Can anyone provide a straightforward solution to the following equation:
$\vec{y}=M\vec{x}+N\vec{x^2}$
where $\vec{x^2}$ is a column vector with each component being the squared value of the components in $\vec{x}$. Also, $M$ and $N$ are constant coefficient matrices? My current problem is a 3-dimensional system, so I know there are 8 solutions, but I cannot come up with an elegant process. I could brute-force it, but that gets very messy very quickly. I've seen similar problems where dealing with a scalar equation, but nothing quite in this form. Thanks in advance for any help!
This might be a good way to start: consider the system $$ y = M x_1 + N x_2 $$ where $x_1$ and $x_2$ are independently chosen. This is now a system of $2n$ linear equations (rather than $n$ non-linear ones). That is, we can think of this as the system $$ \pmatrix{M&N}\pmatrix{x_1\\x_2} = y $$ Once we've solved this, we can make the additional constraint $x_2 = (x_1)^2$ on the solution set.
In particular: let's assume that $M$ and $N$ have full rank, and that our system is $3$-dimensional. By the above method, we'll find the solution looks something like $$ \pmatrix{x_1\\x_2} = \pmatrix{x_{10}\\x_{20}} + a_1 \pmatrix{x_{11}\\x_{21}} + a_2 \pmatrix{x_{12}\\x_{22}} + a_3 \pmatrix{x_{13}\\x_{23}} $$ where $a_i$ are undetermined scalars and $x_{ij}$ are fixed vectors found by the usual solution process. With the constraint in place, we now have the system on the three variables $a_1,a_2,a_3$ given by $$ (x_{10} + a_1 x_{11} + a_2 x_{12} + a_3 x_{13})^2= x_{20} + a_2 x_{21} + a_2 x_{22} + a_3 x_{23} $$