I have (many) equations of the form $$T = z + S \omega,$$ where $T$ and $S$ are complex numbers such that $|T| < 1$ and $|S| = 1$.
I am looking for solutions where $0 < {\rm arg}(z) < \pi/2$; $-\pi/2 < {\rm arg}(z) < 0$; $|z| < 1$; and $|\omega| < 1$. (i.e. $z$ and $\omega$ lie in the first and fourth quarter unit discs).
If solutions do not exist in these regions, I would like to find $z$ and $\omega$ in these regions where the error is minimised.
My first thought was to approximate the quarter discs with simple polygons and calculate the intersection (figure), then find the average of the vertices. This turned out to be very slow.
Is there a 'fast' algorithm to find solutions in the required regions? (Here fast probably means using basic operations and/or a small number of iterations).