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A new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on  evaluations, could achieve optimal convergence order 2−1. Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for =4. Numerical comparisons are made to demonstrate the performance of the methods presented.

Eighth-Order Iterative Methods without Derivatives for Solving Nonlinear Equations