SUBROUTINE ACTION(XI4,S,DS,DDS) - $I\overline{I}$-valley action

Calculation of the $I\overline{I}$-action, ${\rm S}=S^{(I\overline{I})}(\xi )$, as function of the conformal $I\overline{I}$-distance, ${\rm XI4}=\xi$, as well as the 1st, ${\rm DS}=dS^{(I\overline{I})}/d\xi$, and 2nd, ${\rm DDS}=d^2S^{(I\overline{I})}/d\xi^2$, derivatives.

For default settings of the contrôl flags in QIINIT (VALFLAG=.TRUE.), the action is calculated according to the exact valley form [1,2],

$$S^{(I\overline{I})}(\xi ) = 1-\frac{12}{f(\xi )} - \frac{96}{f(\xi )^2} +\frac{48}{f(\xi )^3}... ... 3f(\xi )+8\right] \ln\left[ \frac{1}{2\xi }\bigl( f(\xi ) +4\bigr)\right] \, ,$$ (1)

$$f(\xi ) = \xi^2+\sqrt{\xi^2-4}\xi-4 \, .$$ (2)

For VALFLAG=.FALSE., a simple but good approximation to the exact valley form (1),

$$S^{(I\overline{I})}(\xi ) \approx 1-\frac{6}{(\xi +1/2)^2} ,$$ (3)

is used.