Modified Ostrowski’s method with eighth-order convergence and high efficiency index

Wang, Xia; Liu, Liping

doi:10.1016/j.aml.2010.01.009

Cited by 68 publications

(38 citation statements)

References 24 publications

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“…Neta [13] has used inverse interpolation. Wang and Liu [25] have developed a method based on Hermite interpolation to remove the derivative in the third step. They also use Ostrowski's method which is a special case of King's method when β = 0.…”

Section: Optimal Eighth-order Family Of Methodsmentioning

confidence: 99%

An Analysis of a King-based Family of Optimal Eighth-order Methods

Chun

Neta

2015

AJAC

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In this paper we analyze an optimal eighth-order family of methods based on King's fourth order method to solve a nonlinear equation. This family of methods was developed by Thukral and Petkovićand uses a weight function. We analyze the family using the information on the extraneous ixed points. Two measures of closeness of an extraneous points set to the imaginary axis are considered and applied to the members of the family to ind its best performer. The results are compared to a modi ied version of Wang-Liu method.

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Section: Optimal Eighth-order Family Of Methodsmentioning

confidence: 99%

An Analysis of a King-based Family of Optimal Eighth-order Methods

Chun

Neta

2015

AJAC

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“…In [11] we have compared several members of the family LSSS to the methods cited there and to the method by Wang and Liu [12] denoted WL which showed good results in our previous work (see, for example, Chun and Neta [11,13]. The method WL is as follows:…”

Section: Introductionmentioning

confidence: 99%

“…

a r t i c l e i n f o MSC: 65H05 65B99,

Keywords:Iterative methods Order of convergence Basin of attraction Extraneous fixed points a b s t r a c t Several families of optimal eighth order methods to find simple roots are compared to the best known eighth order method due to Wang and Liu (2010). We have tried to improve their performance by choosing the free parameters of each family using two different criteria.

Published by Elsevier Inc.

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mentioning

confidence: 99%

Comparison of several families of optimal eighth order methods

Chun

Neta

2016

Applied Mathematics and Computation

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“…According to Kunge-Traub [11] conjecture, an iterative method, without memory for solving a single nonlinear equation, could achieve maximum convergence order 2 s−1 , and s is the total number of function evaluations. The iterative methods for solving nonlinear equations [5,9,13,16,19] have attained the proper attention of a large community of researchers. Sometimes it is possible to generalize an iterative method for solving nonlinear equations with an iterative method for solving system of nonlinear equations.…”

Section: Introductionmentioning

confidence: 99%

Frozen jacobian iterative method for solving systems of nonlinear equations application to nonlinear IVPs and BVPs

Ullah¹,

Ahmad²,

Alzahrani³

et al. 2016

J. Nonlinear Sci. Appl.

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Frozen Jacobian iterative methods are of practical interest to solve the system of nonlinear equations. A frozen Jacobian multi-step iterative method is presented. We divide the multi-step iterative method into two parts namely base method and multi-step part. The convergence order of the constructed frozen Jacobian iterative method is three, and we design the base method in a way that we can maximize the convergence order in the multi-step part. In the multi-step part, we utilize a single evaluation of the function, solve four systems of lower and upper triangular systems and a second frozen Jacobian. The attained convergence order per multi-step is four. Hence, the general formula for the convergence order is 3 + 4(m − 2) for m ≥ 2 and m is the number of multi-steps. In a single instance of the iterative method, we employ only single inversion of the Jacobian in the form of LU factors that makes the method computationally cheaper because the LU factors are used to solve four system of lower and upper triangular systems repeatedly. The claimed convergence order is verified by computing the computational order of convergence for a system of nonlinear equations. The efficiency and validity of the proposed iterative method are narrated by solving many nonlinear initial and boundary value problems. c 2016 All rights reserved.Keywords: Frozen Jacobian iterative methods, multi-step iterative methods, systems of nonlinear equations, nonlinear initial value problems, nonlinear boundary value problems. 2010 MSC: 65H10, 65N22.

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Modified Ostrowski’s method with eighth-order convergence and high efficiency index

Cited by 68 publications

References 24 publications

An Analysis of a King-based Family of Optimal Eighth-order Methods

An Analysis of a King-based Family of Optimal Eighth-order Methods

Comparison of several families of optimal eighth order methods

Frozen jacobian iterative method for solving systems of nonlinear equations application to nonlinear IVPs and BVPs

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