Two new efficient sixth order iterative methods for solving nonlinear equations

Solaiman, Obadah Said; Hashim, Ishak

doi:10.1016/j.jksus.2018.03.021

Cited by 20 publications

(9 citation statements)

References 9 publications

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“…The point 0 is called a fixed point for if ( 0 ) = 0 . LetĈ be the Riemann sphere; then for ∈Ĉ, we define the orbit of as orb( ) = { , ( ), [2] We call the closure of the set of its repelling periodic points 'Julia set' ( ) of a nonlinear function ( ). Fatou set ( ) is defined to be the complement of Julia set.…”

Section: Dynamical Comparisonmentioning

confidence: 99%

“…Fatou set ( ) is defined to be the complement of Julia set. If is an attracting orbit which is periodic with period , then the basin of attraction is defined to be the open set ∈Ĉ of all points ∈Ĉ such that the consecutive iterates [ ] ( ), [2 ] ( ), . .…”

Section: Dynamical Comparisonmentioning

confidence: 99%

“…which has second-order of convergence [1]. Many researchers have improved the method of Newton to attain better results and to increase the convergence order; for instance, see [2][3][4] and the references therein. One of the most famous improvements of Newton's scheme is the technique of order three given in [5]:…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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Section: Dynamical Comparisonmentioning

confidence: 99%

Section: Dynamical Comparisonmentioning

confidence: 99%

See 1 more Smart Citation

Efficacy of Optimal Methods for Nonlinear Equations with Chemical Engineering Applications

Solaiman

Hashim

2019

Mathematical Problems in Engineering

Self Cite

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“…al. [23] introduced two novel and efficient rootfinding techniques with sixth-order convergence. The first approach is based on Halley's method and Taylor's expansion, while the second method employs second derivative approximations to enhance the efficiency of the first method.…”

Section: Introductionmentioning

confidence: 99%

Graphical Insights into Dynamical Behavior: Advanced Iterative Methods for Root Finding in Computational Mathematics

Raza,

Toheed,

Haq

et al. 2024

Preprint

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This paper introduces a novel family of Newton-like iterative methods ranging from fourth to seventh order, specifically designed for the efficient determination of simple roots in nonlinear equations. Outperforming existing methods, our proposed techniques exhibit superior stability and expanded basin sizes. Their unprecedented reliability and broad applicability position them as valuable assets for addressing real-world challenges in diverse domains such as Mathematical Modeling, Biomathematics, and Thermodynamics. To illuminate the dynamic behavior of these methods, we employ the graphical tool 'Basin of Attraction'. Through comprehensive analysis, our research demonstrates a significant enhancement in accuracy and convergence rates, presenting a modest yet impactful contribution to the realm of computational mathematics. Mathematics Subject Classification: 65H04, 65H05

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“…In most scientific and engineering applications, the problem of finding the solution of nonlinear equations have become an active area of research. Many researchers have explored various order iterative methods to find solutions of the nonlinear equations using various techniques such as homotopy perturbation technique, variational iterative methods and decomposition technique, for details, see [1][2][3][4][5][6][7][8][9][10][11]. Firstly, Traub [12] initiated the study of the iterative methods for the solution of the nonlinear equations and introduced a basic quadratic convergent Newton iterative method for the solution of the nonlinear equations, which have much significance in the literature.…”

Section: Introductionmentioning

confidence: 99%

On Iterative Methods for Solving Nonlinear Equations in Quantum Calculus

et al. 2021

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Quantum calculus (also known as the q-calculus) is a technique that is similar to traditional calculus, but focuses on the concept of deriving q-analogous results without the use of the limits. In this paper, we suggest and analyze some new q-iterative methods by using the q-analogue of the Taylor’s series and the coupled system technique. In the domain of q-calculus, we determine the convergence of our proposed q-algorithms. Numerical examples demonstrate that the new q-iterative methods can generate solutions to the nonlinear equations with acceptable accuracy. These newly established methods also exhibit predictability. Furthermore, an analogy is settled between the well known classical methods and our proposed q-Iterative methods.

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Two new efficient sixth order iterative methods for solving nonlinear equations

Cited by 20 publications

References 9 publications

Efficacy of Optimal Methods for Nonlinear Equations with Chemical Engineering Applications

Efficacy of Optimal Methods for Nonlinear Equations with Chemical Engineering Applications

Graphical Insights into Dynamical Behavior: Advanced Iterative Methods for Root Finding in Computational Mathematics

On Iterative Methods for Solving Nonlinear Equations in Quantum Calculus

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