Convergence and Dynamics of Schröder’s Method for Zeros of Analytic Functions with Unknown Multiplicity

In this paper, we investigate the local convergence of Schröder’s method for finding zeros of analytic functions with unknown multiplicity. Thus, we obtain a convergence theorem that provides exact domains of initial points together with error estimates to ensure the Q-quadratic convergence of Schröder’s method right from the first step. A comparison with the famous Newton’s method, based on the convergence and dynamics when it is applied to some polynomial and non-polynomial equations, is also provided.