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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.

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  • $\begingroup$ Look at convex optimization if the $f_i$ are convex (or concave, traverse to $-f_i$ in this case). If they're not, you're gonna have a bad time. $\endgroup$ – AlexR Dec 1 '14 at 9:40
  • $\begingroup$ It doesn't matter if they are convex or concave. Nonlinear equations are non-convex. $\endgroup$ – Michael Grant Dec 1 '14 at 14:06
  • $\begingroup$ @MichaelGrant Thank you, I'm not familiar w/ LateX and couldn't have presented the question as well as you did, thanks dude. $\endgroup$ – Manny P Dec 2 '14 at 10:09
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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

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  • $\begingroup$ Leibovic, Thank you, much appreciated. $\endgroup$ – Manny P Dec 2 '14 at 10:10
  • $\begingroup$ You are very welcome ! If fact, this applies to any system of nonlinear equations. The fact that the highest degree is two does not make anything simpler. If you need more, just post. $\endgroup$ – Claude Leibovici Dec 2 '14 at 10:15

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