Simplify non-linear system with linear constraints

Question

I have a set of 6 variables that satisfy 4 linear equations and 2 non-linear equations. In particular, my first four equations can be written as $Ax = 0$, where $x \in \mathbb{R}^6$. This constraint implies that $x \in ker(A)$.

What I think this means is that if I can find the null space of $A$, I can reduce the number of free parameters from 6 to 2 for the non-linear portion of the solution. Where I'm getting stuck is how to use that parameterization to solve the non-linear system.

Concretely, the two non-linear constraints look like this (with known constants $C_1$ and $C_2$):

$${x_1}^2 + {x_2}^2 = C_1$$ $$(x_4 - x_3 - x_1)^2 + (x_6 + x_5 + x_2)^2 = C_2$$

Do I just need to use a non-linear solver here or is there some other trick I can pursue to try to simplify this further and perhaps find an analytical solution?

Answer 1

After expressing everything in terms of two parameters (let's say $s$ and $t$) from the linear portion, you have two equations of the form

$$ \eqalign{(a_1 + b_1 s + c_1 t)^2 + (a_2 + b_2 s + c_2 t)^2 &= C_1 \cr ( a_3 + b_3 s + c_3 t)^2 + (a_4 + b_4 s + c_4 t)^2 &= C_2 \cr}$$ Presumably $C_1, C_2 > 0$ so this is nontrivial.

The resultant of these with respect to one of $s$ and $t$ will be a quartic polynomial. Each real root of that should give you a solution.

Answer 2

If I well understand you can find $x_3,x_4,x_5,x_6$ as linear functions of $x_1$ and $x_2$ so your system become somethig as:

$$ \begin {cases} x_1^2+x_2^2=C_1\\ (ax_1+bx_2)^2+(cx_1+dx_2)^2=C_2 \end{cases} $$ In general this can be reduced to a quartic equation which can be solved analytically but with a lot of work (so it is better to search a numerical solution).