# Second order Taylor method to solve system of equations

How do I use second order Taylor method to solve a system of non-linear equations? Is there a good reference that gives details? I found mentions of it in a dozen of numerical analysis books, but no examples

Specifically, $f:\mathbb{R}^n \to \mathbb{R}^m$, solve $f(\mathbf{x})=\mathbf{0}$ using second order Taylor expansion of $f$ around initial guess $\mathbf{x_0}$

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Which NA books have you been looking at? Note that the multidimensional version of Newton-Raphson involves expanding $f(\mathbf{x})$ up to the Jacobian-containing term (first-order). – J. M. Sep 30 '10 at 4:07
the ones that come up in google books when I search for "higher order Taylor" – Yaroslav Bulatov Sep 30 '10 at 4:14
I don't see any practical generalizations of Halley's method to multidimensional equations, if that's what you're getting at; the quadratic term involves a rank-3 tensor, and it looks unwieldy to manipulate in manner of how one would derive Halley's method from the Taylor expansion. – J. M. Sep 30 '10 at 4:24
It looks unwieldy, which is why I'm looking for some reference that goes through the details – Yaroslav Bulatov Sep 30 '10 at 4:33
Actually, what I was getting at is that there's one question you have to ask first: "how does one 'invert' a rank-3 tensor?" – J. M. Sep 30 '10 at 23:47

I still consider converting simultaneous nonlinear equations into a nonlinear least squares problem as a bit of a cheat, but... :D Anyway, what Yaroslav has originally is a $m$-dimensional vector function with $n$ independent variables, so the first term of the Taylor expansion has a vector, the second (linear) term has a matrix... you get the idea. Rather unfortunate "order" is a very overloaded mathematical term, I must say. – J. M. Sep 30 '10 at 23:44