I have the following equation system:
$A_1 x + B_1 y + C_1 z + D_1 xy + E_1 xz + F_1 yz + G_1 xyz = M_1$
$A_2 x + B_2 y + C_2 z + D_2 xy + E_2 xz + F_2 yz + G_2 xyz = M_2$
$A_3 x + B_3 y + C_3 z + D_3 xy + E_3 xz + F_3 yz + G_3 xyz = M_3$
$A_1$, $B_1$, …, $M_1$, $A_2$, $B_2$, …, $M_2$, $A_3$, $B_3$, …, $M_3$ are known.
Trying to get $x$ based on $y$ and $z$ from the first equation, then substituting it in the second equation, then getting $y$ based on $z$ and substituting it in the third equasion seems a nightmare. How to solve this equation system?
Best Answer
A general cubic system of three equations in three unknown has at most 27 solutions. This can be proved by reducing it to a 27 degree polynomial equation in one variable and apply Gauss theorem. Finding the polynomial give you full control of the number of solutions. However, there is a problem with this scheme since it the polynomial of such high degree might be numerically unstable. Another method is Newtons Raphsons method which will give you a numerical solution of the problem.