# Avoiding initial guess in Newton Method for nonlinear systems

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?

• Have you tried using existing software, e.g. fsolve in Maple? – Robert Israel Aug 29 '14 at 18:26
• I am performing a research project in Java and I am not sure if this good choice helps me. – hossayni Aug 29 '14 at 18:28
• If $(c_1^1,c_2^1)$ and $(c_1^2,c_2^2)$ are close (say as points in $\mathbb{R}^2$) then the corresponding solutions will be close also. So for the starting point of Newton iteration to find $(\gamma^2,\theta^2)$ take $(\gamma^1,\theta^1)$. – Marcin Malogrosz Aug 29 '14 at 19:18
• Reinventing the wheel is usually not a good idea, except maybe as an academic exercise. Good libraries of numerical methods exist. If you don't have access to one in Java, then Java may be a bad choice of language to use. How about python? – Robert Israel Aug 30 '14 at 0:00