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How to use Newton Raphson for more than just one dimension.
Things to look out for in N-Dimensional Newton-Raphson
1.) Stationary points - gradient is 1D or det(jacobian) = 0
2.) Poor initial conditions - Try a few different initial conditions to prevent local minima.
3.) Overshoot - Change that alfa parameter to make sure you don't blow up
4.) There are many other problems but those are the most important.