Finding simple roots by seventh- and eighth-order derivative-free methods
نویسندگان
چکیده
Nonlinear equation solving by without memory iterative methods is taken into account in the present research. Recently, Khattri and Argyros in [S.K. Khattri, I.K. Argyros, Sixth order derivative free family of iterative methods, Appl. Math. Comput. 217 (2011), 5500-5507], proposed a sixth-order family of derivative-free methods including four function evaluations per full cycle to reach the index of efficiency 1.565. In this work, we develop new derivative-free without memory methods, based on the abovementioned work, in which the convergence rates reach the seventhand eighth-order respectively. And subsequently, the index of efficiency will be increased to 1.626 and 1.682. This shows that our proposed methods are more economic than their work in terms of convergence rate and the efficiency index. Moreover, the numerical examples are considered to support the theoretical results and put on show that the contributions in this paper hit the targets. KeywordsNonlinear equations, simple root, iterative methods, derivative-free, efficiency index, order of convergence, Steffensens's method.
منابع مشابه
THIRD-ORDER AND FOURTH-ORDER ITERATIVE METHODS FREE FROM SECOND DERIVATIVE FOR FINDING MULTIPLE ROOTS OF NONLINEAR EQUATIONS
In this paper, we present two new families of third-order and fourth-order methods for finding multiple roots of nonlinear equations. Each of them requires one evaluation of the function and two of its first derivative per iteration. Several numerical examples are given to illustrate the performance of the presented methods.
متن کاملThree-step iterative methods with eighth-order convergence for solving nonlinear equations
A family of eighth-order iterative methods for solution of nonlinear equations is presented. We propose an optimal three-step method with eight-order convergence for finding the simple roots of nonlinear equations by Hermite interpolation method. Per iteration of this method requires two evaluations of the function and two evaluations of its first derivative, which implies that the efficiency i...
متن کاملSeventh-order iterative algorithm free from second derivative for solving algebraic nonlinear equations
متن کامل
A Family of Optimal Derivative Free Iterative Methods with Eighth-Order Convergence for Solving Nonlinear Equations
In this paper, modification of Steffensen’s method with eight-order convergence is presented. We propose a family of optimal three-step methods with eight-order convergence for solving the simple roots of nonlinear equations by using the weight function and interpolation methods. Per iteration this method requires four evaluations of the function which implies that the efficiency index of the d...
متن کاملCorrigendum to "Basins of attraction for optimal eighth-order methods to find simple roots of nonlinear equations"
Several optimal eighth order methods to obtain simple roots are analyzed. The methods are based on two step, fourth order optimal methods and a third step of modified Newton. The modification is performed by taking an interpolating polynomial to replace either f ðznÞ or f ðznÞ. In six of the eight methods we have used a Hermite interpolating polynomial. The other two schemes use inverse interpo...
متن کامل