I have such a nonlinear equation system
$x_i=\frac{\sum_{j\neq i}a_{ij}\times\sum_{k\neq i}x_k}{\sum_{k\neq i}x_k-\sum_{j\neq i}a_{ij}}$
where $a_{ij}$s are known coefficients in $[0,1]$. And $x_i$s are non-negative. All the indicies are in $\{1,2,\dots,N\}$ for some very large, known integer $N$.
Is there any way to solve it analytically? If not, can we solve it numerically? Or will it be helpful to convert it to an optimization problem? I have experience in MATLAB and R, but I have not yet used them to solve non-linear equation system.
I find this post helpful, but I prefer an analytical solution since there might be 10k+ variables.