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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?

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How are $f_1$ and $f_2$ related to the original $f$? What do you mean by "split"? – bubba Aug 14 '13 at 11:01
@bubba $f = f_1+f_2$ – Hugo Aug 14 '13 at 11:50
I don't know if your algorithm would work. I suspect not. But there are many available methods for iteratively solving systems of non-linear equations. An internet search should give you plenty to choose from. For example, both the Newton-Raphson algorithm and the secant method can be generalized to the multi-dimensional case.… – bubba Aug 14 '13 at 12:38
@Amzoti No, not Newton-Raphson algorithm. I want to find a method for nonlinear systems corresponding to the Gauss-Seidel method for linear systems. – Hugo Aug 14 '13 at 13:49
@Hugo: You mean like GS in this?… – Amzoti Aug 16 '13 at 5:40

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