Optimal fourth- and eighth-order of convergence derivative-free modifications of King’s method

Solaiman, Obadah Said; Karim, Samsul Ariffin Abdul; Hashim, Ishak

doi:10.1016/j.jksus.2018.12.001

Cited by 18 publications

(23 citation statements)

References 31 publications

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“…The proposed scheme having fourth-order convergence. Different types of transcendental and algebraic problems solved which show that the proposed scheme is more accurate than the existing schemes presented in [6,7]. From the tables it observed, the proposed scheme is better in terms of CPU time, number of iterations and reducing error more frequently than the existing schemes.…”

Section: Resultsmentioning

confidence: 82%

“…Determining the root of a nonlinear equation is very important; researchers have developed numerical methods by involving derivatives [1]. Many algorithms have been introduced to accelerate the convergence of numerical methods without involving derivative [2,3,4,5,6,7]. By using covenant and suitable selection of parameters to reduce the number of evaluation of numerical method [2,8,9,6].…”

Section: Methods Articlementioning

confidence: 99%

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Comparison of Proposed and Existing Fourth Order Schemes for Solving Non-linear Equations

Rajput

Shaikh

Qureshi

2019

ARJOM

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This paper, investigates the comparison of the convergence behavior of the proposed scheme and existing schemes in literature. While all schemes having fourth-order convergence and derivative-free nature. Numerical approximation demonstrates that the proposed schemes are able to attain up to better accuracy than some classical methods, while still significantly reducing the total number of iterations. This study has considered some nonlinear equations (transcendental, algebraic and exponential) along with two complex mathematical models. For better analysis graphical representation of numerical methods for finding the real root of nonlinear equations with varying parameters has been included. The proposed scheme is better in reducing error rapidly, hence converges faster as compared to the existing schemes.

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Section: Resultsmentioning

confidence: 82%

Section: Methods Articlementioning

confidence: 99%

Comparison of Proposed and Existing Fourth Order Schemes for Solving Non-linear Equations

Rajput

Shaikh

Qureshi

2019

ARJOM

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show abstract

“…Several optimal fourth-order iterative methods were constructed; see, for example, [9][10][11]. The optimal eighth-order of convergence was reached by many authors as presented in [12][13][14]. Also, many sixteenth-order iterative methods were proposed; for instance, see [15][16][17].…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

Efficacy of Optimal Methods for Nonlinear Equations with Chemical Engineering Applications

Solaiman

Hashim

2019

Mathematical Problems in Engineering

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In this study, we propose a modified predictor-corrector Newton-Halley (MPCNH) method for solving nonlinear equations. The proposed sixteenth-order MPCNH is free of second derivatives and has a high efficiency index. The convergence analysis of the modified method is discussed. Different problems were tested to demonstrate the applicability of the proposed method. Some are real life problems such as a chemical equilibrium problem (conversion in a chemical reactor), azeotropic point of a binary solution, and volume from van der Waals equation. Several comparisons with other optimal and nonoptimal iterative techniques of equal order are presented to show the efficiency of the modified method and to clarify the question, are the optimal methods always good for solving nonlinear equations?

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“…After that Kumar et al [23], in 2018, established a new sixth-order parameter based family of algorithm for solving non-linear equations. In 2019, Solaiman et al [38] suggested derivative free optimal fourth-order and eighthorder modifications of King's method by implementing the composition technique combined with rational interpolation, and the idea of Padé approximation. Very recently, Naseem et al [27] presented some new ninth order iterative algorithms for determining the zeros of non-linear scalar equations and then presented their graphical representation by means of polynomiographs using different complex polynomials.…”

Section: Introductionmentioning

confidence: 99%

Novel Iteration Schemes for Computing Zeros of Non-Linear Equations With Engineering Applications and Their Dynamics

et al. 2021

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The task of root-finding of the non-linear equations is perhaps, one of the most complicated problems in applied mathematics especially in a diverse range of engineering applications. The characteristics of the root-finding methods such as convergence rate, performance, efficiency, etc., is directly rely upon the initial guess of the solution to execute the process in most of the systems of non-linear equations. Keeping these facts into mind, based on Taylor's series expansion, we present some new modifications of Halley, Househölder and Golbabai and Javidi's methods and then making them second derivative free by applying Taylor's series. The convergence analysis of the suggested methods is discussed. It is established that the proposed methods possess convergence of orders five and six. Several numerical problems have been tested to demonstrate the validity and applicability of the proposed methods. These test examples also include some real-life problems associated with the chemical and civil engineering such as open channel flow problem, the adiabatic flame temperature equation, conversion of nitrogen-hydrogen feed to ammonia and the van der Wall's equation whose numerical results prove the better performance of the suggested method as compared to other well known existing methods of the same kind in literature. Finally, the dynamics of the presented algorithms in the form of polynomiographs have been shown with the aid of computer program by considering some complex polynomials and compared them with the other wellknown iterative algorithms that revealed the convergence speed and other dynamical aspects of the presented methods.

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Optimal fourth- and eighth-order of convergence derivative-free modifications of King’s method

Cited by 18 publications

References 31 publications

Comparison of Proposed and Existing Fourth Order Schemes for Solving Non-linear Equations

Comparison of Proposed and Existing Fourth Order Schemes for Solving Non-linear Equations

Efficacy of Optimal Methods for Nonlinear Equations with Chemical Engineering Applications

Novel Iteration Schemes for Computing Zeros of Non-Linear Equations With Engineering Applications and Their Dynamics

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