Optimization of adiabatically-contracted dark matter halo density profiles

[abensonca]

In solving for the density profile of adiabatically-contracted dark matter halos we have to solve the equation for the initial radius:

$$M_\mathrm{total,0}(\bar{r}_\mathrm{i}) [ f_\mathrm{i} r_\mathrm{i} - f_\mathrm{f} r_\mathrm{f} ] - V^2_\mathrm{b}(\bar{r}_\mathrm{f}) \bar{r}_\mathrm{f} r_\mathrm{f}/ \mathrm{G} = 0$$

It might be faster to find the root using a gradient-aware method. The gradient in the above is easy enough to compute - the only complicated term is:

$$\frac{\mathrm{d}M_\mathrm{total,0}(\bar{r}_\mathrm{i})}{\mathrm{d}r_\mathrm{i}} = 4 \pi \bar{r}_\mathrm{i}^2 \rho_\mathrm{total,0}(\bar{r}_\mathrm{i}) \frac{\mathrm{d}\bar{r}_\mathrm{i}}{\mathrm{d} r_\mathrm{i}}$$

and

$$\frac{\mathrm{d}\bar{r}_\mathrm{i}}{\mathrm{d} r_\mathrm{i}} = \omega \frac{\bar{r}_\mathrm{i}}{r_\mathrm{i}}$$

It would be useful to benchmark this in cases where we have a good initial guess for the solution, and for cases where we do not.