New iterative methods for nonlinear equations in R

A parameter based on two-step iterative method combining two known third order methods is proposed for solving nonlinear equations in R. The convergence analysis of the method is established to show its fourth order of convergence. In order to enhance the order of convergence from four to seven, its three-step extension is also developed. The increase in the number of function evaluations is reduced by approximating the involved derivative in the third step by the second order divided differences. A number of numerical examples are worked out to demonstrate the efficacy of both the methods. The performance measures used are the number of iterations and the total number of function evaluations required to get an approximation to the root correct up to fifteen decimal places. The informational index and efficiency index of the proposed methods are computed. On comparison with the existing methods, it is observed that our methods give improved performance in terms of computational speed and efficiency.