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

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. 

