I am looking for solving this set of nonlinear equation. \begin{aligned} 2q_1q_3 - 2q_2q_4&=a_1\\ -2q_1q_2 - 2q_3q_4&=a_2\\ - q_1^2 + q_2^2 + q_3^2 - q_4^2&=a3\\ q_1^2 + q_2^2 + q_3^2 + q_4^2&=1\\ \end{aligned} hint:This is IMU quaternion output!
Any Help would be appreciated. Using @FanZheng hint equation would change into \begin{aligned} z_1z_2&=\frac{a_1}{2}-i\frac{a_2}{2}\\ |z_1|^2-|z_2|^2&=-a_3\\ |z_1|^2+|z_2|^2&=1 \end{aligned} more simple \begin{aligned} z_1z_2&=\frac{a_1}{2}-i\frac{a_2}{2}\\ |z_1|^2&=\frac{1-a_3}{2}\\ |z_2|^2&=\frac{1+a_3}{2} \end{aligned}