Darboux' theorem on the convergence of Newton's method

Question

I've only found the text in French, which I just don't read well enough to understand what is going on. Any pointers to a clear proof in English?

The paper is Darboux, "Sur la méthode d'approximation de Newton", Nouvelles annales de mathématiques: journal des candidats aux écoles polytechnique et normale, serie 2, vol 8 (1869), pp.17-27

Answer 1

In fact, concerning Newton method, Darboux theorem is quite simple and, at the same time, extremely impotant in numerical analysis.

To make the story short, what it says is that starting iterations at $x_0$,there will be one overshoot of the solution if $$f(x_0)\, f''(x_0) < 0$$ and none of them if $$f(x_0)\, f''(x_0) > 0$$ This is why we need to test this condition before starting iterations.