A family of simultaneous zero-finding methods

Applying Hansen-Patrickʹs formula for solving the single equation f(z) = 0 to a suitable function appearing in the classical Weierstrassʹ method, two one-parameter families of interation functions for the simultaneous approximation of all simple and multiple zeros of a polynomial are derived. It is shown that all the methods of these families have fourth-order of convergence. Some computational aspects of the proposed methods and numerical examples are given.