A new fifth-order iterative method free from second derivative for solving nonlinear equations

Abdul-Hassan, Noori Yasir; Ali, Ali Hasan; Park, Choonkil

doi:10.1007/s12190-021-01647-1

Cited by 20 publications

(15 citation statements)

References 14 publications

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“…al. [19] introduced a two-step iterative method having fifth-order convergence, avoiding the use of second derivatives of functions, and employing Halley's method and Taylor's expansion, utilizing Hermite orthogonal polynomial basis. In 2022, Sana et al [20] introduced a family of iterative methods using quadrature formula and decomposition technique.…”

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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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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“…Equation ( 2) has a quadratic order of convergence for m ≥ 1. While there exist numerous with and without memory iterative methods for computing simple roots [1][2][3][4][5][6], the task of developing efficient methods for finding multiple roots remains challenging. Despite the availability of several without memory iterative methods for multiple roots (see [7][8][9][10] and the references therein), there is a paucity of methods that utilise multiple points with memory and accelerating parameters for computing multiple roots.…”

Section: Introductionmentioning

confidence: 99%

Novel Parametric Families of with and without Memory Iterative Methods for Multiple Roots of Nonlinear Equations

et al. 2023

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The methods that use memory using accelerating parameters for computing multiple roots are almost non-existent in the literature. Furthermore, the only paper available in this direction showed an increase in the order of convergence of 0.5 from the without memory to the with memory extension. In this paper, we introduce a new fifth-order without memory method, which we subsequently extend to two higher-order with memory methods using a self-accelerating parameter. The proposed with memory methods extension demonstrate a significant improvement in the order of convergence from 5 to 7, making this the first paper to achieve at least a 2-order improvement. In addition to this improvement, our paper is also the first to use Hermite interpolating polynomials to approximate the accelerating parameter in the proposed with memory methods for multiple roots. We also provide rigorous theoretical proofs of convergence theorems to establish the order of the proposed methods. Finally, we demonstrate the potential impact of the proposed methods through numerical experimentation on a diverse range of problems. Overall, we believe that our proposed methods have significant potential for various applications in science and engineering.

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“…The introduction of higher order derivative often presents challenges including increased computational costs and practical application difficulties. Consequently, many researchers seek to find free second derivatives iterative methods with various approaches as evidenced in works such as [3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Modified Householder Method of Fifth Order of Convergence and Its Dynamics on Complex Plane

Putri,

Imran,

Marjulisa

2023

JFMA

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In this paper, a modified Householder method of fifth order is proposed for solving nonlinear equations. The modification is done by adapting a cubic interpolation polynomial to approximate the second derivative in the Householder method. Weprovide a theorem to prove the order of convergence of the proposed method. The simulations reveal that the proposed method needs fewer iterations, even with challenging initial guesses, and excels in sending a large portion of initial points to convergence and exhibits rapid convergence.

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A new fifth-order iterative method free from second derivative for solving nonlinear equations

Cited by 20 publications

References 14 publications

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

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

Novel Parametric Families of with and without Memory Iterative Methods for Multiple Roots of Nonlinear Equations

Modified Householder Method of Fifth Order of Convergence and Its Dynamics on Complex Plane

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