Families of third and fourth order methods for multiple roots of nonlinear equations

This paper presents two families of higher-order iterative methods for solving multiple roots of nonlinear equations. One is of order three and the other is of order four. The presented iterative families all require two evaluations of the function and one evaluation of its first derivative, thus the latter is of optimal order. The third-order family contains several iterative methods known already. And, different from the optimal fourth-order methods for multiple roots known already, the presented fourth-order family use the modified Newton’s method as its first step. Local convergence analyses and some special cases of the presented families are given. We also carry out some numerical examples to show their performance.