On some modified families of multipoint iterative methods for multiple roots of nonlinear equations

In this paper, we propose a new one-parameter family of Schröder's method for finding the multiple roots of nonlinear equations numerically. Further, we derive many new cubically convergent families of Schröder-type methods. Proposed families are derived from the modified Newton's method for multiple roots and one-parameter family of Schröder's method. Furthermore, we introduce new families of third-order multipoint iterative methods for multiple roots free from second-order derivative by semi discrete modifications of the above proposed methods. One of the families requires two evaluations of the function and one evaluation of its first-order derivative and the other family requires one evaluation of the function and two evaluations of its first-order derivative per iteration. Numerical examples are also presented to demonstrate the performance of proposed iterative methods.