Extended Convergence for Two Sixth Order Methods under the Same Weak Conditions

Argyros, Ioannis K.; Regmi, Samundra; John, Jinny Ann; Jayaraman, Jayakumar

doi:10.3390/foundations3010012

Cited by 5 publications

(4 citation statements)

References 16 publications

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“…To improve the efficiency of the one-step iterative schemes, a lot of iterative schemes based on two-step and three-step methods were depicted in [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], and some were based on the multi-composition of the functions [37][38][39][40].…”

Section: Nonlinear Perturbations Of Newton Methodsmentioning

confidence: 99%

Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations

Liu,

Chang

2024

Mathematics

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In the paper, two nonlinear variants of the Newton method are developed for solving nonlinear equations. The derivative-free nonlinear fractional type of the one-step iterative scheme of a fourth-order convergence contains three parameters, whose optimal values are obtained by a memory-dependent updating method. Then, as the extensions of a one-step linear fractional type method, we explore the fractional types of two- and three-step iterative schemes, which possess sixth- and twelfth-order convergences when the parameters’ values are optimal; the efficiency indexes are 6 and 123, respectively. An extra variable is supplemented into the second-degree Newton polynomial for the data interpolation of the two-step iterative scheme of fractional type, and a relaxation factor is accelerated by the memory-dependent method. Three memory-dependent updating methods are developed in the three-step iterative schemes of linear fractional type, whose performances are greatly strengthened. In the three-step iterative scheme, when the first step involves using the nonlinear fractional type model, the order of convergence is raised to sixteen. The efficiency index also increases to 163, and a third-degree Newton polynomial is taken to update the values of optimal parameters.

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Section: Nonlinear Perturbations Of Newton Methodsmentioning

confidence: 99%

Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations

Liu,

Chang

2024

Mathematics

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“…Cordero et al introduced a class of optimal fourth-order iterative methods with weight functions and conducted a dynamic analysis of one of the iterative methods [8]. Argyros et al presented the following sixth-order iterative method (M1) [9]:…”

Section: Introductionmentioning

confidence: 99%

A Class of Sixth-Order Iterative Methods for Solving Nonlinear Systems: The Convergence and Fractals of Attractive Basins

Wang,

2024

Fractal Fract

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In this paper, a Newton-type iterative scheme for solving nonlinear systems is designed. In the process of proving the convergence order, we use the higher derivatives of the function and show that the convergence order of this iterative method is six. In order to avoid the influence of the existence of higher derivatives on the proof of convergence, we mainly discuss the convergence of this iterative method under weak conditions. In Banach space, the local convergence of the iterative scheme is established by using the ω-continuity condition of the first-order Fréchet derivative, and the application range of the iterative method is extended. In addition, we also give the radius of a convergence sphere and the uniqueness of its solution. Finally, the superiority of the new iterative method is illustrated by drawing attractive basins and comparing them with the average iterative times of other same-order iterative methods. Additionally, we utilize this iterative method to solve both nonlinear systems and nonlinear matrix sign functions. The applicability of this study is demonstrated by solving practical chemical problems.

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“…Evaluating the local and semi-local characteristics of iterative techniques offers valuable insights into convergence traits, error limits, and the area of uniqueness for solutions [10][11][12][13][14]. Numerous research endeavors have concentrated on exploring the local and semi-local convergence of effective iterative approaches, yielding noteworthy outcomes like convergence radii, error approximations, and the broadened applicability of these methods [15][16][17][18][19]. These findings are particularly valuable as they shed light on the intricacies involved in selecting appropriate initial points for the iterative process.…”

Section: Introductionmentioning

confidence: 99%

Extended Newton-Traub Method of Order Six

K. Argyros,

Ann John,

Jayaraman

2024

Contemp. Math.

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In various scientific and engineering fields, many applications can be simplified as the task of solving equations or systems of equations within a carefully chosen abstract space. However, analytical solutions for such problems are often difficult or even impossible to find. As a result, iterative methods are widely employed to obtain the desired solutions. This article focuses on introducing a highly efficient three-step iterative method that exhibits sixth order convergence. The analysis presented here thoroughly explores the local and semi-local convergence properties, taking into account the continuity conditions imposed on the operators present in the method. The innovative methodology described in this article is not limited to specific methods but can be extended to a broader range of approaches that involve the utilization of inverse operations on linear operators or matrices.

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Extended Convergence for Two Sixth Order Methods under the Same Weak Conditions

Cited by 5 publications

References 16 publications

Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations

Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations

A Class of Sixth-Order Iterative Methods for Solving Nonlinear Systems: The Convergence and Fractals of Attractive Basins

Extended Newton-Traub Method of Order Six

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