Optimal One-Point Iterative Function Free from Derivatives for Multiple Roots

Abstract

We suggest a derivative-free optimal method of second order which is a new version of a modification of Newton’s method for achieving the multiple zeros of nonlinear single variable functions. Iterative methods without derivatives for multiple zeros are not easy to obtain, and hence such methods are rare in literature. Inspired by this fact, we worked on a family of optimal second order derivative-free methods for multiple zeros that require only two function evaluations per iteration. The stability of the methods was validated through complex geometry by drawing basins of attraction. Moreover, applicability of the methods is demonstrated herein on different functions. The study of numerical results shows that the new derivative-free methods are good alternatives to the existing optimal second-order techniques that require derivative calculations.