- #1

Purplepixie

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I am having difficulty in translating the univariate Newton's approximation {Xn = Xn-1 - [ f(Xn-1) / f'(Xn-1)]} into the multidimensional case. My multidimensional equation system is y = F.x where y and x are (nx1) column vectors and the coefficients matrix F is (nxn), so that (nx1) = (nxn).(nx1) = (nx1).

I have translated the univariate Newton's approximation to the n-variate case as:

x2 = x1 - (F.x1 / F'.x1) <=> x1 - (F.x1).Inv(F'.x1)

But F'.x1 is a nx1 column vector and has no inverse. I then thought that perhaps the x1's cancel out . But if so then we would have x2 = x1 - F.Inv(F') with the last term a nxn matrix, so (nx1) = (nx1) - (nxn), which is not possible.

Your assistance would be greatly appreciated!

(This is my first post incidentally, so pls excuse any breaches of protocol)