I just read here that Jacobian matrices can be used to solve a system of nonlinear equations, and I am wondering exactly how this might work. I have the following system to solve:
$$\frac{(1-x-2y-3z)(3-3x-2y-z)^3}{(4x)^4}-1=0$$ $$\frac{(1-x-2y-3z)(3-3x-2y-z)}{(2y)^2}-1=0$$ $$\frac{(1-x-2y-3z)^3(3-3x-2y-z)}{(4z)^4}-1=0$$ On the article that I linked, it states that a quick way to solve a non-linear system is by solving $$J_F(x_{n+1}-x_n)=-F(x_n)$$ I'm really not too sure what any of this means. I am familiar with the Jacobain matrix, as well as the standard Newton's method using $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$. What does $F(x)$ indicate in the above equation? Can someone provide an example of a simpler problem that uses this method?