Second order derivative free continuation method for solving nonlinear equations in ?

In this paper we develop the second order derivative free variants of a parameter based continuation method combining the Halley´s and the Super-Halley´s methods for solving nonlinear equations of the type f(x) = 0. The convergence analysis of method is discussed in ℝ. This shows that the new method has fourth-order convergence for α = 1 and third order for other values of α. Some numerical examples are worked out to demonstrate the efficiency and performance of the method. On comparison of the error |x_α,n - x*|, Computational order of convergence (COC) and residual f(x_α,n) for the values of the parameter α = 1 by our method with those obtained by the Halley´s and the Super-Halley´s methods. We observed that our method gives improved results. Also, we have compared our method with some existing methods by basins of attraction and observed that the proposed scheme is more efficient.