I am trying to know the conditions under which two non-linear equations, say $f(\textbf{X},\textbf{Y}) = 0$ and $g(\textbf{X},\textbf{Y})=0$ can be solved iteratively. Here both $\textbf{X},\textbf{Y}$ are matrices and the functions $f, g$ contain operations like inverse and determinant.
I can start with a value of $\textbf{X}_0$, find that value of $\textbf{Y}_0$ that satisfies the first equation $f = 0$. Then I use the value of $\textbf{Y}_0$ to solve the equation $g=0$ and find $\textbf{X}_1$. I can continue these iterations to see if the values of $\textbf{X}_n,\textbf{Y}_n$ have converged or not.
I am not sure what are the necessary and sufficient conditions for this sort of approach to have a convergent solution.