Abstract
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 onnevaluations could achieve optimal convergence order of . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for . Numerical comparisons are made to demonstrate the performance of the methods presented.
Highlights
IntroductionWe present a new family of the eighth-order methods to find a simple root α of the nonlinear equation:
In this paper, we present a new family of the eighth-order methods to find a simple root α of the nonlinear equation:f x 0, 1.1 where f : D ⊂ R → R is a scalar function on an open interval D and it is sufficiently smooth in a neighbourhood of α
We present a new family of the eighth-order methods to find a simple root α of the nonlinear equation: f x 0, 1.1 where f : D ⊂ R → R is a scalar function on an open interval D and it is sufficiently smooth in a neighbourhood of α
Summary
We present a new family of the eighth-order methods to find a simple root α of the nonlinear equation:. The eighth-order methods presented in this paper are derivative-free and only use four evaluations of the function per iteration. We present new derivative-free methods which agree with the Kung and Traub conjecture for n 4. The new eighth-order derivativefree methods have a better efficiency index than the sixth-order derivative-free methods presented recently in 6, 7 and in view of this fact, the new methods are significantly better when compared with the established methods. We will define a new family of eighth-order derivative-free methods. The efficiency of the new method is measured by the concept of efficiency index 9, 10 and defined as μ1/β , 2.3 where μ is the order of the method
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