I am looking for an algorithm (numerical or otherwise) which can solve simultaneous equations if those equations are nonlinear and noninvertible.
For example, for the set of equations
$Z_1 = \frac{x}{x+y} e^{xy} + \frac{x+1}{y}$
$Z_2 = y e^{x+y} + \frac{(x+y)^2}{(x+y-1)}$,
I have measured values ($Z_1$, $Z_2$). For each set ($Z_1$, $Z_2$), I would like to know which ($x$, $y$) ordered pair(s) will satisfy the two functions given my measured ($Z_1$, $Z_2$).
I have solved the problem by brute force numerical crunching but I am sure there must be a better method since this is a common problem in physics and geometry.
For a description of the functions and the experimentally measured data points, I have posted the actual problem in a python jupyter notebook here: https://github.com/enlightenedscience/Viscoelastic_modeling_for_QCM/blob/master/voinova_numerical_solver_single_step.ipynb