In Exact solution ?
I know the approximate solution by 1. Iterative method
2. Newton Raphson method
Looking at the possible complexities involved in Nonlinear system of equations, I do not think that there can be any general exact method, like Gauss Elimination method for linear system. For example, in case of nonlinear systems, the equations may contain exponential, trigonometric, logarithmic or any other kind of functions.
But I can point out that there is also one another method, namely, method of optimization. For example, if you have two real valued functions f(X) and g(X) and you want to solve two equations f(X)=c1 and g(X)=c2, where X=[x1,x2], then formulate the objective function
y=(f(X)-c1)^2+(g(X)-c2)^2
and then minimize y. In simple problems, you may apply analytical methods, otherwise use evolutionary algorithms such as GA, PSO, etc. This can be generalized for any number of equations.
I am not sure I understand your question: by "exact solution", do you mean a solution in closed form? In that case the answer is of course that there is no general method. Indeed, it is not even possible to have a closed-form solution for a single general polynomial equation in terms of radicals (see e.g. Abel–Ruffini theorem in wikipedia).
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