Peace be upon you,
I am solving the following system of equations for finding $\gamma$ and $\theta$, while I have about 18000 pairs of $\{c1,c2\}$ constants (i.e. I have about 18000 of such systems for being solved). \begin{align*} &\begin{cases} ln\left(\frac{\gamma^2}{\gamma^2+\theta^2}\right)-c_1+\sum_{I\in I}{a_i\left[\gamma^{-2i}-(\gamma^2+\theta^2)^{-i}\right]} = 0\\ ln\left(\frac{\theta^2}{\gamma^2+\theta^2}\right)-c_2+\sum_{I\in I}{a_i\left[\theta^{-2i}-(\gamma^2+\theta^2)^{-i}\right]} = 0\\ \end{cases},\\ &I=\{1,2,4,6,8,10,12,14\}\\ &a=\{-\frac{1}{2},-\frac{1}{12},\frac{1}{120},-\frac{1}{252},\frac{1}{240},-\frac{1}{132},\frac{691}{32760},-\frac{1}{12}\} \end{align*} Since I have plenty of $\{c1,c2\}$ pairs, I need a global method (without need to initial guess) for solving the above equations (as you know testing the newton method for 18000 potentially different initial guesses is not practical).
For example, by $c_1 = -0.555336088$ and $c_2 = -1.570176901$, I have tested several initial guesses but the Newton method falls to a triple-loop and has a wild treatment.
Can anyone suggest a much suitable method for solving these 18000 systems of equations?
I, successfully, used the Octave Java library for solving the system in Java.