I have to solve a nonlinear system (for some reactions) using newton's method to get molar fractions (positive values), sometimes I get negative values depending on the initial vector, to prevent this, I want to replace variables in the system by their absolute values, but I have no idea if that will change the problem or not.
$\begin{cases} 2x_1+x_5+2x_6+x_8-8\alpha = 0 \\ x_1+2x_2+2x_3+x_4+x_5+x_7-4 = 0\\ 2x_3+x_4+x_9-4\alpha-2 = 0\\ x_4^2-K_{P7}x_3x_2 = 0\\ x_5^2x_6P_T-K_{PT}x_1^2x_{10} = 0 \\ x_6^2x_2P_T-K_{P3}x_1^2x_{10} = 0 \\ x_7^2P_T-K_{P5}x_2x_{10} = 0 \\ x_8^2P_t-K_{P4}x_6x_{10}= 0 \\ x_9^2P_t-K_{P6}x_3x_{10}= 0 \\ \left(\sum\limits_{i=1}^{9}{x_i}\right)-x_{10} = 0 \\ \end{cases}$
where $\alpha, K_{Pi}, PT$ are parameters
I think it is an optimization problem, do that change my initial problem?
Thank you
Replacing variables by their absolute values is likely to give you spurious solutions or trap you in a cycle (I don't know if it will in your particular case). I suspect it's better to try a different initial point if you encounter solutions with negative values. If you are lucky enough to start your Newton's method with an initial point close to a positive solution, you shouldn't encounter solutions with negative values.