Abstract
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 alpha = 1 and third order for other values of alpha. Some numerical examples are worked out to demonstrate the efficiency and performance of the method. On comparison of the error vertical bar x(alpha,n) - x*vertical bar, Computational order of convergence (COC) and residual f (x(alpha,n)) for the values of the parameter alpha = 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.