Abstract
New four-point derivative-free sixteenth-order iterative methods for solving nonlinear equations are constructed. 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 <i>n</i> evaluations, could achieve optimal convergence order Thus, we present new derivative-free methods which agree with the Kung and Traub conjecture for Numerical comparisons are made with other existing methods to show the performance of the presented methods.
Highlights
Multipoint iterative methods for solving nonlinear equations are of great practical importance since they overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency
The prime motive of this study is to develop a class of very efficient four-step derivative-free methods for solving nonlinear equations
After an extensive experimentation we were not able to designate a specific iterative method which always produces the best results for all tested nonlinear equations
Summary
Multipoint iterative methods for solving nonlinear equations are of great practical importance since they overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency. We present new derivative-free methods which agree with the Kung and Traub conjecture for n = 5 These new sixteenth-order derivative-free methods have an equivalent efficiency index to the recently established methods presented in[5,6]. The typo errors occur in the weight functions of the eighth-order derivative-free methods in (2.27) and (2.32),[16] Since these eighth-order derivative-free methods have been proved to converge of the order eight, we shall use and simplify various expressions given in[16]. The total number of function evaluations of the proposed four-point derivative-free methods is five and according to the Kung-Traub conjecture is of the optimal order[9,18]. The purpose of this paper is to establish new derivative-free methods with optimal order; we reduce the number of evaluations to five by using some suitable approximation of the derivatives.
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