A family of second derivative free fourth order continuation method for solving nonlinear equations

Abstract

In this paper, we present a parameter based iterative method free from the second derivative for solving nonlinear equations of type f(x)=0. Ezquerro and Hernández (1999) discussed the convergence analysis of a uniparametric family of an iterative method in Banach space. Based on this idea, we propose a uniparametric family of second derivative free iterative method in R. We discuss the convergence analysis and observe that it possesses a fourth-order convergence for solving nonlinear equations in α∈R. Several numerical examples are worked out with our proposed methods for parameter α=1, α=2 and the existing fourth-order iterative method proposed in King (1973), Chun (2007), Soleymani et al. (2012) and Maheshwari (2009). Finally, from the comparison, we observe that our method is more efficient than existing methods. Finally, we compare basins of attraction of our methods with the second derivative free fourth-order iterative method proposed in King (1973), Chun (2007), Soleymani et al. (2012) and Maheshwari (2009) observe that the proposed scheme is more efficient.