Inverting a Linear Fractional Transformation I have a particular Linear Fractional Transformation from $\mathbb{C}$ to $\mathbb{C}$, I am using to solve Laplace's equation, and I was hoping to find the inverse of this transformation.  Is there a algorithm that can be used to find the inverse of a Linear Fractional Transformation?  Are there any useful algorithms for special cases?
The inverse of the linear fractional transformation 
$$z\to \frac{a z + b}{c z + d}$$ is the linear fractional transformation
$$z\to \frac{e z + f}{g z + h}$$ where the matrix
$$\begin{pmatrix}e &amp; f\\ g &amp; h\end{pmatrix}$$ is the inverse of the matrix
$$\begin{pmatrix}a &amp; b\\ c &amp; d\end{pmatrix}.$$
Answer is here:

https://ieeexplore.ieee.org/document/7106488/

There is a simple expression to compute the inverse LFT...