New Families of Fourth-Order Derivative-Free Methods for Solving Nonlinear Equations with Multiple Roots

In this paper, two new fourth-order derivative-free methods for finding multiple zeros of nonlinear equations are presented. In terms of computational cost the family requires three evaluations of functions per iteration. It is proved that the each of the methods has a convergence of order four. In this way it is demonstrated that the proposed class of methods supports the Kung-Traub hypothesis (1974) on the upper bound 2 of the order of multipo int methods based on 1 n + function evaluations. Numerical examples suggest that the new methods are competitive to other fourth-order methods for multip le roots.