Optimization of Möbius transformation

Question

Say I have a family of points $(w_i, z_i)$ for $i=1,2,...,n$, and I wish to find $a,b,c,d$ such that $\sum_i \left|\frac{a z_i -b}{c z_i - d} - w_i \right|^2 $ is minimized. I realize there are things like nonlinear least squares, hill climbing, etc, but is there a more elegant way to do this? One thing I noticed is that once one finds a good transformation $M_0$ (by solving exactly for a transform sending 3 $w_i$s to their corresponding $z_i$s, one can come up with a transformation $M_1$ such that $M_1(M_0(z_j)) = w_j$ for 3 $j \neq i$, and then treat it in the lie algebra way by treating $(1+M_1\epsilon)M_0$ ($M$ are matrix reps, small $\epsilon$) as a new transformation if it's better, moving slowly towards the ideal map. Any other ideas?