A family of optimal fourth-order methods for multiple roots of nonlinear equations

Abstract

[EN] Newton-Raphson method has always remained as the widely used method for finding simple and multiple roots of nonlinear equations. In the past years, many new methods have been introduced for finding multiple zeros that involve the use of weight function in the second step, thereby, increasing the order of convergence and giving a flexibility to generate a family of methods satisfying some underlying conditions. However, in almost all the schemes developed over the past, the usual way is to use Newton-type method at the first step. In this paper, we present a new two-step optimal fourth-order family of methods for multiple roots (m > 1). The proposed iterative family has the flexibility of choice at both steps. The development of the scheme is based on using weight functions. The first step can not only recapture Newton's method for multiple roots as special case but is also capable of defining new choices of first step. A stability analysis of some particular cases is also given to explain the dynamical behavior of the new methods around the multiple roots and decide the best values of the free parameters involved. Finally, we compare our methods with the existing schemes of the same order with a real life application as well as standard test problems. From the numerical results, we find that our methods can be considered as a better alternative for the existing procedures of same order.