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21 pages, 1452 KB

Open AccessArticle

Extending the Applicability of Newton-Jarratt-like Methods with Accelerators of Order 2m + 1 for Solving Nonlinear Systems

by Ioannis K. Argyros, Stepan Shakhno and Mykhailo Shakhov

Axioms 2025, 14(10), 734; https://doi.org/10.3390/axioms14100734 - 28 Sep 2025

Viewed by 505

Abstract

The local convergence analysis of the

m + 1

-step Newton-Jarratt composite scheme with order

2 m + 1

has been shown previously. But the convergence order

2 m + 1

is obtained using Taylor series and assumptions on the existence of at [...] Read more.

The local convergence analysis of the

m + 1

-step Newton-Jarratt composite scheme with order

2 m + 1

has been shown previously. But the convergence order

2 m + 1

is obtained using Taylor series and assumptions on the existence of at least the fifth derivative of the mapping involved, which is not present in the method. These assumptions limit the applicability of the method. A priori error estimates or the radius of convergence or uniqueness of the solution results have not been given either. These drawbacks are addressed in this paper. In particular, the convergence is based only on the operators on the method, which are the operator and its first derivative. Moreover, the radius of convergence is established, a priori estimates and the isolation of the solution is discussed using generalized continuity assumptions on the derivative. Furthermore, the more challenging semi-local convergence analysis, not previously studied, is presented using majorizing sequences. The convergence for both analyses depends on the generalized continuity of the Jacobian of the mapping involved, which is used to control it and sharpen the error distances. Numerical examples validate the sufficient convergence conditions presented in the theory. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization, 2nd Edition)

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25 pages, 746 KB

Open AccessArticle

Convergence Analysis of Jarratt-like Methods for Solving Nonlinear Equations for Thrice-Differentiable Operators

by Indra Bate, Kedarnath Senapati, Santhosh George, Ioannis K. Argyros and Michael I. Argyros

AppliedMath 2025, 5(2), 38; https://doi.org/10.3390/appliedmath5020038 - 3 Apr 2025

Cited by 1 | Viewed by 1062

Abstract

The main goal of this paper is to study Jarratt-like iterative methods to obtain their order of convergence under weaker conditions. Generally, obtaining the

p^{t h}

-order convergence using the Taylor series expansion technique needed at least

p + 1

times differentiability [...] Read more.

The main goal of this paper is to study Jarratt-like iterative methods to obtain their order of convergence under weaker conditions. Generally, obtaining the

p^{t h}

-order convergence using the Taylor series expansion technique needed at least

p + 1

times differentiability of the involved operator. However, we obtain the fourth- and sixth-order for Jarratt-like methods using up to the third-order derivatives only. An upper bound for the asymptotic error constant (AEC) and a convergence ball are provided. The convergence analysis is developed in the more general setting of Banach spaces and relies on Lipschitz-type conditions, which are required to control the derivative. The results obtained are examined using numerical examples, and some dynamical system concepts are discussed for a better understanding of convergence ideas. Full article

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32 pages, 1051 KB

Open AccessArticle

Convergence Order of a Class of Jarratt-like Methods: A New Approach

by Ajil Kunnarath, Santhosh George, Jidesh Padikkal and Ioannis K. Argyros

Symmetry 2025, 17(1), 56; https://doi.org/10.3390/sym17010056 - 31 Dec 2024

Viewed by 1018

Abstract

Symmetry and anti-symmetry appear naturally in the study of systems of nonlinear equations resulting from numerous fields. The solutions of such equations can be obtained in analytical form only in some special situations. Therefore, algorithms or iterative schemes are mostly studied, which approximate [...] Read more.

Symmetry and anti-symmetry appear naturally in the study of systems of nonlinear equations resulting from numerous fields. The solutions of such equations can be obtained in analytical form only in some special situations. Therefore, algorithms or iterative schemes are mostly studied, which approximate the solution. In particular, Jarratt-like methods were introduced with convergence order at least six in Euclidean spaces. We study the methods in the Banach-space setting. Semilocal convergence is studied to obtain the ball containing the solution. The local convergence analysis is performed without the help of the Taylor series with relaxed differentiability assumptions. Our assumptions for obtaining the convergence order are independent of the solution; earlier studies used assumptions involving the solution for local convergence analysis. We compare the methods numerically with similar-order methods and also study the dynamics. Full article

(This article belongs to the Section Mathematics)

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14 pages, 777 KB

Open AccessArticle

Perturbed Newton Methods for Solving Nonlinear Equations with Applications

by Ioannis K. Argyros, Samundra Regmi, Stepan Shakhno and Halyna Yarmola

Symmetry 2022, 14(10), 2206; https://doi.org/10.3390/sym14102206 - 20 Oct 2022

Cited by 6 | Viewed by 2269

Abstract

Symmetries play an important role in the study of a plethora of physical phenomena, including the study of microworlds. These phenomena reduce to solving nonlinear equations in abstract spaces. Therefore, it is important to design iterative methods for approximating the solutions, since closed [...] Read more.

Symmetries play an important role in the study of a plethora of physical phenomena, including the study of microworlds. These phenomena reduce to solving nonlinear equations in abstract spaces. Therefore, it is important to design iterative methods for approximating the solutions, since closed forms of them can be found only in special cases. Several iterative methods were developed whose convergence was established under very general conditions. Numerous applications are also provided to solve systems of nonlinear equations and differential equations appearing in the aforementioned areas. The ball convergence analysis was developed for the King-like and Jarratt-like families of methods to solve equations under the same set of conditions. Earlier studies have used conditions up to the fifth derivative, but they failed to show the fourth convergence order. Moreover, no error distances or results on the uniqueness of the solution were given either. However, we provide such results involving the derivative only appearing on these methods. Hence, we have expanded the usage of these methods. In the case of the Jarratt-like family of methods, our results also hold for Banach space-valued equations. Moreover, we compare the convergence ball and the dynamical features both theoretically and in numerical experiments. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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24 pages, 4556 KB

Open AccessArticle

Development of a Family of Jarratt-Like Sixth-Order Iterative Methods for Solving Nonlinear Systems with Their Basins of Attraction

by Min-Young Lee and Young Ik Kim

Algorithms 2020, 13(11), 303; https://doi.org/10.3390/a13110303 - 20 Nov 2020

Cited by 7 | Viewed by 2982

Abstract

We develop a family of three-step sixth order methods with generic weight functions employed in the second and third sub-steps for solving nonlinear systems. Theoretical and computational studies are of major concern for the convergence behavior with applications to special cases of rational [...] Read more.

We develop a family of three-step sixth order methods with generic weight functions employed in the second and third sub-steps for solving nonlinear systems. Theoretical and computational studies are of major concern for the convergence behavior with applications to special cases of rational weight functions. A number of numerical examples are illustrated to confirm the convergence behavior of local as well as global character of the proposed and existing methods viewed through the basins of attraction. Full article

(This article belongs to the Special Issue Iterative Algorithms for Nonlinear Problems: Convergence and Stability)

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13 pages, 831 KB

Open AccessArticle

Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders

by Min-Young Lee and Young Ik Kim

Mathematics 2020, 8(1), 51; https://doi.org/10.3390/math8010051 - 1 Jan 2020

Cited by 1 | Viewed by 2235

Abstract

Bifurcations have been studied with an extensive analysis of boundary curves of red, fixed components in the parametric space for a uniparametric family of simple-root finders under the Möbius conjugacy map applied to a quadratic polynomial. An elementary approach from the perspective of [...] Read more.

Bifurcations have been studied with an extensive analysis of boundary curves of red, fixed components in the parametric space for a uniparametric family of simple-root finders under the Möbius conjugacy map applied to a quadratic polynomial. An elementary approach from the perspective of a plane curve theory properly describes the geometric figures resembling a circle or cardioid to characterize the underlying boundary curves that are parametrically expressed. Moreover, exact bifurcation points for satellite components on the boundaries have been found, according to the fact that the tangent line at a bifurcation point simultaneously touches the red fixed component and the satellite component. Computational experiments implemented with examples well reflect the significance of the theoretical backgrounds pursued in this paper. Full article

(This article belongs to the Special Issue Multipoint Methods for the Solution of Nonlinear Equations)

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16 pages, 484 KB

Open AccessFeature PaperArticle

Local Convergence and Attraction Basins of Higher Order, Jarratt-Like Iterations

by Janak Raj Sharma, Deepak Kumar and Ioannis K. Argyros

Mathematics 2019, 7(12), 1203; https://doi.org/10.3390/math7121203 - 8 Dec 2019

Viewed by 2152

Abstract

We studied the local convergence of a family of sixth order Jarratt-like methods in Banach space setting. The procedure so applied provides the radius of convergence and bounds on errors under the conditions based on the first Fréchet-derivative only. Such estimates are not [...] Read more.

We studied the local convergence of a family of sixth order Jarratt-like methods in Banach space setting. The procedure so applied provides the radius of convergence and bounds on errors under the conditions based on the first Fréchet-derivative only. Such estimates are not proposed in the approaches using Taylor expansions of higher order derivatives which may be nonexistent or costly to compute. In this sense we can extend usage of the methods considered, since the methods can be applied to a wider class of functions. Numerical testing on examples show that the present results can be applied to the cases where earlier results are not applicable. Finally, the convergence domains are assessed by means of a geometrical approach; namely, the basins of attraction that allow us to find members of family with stable convergence behavior and with unstable behavior. Full article

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16 pages, 2202 KB

Open AccessArticle

On Locating and Counting Satellite Components Born along the Stability Circle in the Parameter Space for a Family of Jarratt-Like Iterative Methods

by Young Hee Geum and Young Ik Kim

Mathematics 2019, 7(9), 839; https://doi.org/10.3390/math7090839 - 11 Sep 2019

Cited by 2 | Viewed by 2144

Abstract

This paper is devoted to an analysis on locating and counting satellite components born along the stability circle in the parameter space for a family of Jarratt-like iterative methods. An elementary theory of plane geometric curves is pursued to locate bifurcation points of [...] Read more.

This paper is devoted to an analysis on locating and counting satellite components born along the stability circle in the parameter space for a family of Jarratt-like iterative methods. An elementary theory of plane geometric curves is pursued to locate bifurcation points of such satellite components. In addition, the theory of Farey sequence is adopted to count the number of the satellite components as well as to characterize relationships between the bifurcation points. A linear stability theory on local bifurcations is developed based upon a small perturbation about the fixed point of the iterative map with a control parameter. Some properties of fixed and critical points under the Möbius conjugacy map are investigated. Theories and examples on locating and counting bifurcation points of satellite components in the parameter space are presented to analyze the bifurcation behavior underlying the dynamics behind the iterative map. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

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