It's great that you clarified the root-finding / minimization perspectives. Shouldn't your update equations be written in terms of F, not f? (Alternatively, wr

It's great that you clarified the root-finding / minimization perspectives. Shouldn't your update equations be written in terms of F, not f? (Alternatively, write f'(x_k) / f''(x_k).) To minimize f, find zeros of f' = F. To find a zero of F, construct a linear approximation using f''=F' If you're using Newton's method to minimize a function f, you need the first- and second-order derivatives (hence the same "second-order method").