a CASE integral. All integrals for which $\mathrm{T_{crit}} < \mathrm{T}_{\omega}$ do not need to be computed. The value of $\mathrm{T}_{\omega}$ cannot be used to find the significant integrals unfortunately, as it requires the computation of R for each integral, which is O(N²) work. Another way of screening out integrals is needed, and this is the topic of the next section.