I'd like to know whether there are any methods like the Gauss-Seidel method to solve nonlinear equations.
For example, I'd like to solve $f(\textbf x) = 0$, where $f(\textbf x)$ is a nonlinear function and can be split into two parts $f(\textbf x)=f_1(\textbf x)+f_2(\textbf x)$. Solving $f_1(\textbf x) = c$ is quite simple, where $c$ is an arbitrary constant.
As a result, can I solve $f(\textbf x) = 0$ iteratively by solving
$f_1(\textbf x^{(i+1)}) = - f_2(\textbf x^{(i)})$
in each iteration $i$?
When does this method converge?