Simultaneous determination of multiple zeros via Schröder–Ehrlich-type corrections

In this work, we introduce new simultaneous iterative methods for computing multiple roots of nonlinear equations without the knowledge of the multiplicity. The proposed schemes are obtained by composing Schröder-type predictors with Ehrlich-type correction steps. We first show that the combination of Schröder’s method with a single Ehrlich correction yields a fourth-order simultaneous method for multiple roots. More generally, we prove that if a method of order p for finding multiple roots is used as a predictor, then the subsequent Ehrlich correction produces a method of order 2 p . This result holds both when Ehrlich’s summation is constructed using the nodes from the previous iterate and when it is built from the intermediate predictor step. The stability of the resulting families is investigated through a complex dynamical analysis adapted to the simultaneous multiple-root setting, showing favorable stability properties. Finally, several numerical experiments on polynomial and nonlinear problems arising in science and engineering confirm the theoretical convergence orders and demonstrate the efficiency and robustness of the proposed methods in the presence of multiple roots.