Methods with prefixed order for approximating square roots with global and general convergence

摘要

A family of Newton-like methods is constructed to approximate the square root of a positive real number. The iterative methods of the family are global or generally convergent depending on the prefixed order of convergence is even or odd. We show the dynamical behaviour of some of these methods by means of several Julia sets and the intricate fractal structures which arise from the order of the iterative methods are displayed.