Closed-Form solution for system of simple nonlinear equations I am interested in analytical solutions for a system of nonlinear equations.
Motivation: The source of the question is a very convinient method to create random matrices with special properties. Mathematica can give me solutions up to certain sizes of the linear system, but I would like to have it for arbitrary size N. I can also use numerical algorithms (which I am doing at the moment), but for N in the order of $N\approx10.000$, they are quite slow.
System of nonlinear equations: 
$$
(w_i \cdot \sum_{j=1}^N w_j) - w_i^2 = d_i
$$
for $i=1...N$, and $w$ and $d$ are vectors with $N$ dimensions, and $w_i$ and $d_i$ is the $i$-th component of the vector. Both $d_i$ and $w_i \in \mathbb{R_+}$. I am providing the vector $d$ (i.e. N real non-negative numbers), and want to solve for $w_i$. Is there a way to solve this system analytically for arbitrary N?

Edit: For clarification, if N=3 we have the following system of equations:
$$
w_1 \cdot (w_2 + w_3) = d_1 \\
w_2 \cdot (w_1 + w_3) = d_2 \\
w_3 \cdot (w_1 + w_2) = d_3
$$
with $w_i, d_i \in \mathbb{R}$. For a given vector $d=(d_1,d_2,d_3)$, I want to get $w=(w_1,w_2,w_3)$.

Edit2: I think I see a way how it could be solved, but I'm not certain:
Let's set $c=\sum_{j=1}^N w_j$, which is the sum of all weights. What we have now:
$$c \cdot w_i - w_i^2 = d_i \\
w_i^2 - c \cdot w_i + d_i = 0
$$
which has two solutions:
$$w_{i_{1,2}} = \frac{c}{2} \pm \sqrt{ \left(\frac{c}{2}\right)^2 - d_i}
$$
and the normalisation constant $c$ can be calculated by the sum of all weights:
$$\sum_{j=1}^N w_j = \sum_{j=1}^N \left(\frac{c}{2} \pm \sqrt{ \left(\frac{c}{2}\right)^2 - d_j} \right) = c
$$
Is this correct? Do you know an analytical solution for c?
 A: $n=3$ is rather easy: $w_i$ satisfies a quadratic polynomial.
For $n=4$, each $w_i$ satisfies a rather nasty polynomial of degree $8$
(but involving only even powers).  Thus there is a solution in terms of radicals, but it won't be pleasant.
For $n=5$, it seems each $w_i$ satisfies a polynomial of degree $22$.
A solution in radicals is not to be expected.  Thus with $d_1 = -7, d_2 = 3, d_3 = 9, d_4 = 7, d_5 = 8$, $w_5$ satisfies
$$ 81\,{w_{{5}}}^{22}+7074\,{w_{{5}}}^{20}+198792\,{w_{{5}}}^{18}+2887764
\,{w_{{5}}}^{16}+23487600\,{w_{{5}}}^{14}+99587008\,{w_{{5}}}^{12}+
69082752\,{w_{{5}}}^{10}-1402988992\,{w_{{5}}}^{8}-6995300352\,{w_{{5}
}}^{6}-14191984640\,{w_{{5}}}^{4}-13095665664\,{w_{{5}}}^{2}-
4492099584 = 0
$$
EDIT: This is an irreducible polynomial of degree $11$ in $x = w_5^2$.  Maple doesn't do Galois groups of polynomials of degree $11$, but GAP does, and confirms that its Galois group is $S_{11}$.  In particular, there is no
solution in radicals.
A: For $$c^2\gt 4d_j$$
$$\sqrt{c^2-4d_j}=c\sqrt{1-\frac{4d_j}{c^2}}\approx c\left(1-\frac{2d_j}{c^2}\right)=c-\frac{2d_j}c,$$ with a relatively good approximation.
Summing,
$$\sum_{j=1}^N\left(c\pm\left(c-\frac{2d_j}c\right)\right)=2c.$$
With $M$ positive signs,
$$Mc-\frac{d}c=c,\\c=\pm\sqrt{-\frac{d}{M-1}},$$
where $d=\sum_{j=1}^N\pm d_j$.
This might give you reasonable approximations to start Newton's iterations or maybe just fixed-point.
