Complex nonlinear system of four variables

Question

I am stuck on the following problem. Given the system: $$\displaystyle S_1=w_1e^{i(\alpha\omega_x+\beta\omega_y)}+w_2e^{i(\gamma\omega_x+\delta\omega_y)}$$ $$\displaystyle S_2=w_3e^{i(\eta\omega_x+\mu\omega_y)}+w_4e^{i(\psi\omega_x+\zeta\omega_y)}$$ I would like to solve it for the two variables: $\omega_x,\omega_y$ assuming: $\alpha,\beta,\gamma,\delta,\eta,\zeta$ real and $S_1,S_2,w_1,w_2,w_3,w_4$ complex. Is there some method to solve it analytically? If not, what is the best method to solve it numerically?

Answer 1

Let $u=\alpha\omega_x+\beta\omega_y+\angle w_1-\angle S_1,v=\gamma\omega_x+\delta\omega_y+\angle w_2-\angle S_1$.

Then the first equation expresses

$$|w_1|\cos u+|w_2|\cos v=|S_1|,\\|w_1|\sin u+|w_2|\sin v=|S_1|.$$

We eliminate $v$ by

$$(|S_1|-|w_1|\cos u)^2+(|S_1|-|w_1|\sin u)^2=|w_2|^2.$$

After transformation, this is a classical

$$2|w_1|(\cos u+\sin u)=2|S_1|^2+|w_1|^2-|w_2|^2,$$ which you can solve for $u$, and then

$$\tan v=\frac{|S_1|-|w_1|\sin u}{|S_1|-|w_1|\cos u}.$$

From $u$ a,d $v$ you deduce the $\omega$'s by solving a linear system. Plugging in the second equation will tell you about the compatibility.

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Geometrically, you are asking how to rotate two vectors in such a way that they sum to a third one. This is equivalent to the resolution of a triangle of three known sides.