ericagol
commented
6 years ago I worked out some (all?) of these derivatives for the s_2 special cases. I still need to check that these are correct, and code up. The other cases are either trivial (derivatives are zero), or they can be differentiated with auto-diff (or we could just work out the full, complicated expressions).
\frac{d \Lambda}{d b} &= \begin{cases} 0 & b = 0 \\ -\frac{2}{3\pi} & b=r=\tfrac{1}{2} \\ \frac{4r}{3\pi}\mathrm{cel}(k_c,1,-1,k_c^2) & b=r < 1/2\\ -\frac{2}{3\pi} \mathrm{cel}(k_c,1,1,2k_c^2) & b=r > 1/2\\ -\frac{8r}{3\pi}\sqrt{r(1-r)} & b+r = 1 \\ \end{cases}\\ \frac{d \Lambda}{d r} &= \begin{cases} 2 r\sqrt{1-r^2} & b=0\\ \frac{2}{\pi} & b = r = \tfrac{1}{2}\\ \frac{4r}{\pi} E(4r^2) & b=r < 1/2\\ \frac{2}{\pi} \mathrm{cel}(k_c,1,1,0) & b=r > 1/2\\ \frac{8r}{\pi}\sqrt{r(1-r)} & b+r = 1\\ \end{cases}
rodluger
Since the special cases such as b=0, b=r, and b=1-r only depend on one variable, the computed partial derivatives of s_2 are incorrect.
A couple of options exist: 1). We could add the Taylor series expansion to the evaluation, and then use AutoDiff (but this wouldn't allow arbitrary order to be computed). 2). We could compute the derivatives analytically for these cases (these are already given in Pal 2008).