Ball Comparison between Three Sixth Order Methods for Banach Space Valued Operators

Behl, Ramandeep; Argyros, Ioannis K.; Machado, J. A. Tenreiro

doi:10.3390/math8050667

Cited by 3 publications

(2 citation statements)

References 16 publications

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“…We choose λ = 0, λ = 0.5 and λ = 1 in our scheme (2), called by (PS1), (PS2) and (PS3), respectively. Now, we compare our schemes with a 6th-order iterative methods suggested by Abbasbandy et al [19] and Hueso et al [20], among them we picked the methods (8) and (14)(15) for t 1 = − 9 4 and s 2 = 9 8 , respectively, known as (AS) and (HS). Moreover, a comparison of them has been done with the 6th-order iterative methods given by Wang and Li [21], among their method we chose expression (6), denoted by and (WS).…”

Section: Applications With Large Systemsmentioning

confidence: 99%

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Some High-Order Iterative Methods for Nonlinear Models Originating from Real Life Problems

2020

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We develop a sixth order Steffensen-type method with one parameter in order to solve systems of equations. Our study’s novelty lies in the fact that two types of local convergence are established under weak conditions, including computable error bounds and uniqueness of the results. The performance of our methods is discussed and compared to other schemes using similar information. Finally, very large systems of equations (100×100 and 200×200) are solved in order to test the theoretical results and compare them favorably to earlier works.

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Section: Applications With Large Systemsmentioning

confidence: 99%

“…Here, F : Ω ⊂ B → B, is differentiable, B is a Banach space and Ω is nonempty and open. Closed form solutions are rarely found, so iterative methods [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] are used converging to the solution u * .…”

Section: Introductionmentioning

confidence: 99%

Some High-Order Iterative Methods for Nonlinear Models Originating from Real Life Problems

2020

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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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Extending the Applicability of Highly Efficient Iterative Methods for Nonlinear Equations and Their Applications

et al. 2022

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Numerous three-step methods of high convergence order have been developed to produce sequences approximating solutions of equations usually defined on the Euclidean space with a finite dimension. The local convergence order is determined by Taylor expansions requiring the existence of derivatives that are not present on the methods. The more interesting semi-local convergence analysis for these methods has not been considered before. The semi-local is also provided based on generalized ω-continuity conditions on the derivative of the operator involved and the majorizing sequences, thus limiting their usage to only solving equations with operators that are many times differentiable. However, these methods may convergence to a solution of the equation even if these high-order derivatives do not exist. That is why a methodology is utilized on two sixth convergence order methods and in the more general setting of a Banach space. This time, the convergence depends only on the operators and the first derivative on the method. Therefore, by this methodology the applicability of the methods is in the extended area. Although this methodology is demonstrated on two competing and efficient methods, it can also be utilized for the same reasons on other methods involving inverses of operators that are linear. This is the motivation and novelty of the paper. The numerical applications further validate the theoretical results both in the local as well as the semi-local convergence case.

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Ball Comparison between Three Sixth Order Methods for Banach Space Valued Operators

Cited by 3 publications

References 16 publications

Some High-Order Iterative Methods for Nonlinear Models Originating from Real Life Problems

Some High-Order Iterative Methods for Nonlinear Models Originating from Real Life Problems

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

Extending the Applicability of Highly Efficient Iterative Methods for Nonlinear Equations and Their Applications

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