# Solving a coupled equation (non-linear)

I have a function $F[x]= \sum_i G_i[x]$ (in term of some other known functions of $G_i[x]$), except $G_a[x]=g[x](\frac{F[x]}{F[x]^2+\epsilon^2})$ (g[x] is known and well-behave)

Now, of course I cannot define my $F$[$x$_]$:=\sum_i G_i[x]$.

So, I decided to use Solve command,$Func[x\_]:= F[x]/. Solve[F[x]==\sum_i G_i [x],F[x],WorkingPrecision -> MachinePrecision][]$

but it really doesn't look like a right command or at least it needs a method to avoid inconsistent results. Any idea please?

You can use a zero equation for the implicit definition and, if possible, solve it, but only if you have a finite summatino number

n = 5;
G[n] = g*(F/F^2 + e^2);
eq = F - Sum[G[i], {i, 1, n}] == 0;
Fs = Solve[eq, F] • Thanks, but still I get imaginary numbers in some regions! and I am not sure if it's Mathematica's error or my function just turns out to be like this! – ShS Dec 20 '16 at 18:48
• But there is no method for solve command that can solve it like point by point, or Monte Carlo? – ShS Dec 20 '16 at 18:51
• You just might have bad luck with your function, I mean, imaginary solutions are just solutions. You might restrict the solutions with Solve[eq,F,Reals]. – Mauricio Fernández Dec 21 '16 at 10:09