Solving nonlinear equations by a derivative-free form of the King’s family with memory

Sharifi, Somayeh; Siegmund, Stefan; Salimi, Mehdi

doi:10.1007/s10092-015-0144-1

Cited by 24 publications

(16 citation statements)

References 17 publications

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“…We construct in this section a new optimal three-point method for solving nonlinear equations by using a Newton-step and Newton's interpolation polynomial of degree three which was also applied in [22].…”

Section: Description Of the Methods And Convergence Analysismentioning

confidence: 99%

An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics

Matthies

Salimi

Sharifi

et al. 2016

Japan J. Indust. Appl. Math.

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We present a three-point iterative method without memory for solving nonlinear equations in one variable. The proposed method provides convergence order eight with four function evaluations per iteration. Hence, it possesses a very high computational efficiency and supports Kung and Traub's conjecture. The construction, the convergence analysis, and the numerical implementation of the method will be presented. Using several test problems, the proposed method will be compared with existing methods of convergence order eight concerning accuracy and basin of attraction. Furthermore, some measures are used to judge methods with respect to their performance in finding the basin of attraction.

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Section: Description Of the Methods And Convergence Analysismentioning

confidence: 99%

An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics

Matthies

Salimi

Sharifi

et al. 2016

Japan J. Indust. Appl. Math.

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“…In this part, we will derive the convergence analysis of the proposed schemes in Equations (5), (9), and (13) with the help of MATHEMATICA software.…”

Section: Convergence Analysismentioning

confidence: 99%

“…The method provides a convergence order of eight with four function evaluations per iteration. Sharifi et al [9] presented an iterative method with memory based on the family of King's methods to solve nonlinear equations. The method has eighth-order convergence and costs only four function evaluations per iteration.…”

Section: Introductionmentioning

confidence: 99%

Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction

Li¹,

Madhu

2019

Mathematics

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Based on the Steffensen-type method, we develop fourth-, eighth-, and sixteenth-order algorithms for solving one-variable equations. The new methods are fourth-, eighth-, and sixteenth-order converging and require at each iteration three, four, and five function evaluations, respectively. Therefore, all these algorithms are optimal in the sense of Kung–Traub conjecture; the new schemes have an efficiency index of 1.587, 1.682, and 1.741, respectively. We have given convergence analyses of the proposed methods and also given comparisons with already established known schemes having the same convergence order, demonstrating the efficiency of the present techniques numerically. We also studied basins of attraction to demonstrate their dynamical behavior in the complex plane.

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“…In Tables 1, 2, 3 and 4, the proposed methods (13), (15) and (17) with the methods (18), (20) and (22) have been tested on different nonlinear equations. It is clear that these methods are in accordance with the developed theory.…”

Section: Numerical Performancementioning

confidence: 99%

“…Since then, there have been many attempts to construct optimal multi-point methods, utilizing e.g. weight functions [8][9][10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

New modification of Maheshwari’s method with optimal eighth order convergence for solving nonlinear equations

et al. 2016

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Abstract:In this paper, we present a family of three-point with eight-order convergence methods for finding the simple roots of nonlinear equations by suitable approximations and weight function based on Maheshwari's method. Per iteration this method requires three evaluations of the function and one evaluation of its first derivative. These class of methods have the efficiency index equal to 8 1 4 1:682. We describe the analysis of the proposed methods along with numerical experiments including comparison with the existing methods. Moreover, the attraction basins of the proposed methods are shown with some comparisons to the other existing methods.

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Solving nonlinear equations by a derivative-free form of the King’s family with memory

Cited by 24 publications

References 17 publications

An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics

An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics

Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction

New modification of Maheshwari’s method with optimal eighth order convergence for solving nonlinear equations

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