New Sixteenth-Order Derivative-Free Methods for Solving Nonlinear Equations

New four-point derivative-free sixteenth-order iterative methods for solving nonlinear equations are con- structed. It is proved that these methods have the convergence order of sixteen requiring only five function evaluations 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 n evaluations, could achieve optimal convergence order 1 2. n− Thus, we present new derivative-free methods which agree with the Kung and Traub conjecture for 5. n = Numerical comparisons are made with other existing methods to show the performance of the presented methods.

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