Abstract
In this paper, we present a new three-step derivative-free family based on Potra-Pták’s method for solving nonlinear equations numerically. In terms of computational cost, each member of the proposed family requires only four functional evaluations per full iteration to achieve optimal eighth-order convergence. Further, computational results demonstrate that the proposed methods are highly efficient as compared with many well-known methods.
Highlights
IntroductionMultipoint iterative methods for solving nonlinear equation are of great practical importance since they overcome the limitations of one-point methods regarding the convergence order and computational efficiency
One of the most basic and earliest problem of numerical analysis concerns with finding efficiently and accurately the simple roots of a nonlinear equation of the form f (x) = 0, (1)where f : D ⊆ R → R is a nonlinear continuous function
Multipoint iterative methods for solving nonlinear equation are of great practical importance since they overcome the limitations of one-point methods regarding the convergence order and computational efficiency
Summary
Multipoint iterative methods for solving nonlinear equation are of great practical importance since they overcome the limitations of one-point methods regarding the convergence order and computational efficiency. According to the Kung-Traub conjecture [2], the order of convergence of any multipoint method without memory requiring n function evaluations per iteration, cannot exceed the bound 2n−1 , called the optimal order. The efficiency of an iterative method is measured by the efficiency index defined by Ostrowski in [3] as p1/d , where p is the order of convergence and d is the number of functional evaluations per step. As a matter of fact, both methods maintain quadratic convergence using only two functional evaluations per full step, but Steffensen method is derivative free, which is very useful in optimization problems. We do not have any higher-order derivative-free modifications of Potra-Pták’s method till date. It is found by way of illustrations that the proposed methods are very useful in high precision computations
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