- #1

PFuser1232

- 479

- 20

https://en.m.wikipedia.org/wiki/Newton-Raphson#Failure_of_the_method_to_converge_to_the_root

According to this wikipedia article, "if the first derivative is not well behaved in the neighborhood of a particular root, the method may overshoot, and diverge from that root."

However, the example cited, namely ##|x|^a## for ##0 < a < \frac{1}{2}##, is a case where the first derivative is not well behaved at the root, despite being defined for all ##x ≠ 0##. So my question is, will Newton's method always fail if the first derivative doesn't exist at the root of the function?

Also, suppose we're given a function ##f(x)## and we're asked to find the root of the function, which happens to lie in the interval ##[a, b]##. Is there a general rule of thumb for finding a good initial guess that won't result in Newton's method diverging? What if ##f(x)## has a critical number in ##[a, b]##? Will the method always fail in that case? Or is there some chance it won't (depending on the "direction of convergence", for lack of a better term)?