If there is a system of nonlinear equations and all variables are unbounded real numbers and the functions are continuous, is there a general result on the complexity of solving it? More specifically, can a statement be made about how the computational time to numerically solve the system increases with the number of variables and equations? I am particularly looking for a literature reference for such a statement.



Solving a nonlinear equation system: is there a general result on complexity?
There is not even an algorithm to test whether a solution exists. See e.g. [Richardson's theorem](https://en.wikipedia.org/wiki/Richardson's_theorem).


To have something sensible, you need at least to be able to restrict the domain to a bounded region.


EDIT: ... and even then, depending on the precise formulation of the problem, it can be impossible. Consider the following family of functions. Given Turing machines $T\_0$ and $T\_1$ (including specified inputs), let 
$$f(x) = \sum\_{n=1}^\infty \left(a\_n x^{n} + b\_n (x-1)^{n}\right)$$
where 


1. If $T\_0$ halts in exactly $n$ steps and $T\_1$ has not halted by $n$ steps, $a\_n = 1/n!$ and $ b\_n = 0$.
2. If $T\_1$ halts in exactly $n$ steps and $T\_0$ has not halted by $n$ steps, $a\_n = 0$ and $b\_n = 1/n!$.
3. Otherwise $a\_n = b\_n = 0$.


If $T\_0$ halts before $T\_1$, $f(x)=0$ has the unique solution $x=0$; if $T\_1$ halts before $T\_0$, it has the unique solution $x=1$; if neither halts, or if they halt at the same step, all $x$ are solutions. 


Note that $f(x)$ is a perfectly nice function, in fact a polynomial. The value of $f$ or any of its derivatives at a given point can be approximated with arbitrary precision by simulating sufficiently many steps of the Turing machines. But since there is no algorithm to determine whether a Turing machine halts on given input, there is no algorithm to find (or approximate) a solution to $f(x)=0$.


