How to choose initial point of root finding algorithm?

Question

I want to solve the following system of non-linear equations:

That is, three equations in two unknowns ($x$ and $y$).

These, of course, describe hyperbola. The solution, as I understand it, involves the intersection of hyperbola. I want to solve this using the GNU Scientific Library's multi-root finder (https://www.gnu.org/software/gsl/doc/html/multiroots.html). This requires that I choose some initial point for the root finder. How do I go about determining an initial point?

Answer 1

Let $r_i=\sqrt{(x-x_i)^2+(y-y_i)^2}$ be the radius and $\hat r_i=r_i-st_i,~r_i=\hat r_i+st_i$ be the "shifted radius" of the circle centered at $(x_i,y_i)$ with $(x,y)$ on its edge. We can interpret the problem as finding the point where all shifted radii are equal. A decent and simple initial estimate may be to use the average of $(x_i,y_i)$.

It's worth noting that it is very easy to algebraically solve for $r_i$. This then boils the problem down to circle-circle intersection.