Suppose you need to solve $f(\mathbf{x})=\mathbf{0}$ where $f:\mathbb{R}^n \to \mathbb{R}^m$, $m,n>1$. Newton's method relies on first order Taylor expansion of f. Where can I find details of analogous method using second order Taylor expansion? I found at least a dozen numerical analysis books which mention this method, but give no details or applications
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The singlevariable version of what you are looking for is called Halley's method. (See, for example, MathWorld's article on Halley's method.) Maybe there's a fairly straightforward way to generalize it to multivariable functions. Or, if nothing else, this gives you another search term. 


I haven't gotten around to downloading and reading it (and I'm wondering how I missed this when I was searching for results related to Halley's method), but apparently a multivariate version of the Halley iteration has already been developed decades ago. Maybe this might be of use. 

