A new class of three-point methods with optimal convergence order eight and its dynamics

Lotfi, Taher; Sharifi, Somayeh; Salimi, Mehdi; Siegmund, Stefan

doi:10.1007/s11075-014-9843-y

Cited by 52 publications

(34 citation statements)

References 22 publications

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“…Assume that all assumptions of Theorem 1 are fulfilled. Then the family of three-point iterative methods (2) has an eighth-order of convergence if and only if the parameters τ n and α n are given by (6), (8) and…”

Section: Proper Representation Of the Optimal Order Three-point Iteramentioning

confidence: 99%

“…In recent years, a number of higher-order iterative methods have been developed and analyzed on this issue, see [1][2][3][4][5][6][7][8][9][10][11] and references therein. Motivated by the recent results in [11], in this paper we introduce a generating function method for the construction of new two and three-point iterations with p-th order of convergence.…”

Section: Introductionmentioning

confidence: 99%

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Generating Function Approach to the Derivation of Higher-Order Iterative Methods for Solving Nonlinear Equations

2018

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Abstract. In this paper we propose a generating function method for constructing new two and three-point iterations with p (p = 4, 8) order of convergence. This approach allows us to derive a new family of optimal order iterative methods that include well known methods as special cases. Necessary and sufficient conditions for p-th (p = 4, 8) order convergence of the proposed iterations are given in terms of parameters τ n and α n . We also propose some generating functions for τ n and α n . We develop a unified representation of all optimal eighth-order methods. The order of convergence of the proposed methods is confirmed by numerical experiments.

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Section: Proper Representation Of the Optimal Order Three-point Iteramentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generating Function Approach to the Derivation of Higher-Order Iterative Methods for Solving Nonlinear Equations

2018

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“…The basin of attraction is a method to visually comprehend how an algorithm behaves as a function of the various starting points [12,13]. In this section, polynomial and rational functions have been considered, which are the same functions that appear in [14][15][16]:…”

Section: Dynamical Analysismentioning

confidence: 99%

A Novel Geometric Modification to the Newton-Secant Method to Achieve Convergence of Order1+2and Its Dynamics

Fernández-Torres

2015

Modelling and Simulation in Engineering

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A geometric modification to the Newton-Secant method to obtain the root of a nonlinear equation is described and analyzed. With the same number of evaluations, the modified method converges faster than Newton's method and the convergence order of the new method is 1 + √ 2 ≈ 2.4142. The numerical examples and the dynamical analysis show that the new method is robust and converges to the root in many cases where Newton's method and other recently published methods fail.

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“…Lotfi et al [5] have developed an eighth order family of optimal methods (denoted LSSS) y n = x n − v n , z n = y n − v n t n 1 − 2t n ,…”

Section: Introductionmentioning

confidence: 99%

“…(8) They also include several methods and the following families of methods in their comparative study [5]:…”

Section: Introductionmentioning

confidence: 99%

Comparison of several families of optimal eighth order methods

Chun

Neta

2016

Applied Mathematics and Computation

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A new class of three-point methods with optimal convergence order eight and its dynamics

Cited by 52 publications

References 22 publications

Generating Function Approach to the Derivation of Higher-Order Iterative Methods for Solving Nonlinear Equations

Generating Function Approach to the Derivation of Higher-Order Iterative Methods for Solving Nonlinear Equations

A Novel Geometric Modification to the Newton-Secant Method to Achieve Convergence of Order1+2and Its Dynamics

Comparison of several families of optimal eighth order methods

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