In reply to gordon Keener.

Yes, as the power gets higher, the function x**n - S is really really flat on [0,1] and really really steep for x greater than 2. Therefore an initial guess less than 1 shoots out and results in a next guess that is very large. (As it says in the fourth bullet, "avoid inputs where the derivative is close to zero.") When x is large, the function is so steep that it takes many iterations to march back towards the root.

On the other hand, one you get close to the root, Newton's method will converge REALLY fast! So use a bounding method to search for an initial guess and then use two or three iterates of Newton's method to home in on the accuracy.