Choosing starting values for certain Newton–Raphson iterations

We aim at finding the best possible seed values when computing a1/p using the Newton–Raphson iteration in a given interval. A natural choice of the seed value would be the one that best approximates the expected result. It turns out that in most cases, the best seed value can be quite far from this natural choice. When we evaluate a monotone function f(a) in the interval [amin,amax], by building the sequence xn defined by the Newton–Raphson iteration, the natural choice consists in choosing x0 equal to the arithmetic mean of the endpoint values. This minimizes the maximum possible distance between x0 and f(a). And yet, if we perform n iterations, what matters is to minimize the maximum possible distance between xn and f(a). In several examples, the value of the best starting point varies rather significantly with the number of iterations.