Assume that I have a set of $N$ complex numbers $\{z_1,...,z_N\}$ with corresponding images $\{w_1,...,w_N\}$. A Möbius transformation can be defined by specifying three points and their images. If $N>3$, no exact mapping for all points will exist in the general case. It is my goal to find one (or, in case of an underdetermined system, several) Möbius transformations which minimize the mismatch between the Möbius transformation
$$M(z)=\frac{az+b}{cz+d}$$
and the target images $\{w_1,...,w_N\}$ in terms of (for example) least-squares:
$$\frac{1}{N}\sum_{n=1}^{N}|M(z_n)-w_n|^2$$
This is a problem which can probably be solved iteratively, but what is the most elegant way of finding a solution to this problem?
Note: a very similar question was already asked here, but unfortunately it wasn't answered back then.