I am faced with the system depicted below
We have a the following system: $$ [A]\left\{q\right\} + [B]\left\{q^2 \right\} + [C]\left\{ Q \right\} = \left\{ F\right\} $$ where $\left\{ q \right\}$ is a vector of four unknowns ($size(\left\{q\right\})=(4,1)$). The vector $\left\{Q\right\}$ is composed of all the possible crossed terms of $q$ ($size(\left\{ Q \right\})$= (6,1)), they are defined by $$ \left\{ q \right\} = \left[ \begin{array}{c} q_{1x} \\ q_{1y} \\ q_{2x} \\ q_{2y} \end{array} \right] \qquad \left\{ q^2 \right\} = \left[ \begin{array}{c} q_{1x}^2 \\ q_{1y}^2 \\ q_{2x}^2 \\ q_{2y}^2 \end{array} \right] \qquad \left\{ Q \right\} = \left[ \begin{array}{c} q_{1x}q_{1y} \\ q_{1x}q_{2x} \\ q_{1x}q_{2y} \\ q_{1y}q_{2x} \\ q_{1y}q_{2y} \\ q_{2x}q_{2y} \end{array} \right] $$ The matrices $[A]$ , $[B]$ and $[C]$ are known and constant matrices. $size([A])=size[B]=(4,4)$ and $size([C])=(4,6)$.
The main problem is, that I want to find {q} given the matrices [A], [B] and [C] and the vector {F}.
So my Questions are:
1) Is there any direct method that can be used to solve this problem.
2) If not, is there any numerical method that can solve this problem (I have found Newton).
Thanks in advance.