How would I go about solving a system of nonlinear equations where the highest degree is two?
For example:
$$f_1(x) = f_1(x_1, x_2,\dots, x_n) = 0,$$
$$f_2(x) = f_2(x_1, x_2,\dots, x_n) = 0,$$
$$\vdots$$
$$f_n(x) = f_n(x_1, x_2,\dots, x_n) = 0$$
If you could direct me to a pdf or anything it'd be greatly appreciated.
Best Answer
As already said in comments and answers, optimization is the way to go. You can formulate it as the minimization of $$\Phi(x_1, x_2,\dots, x_n) = \sum_{i=1}^nf_i^2(x_1, x_2,\dots, x_n)$$ hoping that you arrive to something close to zero.
In any manner, except if you use global optimization, this will require a "good" starting point. If you have such a point, you could linearize each of the $f_i(x_1, x_2,\dots, x_n)$ and solve the problem as a linear problem; each iteration would hopefully improve the guess. This is Newton-Raphson method.
You could be interested by http://fourier.eng.hmc.edu/e161/lectures/ica/node13.html, http://tx.technion.ac.il/~dlewin/054374/Day_5.pdf