An efficient twelfth-order iterative method for finding all the solutions of nonlinear equations

Soleymani, Fazlollah

doi:10.3233/jcm-2012-0447

Cited by 3 publications

(5 citation statements)

References 18 publications

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“…Convergence order of the scheme is proved using matrix method of Herzburger. Our presented scheme is less time consuming with higher efficiency index as compared to other iterative schemes [17]- [18]. The main advantage of this high order scheme is that it is totally derivative free.…”

Section: Discussionmentioning

confidence: 93%

Numerical Solution of Nonlinear Equations by a Twelth-Order Iterative Method with Memory

Hassan

Aslam²,

Asghar³

et al. 2022

ARJOM

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Some real life mathematical problems can be converted in the form of nonlinear equations. Solving such problems by analytical approaches is dificult in many situations. Hence numerical solution is the best way in this case. In this paper, a twelth-order iterative scheme for solvingnonlinear equations is presented and analyzed in terms of eciency. The new scheme is derived from the well-known King's method with order of convergence eight. We extend eighth-order King's method to an iterative method with memory of order 12:16 by using famous Newton'sinterpolating polynomial of degree 6 to avoid the derivative used in King's method. The new derived method is a three-step and is totally derivative free with twelth order of convergence. The method requires four functional evaluations at each iteration introducing high efficiency index of (12:16) 1/4 = 1:8673: Convergence order of new method is also studied. It is achieved by using matrix method of Herzburger. Numerical results are also provided to support theoretical analysis. Comparison of the derived scheme with previously well-known iterative schemes of the same order is also presented. As diffierent schemes of same order has efficiency index of (12)1/6 = 1:5131 because they requires six functional evaluations at each iteration, hence the proposed scheme is better than other schemes.

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Section: Discussionmentioning

confidence: 93%

Numerical Solution of Nonlinear Equations by a Twelth-Order Iterative Method with Memory

Hassan

Aslam²,

Asghar³

et al. 2022

ARJOM

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“…A question might arise that how the weight functions in (12) were chosen to attain as high as possible convergence order with as small as possible number of functional evaluations. Although we have tried to suggest a simple family of iterations in (12), the weight function should be chosen generally in what follows:…”

Section: Remarkmentioning

confidence: 99%

“…Thus, we have considered the initial approximations close enough to the sought zeros in numerical examples to reach the convergence. A clear hybrid algorithm written in Mathematica [11] has recently been given in [12] to provide robust initial guesses for all the real zeros of nonlinear functions in an interval. Thus, the convergence of such iterative methods could be guaranteed by following such hybrid algorithms for providing robust initial approximations.…”

Section: Numerical Reportsmentioning

confidence: 99%

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Some Iterative Methods Free from Derivatives and Their Basins of Attraction for Nonlinear Equations

Soleimani¹,

Soleymani²,

Shateyi³

2013

Discrete Dynamics in Nature and Society

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First, we make the Jain's derivative-free method optimal and subsequently increase its efficiency index from 1.442 to 1.587. Then, a novel three-step computational family of iterative schemes for solving single variable nonlinear equations is given. The schemes are free from derivative calculation per full iteration. The optimal family is constructed by applying the weight function approach alongside an approximation for the first derivative of the function in the last step in which the first two steps are the optimized derivative-free form of Jain's method. The convergence rate of the proposed optimal method and the optimal family is studied. The efficiency index for each method of the family is 1.682. The superiority of the proposed contributions is illustrated by solving numerical examples and comparing them with some of the existing methods in the literature. In the end, we provide the basins of attraction for some methods to observe the beauty of iterative nonlinear solvers in providing fractals and also choose the best method in case of larger attraction basins.

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“…Soleymani proposes a hybrid method for finding all the real solutions of nonlinear equations and dicusses the sixteenth-order method for nonlinear equations [2]. Botzoris etcs.…”

Section: Introductionmentioning

confidence: 99%

Research on enterprise information utility indifference curve of big data time

Yang

2015

JCM

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Based on the characters of information, a model of information utilities indifference curve has been build which combined information abundance and rechness. The information utilities indifference curve has been explained and discussed, and then analyses how enterprise choose information combination based on cost bind, that is the equilibrium condition of enterprise realizing the most satisfactory information utilities combination, and points out that the formed equilibrium must be broked by the advancement of information technology, this impels the change of enterprise's behavior model and organization structure, the new equilibrium comes into being.

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An efficient twelfth-order iterative method for finding all the solutions of nonlinear equations

Cited by 3 publications

References 18 publications

Numerical Solution of Nonlinear Equations by a Twelth-Order Iterative Method with Memory

Numerical Solution of Nonlinear Equations by a Twelth-Order Iterative Method with Memory

Some Iterative Methods Free from Derivatives and Their Basins of Attraction for Nonlinear Equations

Research on enterprise information utility indifference curve of big data time

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