I want to solve this system of N non-linear equations without using a numerical method:


$x\_{k}^{2}= \alpha\_{k }+ \sum\limits\_{m=1}^{N} (\beta\_{km} x\_{m} + \psi\_{km} x\_{m}^{\*})$


With


$\left| x\_{k}-1 \right| \leq 1 $


All values are complex and $x^{\*} $ is the conjugate of x. Any idea? An approximated solution could be usefull



System of quadratic complex equations
Put $X=[x\_1,\cdots,x\_N]^T$. We consider an equation in the form $X\circ X+A'+B'X+C'\overline{X}=0$ where $\circ$ is the Hadamard product, $A'$ is a vector and $B',C'$ are square matrices.


Step 1. Put, for every $k$, $x\_k=1+y\_k$. Then $Y$ satisfies an equation in the previous form. Then we may assume that $|x\_k|\leq 1$.


Step 2. Separate real and imaginary parts with $X=U+iV$. We obtain a system (in the unknown REAL vectors $U,V$) in the following form $\{f(U,V)=U\circ U-V\circ V+A+BU+CV=0,\;\;g(U,V)=2U\circ V+D+EU+FV=0\}$. 


Solving (exactly) this system is hopeless. For $N=2$, there are $2^4$ solutions in $\mathbb{C}$ (using the Grobner basis method, we must solve an irreducible polynomial of degree $2^4$). I think that, more generally, there are $2^{2N}$ solutions in $\mathbb{C}$. Obviously, we keep only the real solutions!


You search local solutions ; then you can use the Newton's method. Let $F=[f,g]^T$. The recurrence formula is well-known: $[U\_{n+1},V\_{n+1}]^T=[U\_n,V\_n]^T-(DF([U\_n,V\_n]^T))^{-1}([f(U\_n,V\_n),g(U\_n,V\_n)]^T$. Here $DF([U,V]^T)=\begin{pmatrix}2\tilde{U}+B&-2\tilde{V}+C\\2\tilde{V}+E&2\tilde{U}+F\end{pmatrix}$ where $\tilde{U}=diag(u\_1,\cdots,u\_N)$ and $\tilde{V}=diag(v\_1,\cdots,v\_N)$


