Research

14 pages, 759 KiB

Open AccessArticle

Ball Comparison between Two Efficient Weighted-Newton-like Solvers for Equations

by Ioannis K. Argyros, Samundra Regmi, Christopher I. Argyros and Debasis Sharma

Foundations 2022, 2(4), 1031-1044; https://doi.org/10.3390/foundations2040069 - 01 Nov 2022

Viewed by 989

We compare the convergence balls and the dynamical behaviors of two efficient weighted-Newton-like equation solvers by Sharma and Arora, and Grau-Sánchez et al. First of all, the results of ball convergence for these algorithms are established by employing generalized Lipschitz constants and assumptions [...] Read more.

We compare the convergence balls and the dynamical behaviors of two efficient weighted-Newton-like equation solvers by Sharma and Arora, and Grau-Sánchez et al. First of all, the results of ball convergence for these algorithms are established by employing generalized Lipschitz constants and assumptions on the first derivative only. Consequently, outcomes for the radii of convergence, measurable error distances and the existence–uniqueness areas for the solution are discussed. Then, the complex dynamical behaviors of these solvers are compared by applying the attraction basin tool. It is observed that the solver suggested by Grau-Sánchez et al. has bigger basins than the method described by Sharma and Arora. Lastly, our ball analysis findings are verified on application problems and the convergence balls are compared. It is found that the method given by Grau-Sánchez et al. has larger convergence balls than the solver of Sharma and Arora. Hence, the solver presented by Grau-Sánchez et al. is more suitable for practical application. The convergence analysis uses the first derivative in contrast to the aforementioned studies, utilizing the seventh derivative not on these methods. The developed process can be used on other methods in order to increase their applicability. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

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9 pages, 284 KiB

Open AccessArticle

Semi-Local Convergence of Two Derivative-Free Methods of Order Six for Solving Equations under the Same Conditions

by Ioannis K. Argyros, Christopher I. Argyros, Jinny Ann John and Jayakumar Jayaraman

Foundations 2022, 2(4), 1022-1030; https://doi.org/10.3390/foundations2040068 - 01 Nov 2022

Cited by 1 | Viewed by 1007

Abstract

We propose the semi-local convergence of two derivative-free, competing methods of order six to address non-linear equations. The sufficient convergence criteria are the same, making a direct comparison between them possible. The existing convergence technique uses the standard Taylor series approach, which requires [...] Read more.

We propose the semi-local convergence of two derivative-free, competing methods of order six to address non-linear equations. The sufficient convergence criteria are the same, making a direct comparison between them possible. The existing convergence technique uses the standard Taylor series approach, which requires derivatives up to order seven. The novelty and originality of our work lies in the fact that in contrast to previous research works, our convergence theorems only demand the first derivative. In addition, formulas for determining the region of uniqueness for solution, convergence radii, and error estimations are suggested. Such results cannot be found in works relying on the seventh derivatives. As a consequence, we are able to broaden the utility of these productive methods. The confirmation of our convergence findings through application problems brings this research to a close. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

12 pages, 304 KiB

Open AccessArticle

Semi-Local Convergence of a Seventh Order Method with One Parameter for Solving Non-Linear Equations

by Christopher I. Argyros, Ioannis K. Argyros, Samundra Regmi, Jinny Ann John and Jayakumar Jayaraman

Foundations 2022, 2(4), 827-838; https://doi.org/10.3390/foundations2040056 - 21 Sep 2022

Cited by 3 | Viewed by 1410

Abstract

The semi-local convergence is presented for a one parameter seventh order method to obtain solutions of Banach space valued nonlinear models. Existing works utilized hypotheses up to the eighth derivative to prove the local convergence. But these high order derivatives are not on [...] Read more.

The semi-local convergence is presented for a one parameter seventh order method to obtain solutions of Banach space valued nonlinear models. Existing works utilized hypotheses up to the eighth derivative to prove the local convergence. But these high order derivatives are not on the method and they may not exist. Hence, the earlier results can only apply to solve equations containing operators that are at least eight times differentiable although this method may converge. That is why, we only apply the first derivative in our convergence result. Therefore, the results on calculable error estimates, convergence radius and uniqueness region for the solution are derived in contrast to the earlier proposals dealing with the less challenging local convergence case. Hence, we enlarge the applicability of these methods. The methodology used does not depend on the method and it is very general. Therefore, it can be used to extend other methods in an analogous way. Finally, some numerical tests are performed at the end of the text, where the convergence conditions are fulfilled. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

7 pages, 355 KiB

Open AccessArticle

Approximating Solutions of Nonlinear Equations Using an Extended Traub Method

by Santhosh George, Ioannis K. Argyros, Christopher I. Argyros and Kedarnath Senapati

Foundations 2022, 2(3), 617-623; https://doi.org/10.3390/foundations2030042 - 27 Jul 2022

Viewed by 970

Abstract

The Traub iterates generate a sequence that converges to a solution of a nonlinear equation given certain conditions. The order of convergence has been shown provided that the fifth Fréchet-derivative exists. Notice that this derivative does not appear on the Traub method. Therefore, [...] Read more.

The Traub iterates generate a sequence that converges to a solution of a nonlinear equation given certain conditions. The order of convergence has been shown provided that the fifth Fréchet-derivative exists. Notice that this derivative does not appear on the Traub method. Therefore, according to the earlier results, there is no guarantee that the Traub method converges if the operator is not five times Fréchet-differentiable or more. However, the Traub method can converge, since these assumptions are only sufficient. The novelty of our new technique is the fact that only the Fréchet-derivative on the method is assumed to exist to prove convergence. Moreover, the new results does not depend on the Traub method. Consequently, the same technique can be applied on other methods. The dynamics of this method are also studied. Examples further explain the theoretical results. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

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11 pages, 289 KiB

Open AccessArticle

On the Semi-Local Convergence of a Noor–Waseem-like Method for Nonlinear Equations

by Ioannis K. Argyros, Jai Prakash Jaiswal, Akanksha Saxena and Michael I. Argyros

Foundations 2022, 2(2), 512-522; https://doi.org/10.3390/foundations2020034 - 20 Jun 2022

Viewed by 1149

Abstract

The significant feature of this paper is that the semi-local convergence of high order methods for solving nonlinear equations defined on abstract spaces has not been studied extensively as done for the local convergence by a plethora of authors which is certainly a [...] Read more.

The significant feature of this paper is that the semi-local convergence of high order methods for solving nonlinear equations defined on abstract spaces has not been studied extensively as done for the local convergence by a plethora of authors which is certainly a more interesting case. A process is developed based on majorizing sequences and the notion of restricted Lipschitz condition to provide a semi-local convergence analysis for the third convergent order Noor–Waseem method. Due to the generality of our technique, it can be used on other high order methods. The convergence analysis is enhanced. Numerical applications complete are used to test the convergence criteria. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

8 pages, 248 KiB

Open AccessArticle

Extended Newton–Kantorovich Theorem for Solving Nonlinear Equations

by Samundra Regmi, Ioannis K. Argyros, Santhosh George and Christopher I. Argyros

Foundations 2022, 2(2), 504-511; https://doi.org/10.3390/foundations2020033 - 08 Jun 2022

Viewed by 1371

Abstract

The Newton–Kantorovich theorem for solving Banach space-valued equations is a very important tool in nonlinear functional analysis. Several versions of this theorem have been given by Adley, Argyros, Ciarlet, Ezquerro, Kantorovich, Potra, Proinov, Wang, et al. This result, e.g., establishes the existence and [...] Read more.

The Newton–Kantorovich theorem for solving Banach space-valued equations is a very important tool in nonlinear functional analysis. Several versions of this theorem have been given by Adley, Argyros, Ciarlet, Ezquerro, Kantorovich, Potra, Proinov, Wang, et al. This result, e.g., establishes the existence and uniqueness of the solution. Moreover, the Newton sequence converges to the solution under certain conditions of the initial data. However, the convergence region in all of these approaches is small in general; the error bounds on the distances involved are pessimistic, and information about the location of the solutions appears improvable. The novelty of our study lies in the fact that, motivated by optimization concerns, we address all of these. In particular, we introduce a technique that extends the convergence region; provides weaker sufficient semi-local convergence criteria; offers tighter error bounds on the distances involved and more precise information on the location of the solution. These advantages are achieved without additional conditions. This technique can be used to extend other iterative methods along the same lines. Numerical experiments illustrate the theoretical results. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

18 pages, 333 KiB

Open AccessArticle

Location, Separation and Approximation of Solutions for Quadratic Matrix Equations

by Miguel Á. Hernández-Verón and Natalia Romero

Foundations 2022, 2(2), 457-474; https://doi.org/10.3390/foundations2020030 - 12 May 2022

Viewed by 1217

Abstract

In this work, we focus on analyzing the location and separation of the solutions of the simplest quadratic matrix equation. For this, we use the qualitative properties that we can deduce of the study of the convergence of iterative processes. This study allow [...] Read more.

In this work, we focus on analyzing the location and separation of the solutions of the simplest quadratic matrix equation. For this, we use the qualitative properties that we can deduce of the study of the convergence of iterative processes. This study allow us to determine domains of existence and uniqueness of solutions, and therefore to locate and separate the solutions. Another goal is to approximate a solution of the quadratic matrix equation. For this, we consider iterative processes of fixed point type. So, analyzing the convergence of these iterative processes of fixed point type, we locate, separate and approximate solutions of quadratic matrix equations. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

14 pages, 284 KiB

Open AccessArticle

Extending King’s Method for Finding Solutions of Equations

by Samundra Regmi, Ioannis K. Argyros, Santhosh George and Christopher I. Argyros

Foundations 2022, 2(2), 348-361; https://doi.org/10.3390/foundations2020024 - 18 Apr 2022

Viewed by 1310

Abstract

King’s method applies to solve scalar equations. The local analysis is established under conditions including the fifth derivative. However, the only derivative in this method is the first. Earlier studies apply to equations containing at least five times differentiable functions. Consequently, these articles [...] Read more.

King’s method applies to solve scalar equations. The local analysis is established under conditions including the fifth derivative. However, the only derivative in this method is the first. Earlier studies apply to equations containing at least five times differentiable functions. Consequently, these articles provide no information that can be used to solve equations involving functions that are less than five times differentiable, although King’s method may converge. That is why the new analysis uses only the operators and their first derivatives which appear in King’s method. The article contains the semi-local analysis for complex plane-valued functions not presented before. Numerical applications complement the theory. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

10 pages, 362 KiB

Open AccessArticle

Extending the Local Convergence of a Seventh Convergence Order Method without Derivatives

by Ioannis K. Argyros, Debasis Sharma, Christopher I. Argyros and Sanjaya Kumar Parhi

Foundations 2022, 2(2), 338-347; https://doi.org/10.3390/foundations2020023 - 12 Apr 2022

Viewed by 1334

Abstract

For the purpose of obtaining solutions to Banach-space-valued nonlinear models, we offer a new extended analysis of the local convergence result for a seventh-order iterative approach without derivatives. Existing studies have used assumptions up to the eighth derivative to demonstrate its convergence. However, [...] Read more.

For the purpose of obtaining solutions to Banach-space-valued nonlinear models, we offer a new extended analysis of the local convergence result for a seventh-order iterative approach without derivatives. Existing studies have used assumptions up to the eighth derivative to demonstrate its convergence. However, in our convergence theory, we only use the first derivative. Thus, in contrast to previously derived results, we obtain conclusions on calculable error estimates, convergence radius, and uniqueness region for the solution. As a result, we are able to broaden the utility of this efficient method. In addition, the convergence regions of this scheme for solving polynomial equations with complex coefficients are illustrated using the attraction basin approach. This study is concluded with the validation of our convergence result on application problems. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

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9 pages, 280 KiB

Open AccessFeature PaperArticle

On the Local Convergence of a (p + 1)-Step Method of Order 2p + 1 for Solving Equations

by Janak Raj Sharma, Ioannis K. Argyros, Harmandeep Singh and Michael I. Argyros

Foundations 2022, 2(1), 242-250; https://doi.org/10.3390/foundations2010018 - 20 Feb 2022

Cited by 2 | Viewed by 1654

Abstract

The local convergence of a generalized

(p + 1)

-step iterative method of order

2 p + 1

is established in order to estimate the locally unique solutions of nonlinear equations in the Banach spaces. In earlier studies, convergence analysis for [...] Read more.

The local convergence of a generalized

(p + 1)

-step iterative method of order

2 p + 1

is established in order to estimate the locally unique solutions of nonlinear equations in the Banach spaces. In earlier studies, convergence analysis for the given iterative method was carried out while assuming the existence of certain higher-order derivatives. In contrast to this approach, the convergence analysis is carried out in the present study by considering the hypothesis only on the first-order Fréchet derivatives. This study further provides an estimate of convergence radius and bounds of the error for the considered method. Such estimates were not provided in earlier studies. In view of this, the applicability of the given method clearly seems to be extended over the wider class of functions or problems. Moreover, the numerical applications are presented to verify the theoretical deductions. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

8 pages, 266 KiB

Open AccessFeature PaperArticle

On the Semi-Local Convergence of a Jarratt-Type Family Schemes for Solving Equations

by Christopher I. Argyros, Ioannis K. Argyros, Stepan Shakhno and Halyna Yarmola

Foundations 2022, 2(1), 234-241; https://doi.org/10.3390/foundations2010017 - 17 Feb 2022

Viewed by 1545

Abstract

We study semi-local convergence of two-step Jarratt-type method for solving nonlinear equations under the classical Lipschitz conditions for first-order derivatives. To develop a convergence analysis we use the approach of restricted convergence regions in combination to majorizing scalar sequences and our technique of [...] Read more.

We study semi-local convergence of two-step Jarratt-type method for solving nonlinear equations under the classical Lipschitz conditions for first-order derivatives. To develop a convergence analysis we use the approach of restricted convergence regions in combination to majorizing scalar sequences and our technique of recurrent functions. Finally, the numerical example is given. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

11 pages, 287 KiB

Open AccessFeature PaperArticle

On the Semi-Local Convergence of a Fifth-Order Convergent Method for Solving Equations

by Christopher I. Argyros, Ioannis K. Argyros, Stepan Shakhno and Halyna Yarmola

Foundations 2022, 2(1), 140-150; https://doi.org/10.3390/foundations2010008 - 20 Jan 2022

Viewed by 1741

Abstract

We study the semi-local convergence of a three-step Newton-type method for solving nonlinear equations under the classical Lipschitz conditions for first-order derivatives. To develop a convergence analysis, we use the approach of restricted convergence regions in combination with majorizing scalar sequences and our [...] Read more.

We study the semi-local convergence of a three-step Newton-type method for solving nonlinear equations under the classical Lipschitz conditions for first-order derivatives. To develop a convergence analysis, we use the approach of restricted convergence regions in combination with majorizing scalar sequences and our technique of recurrent functions. Finally, a numerical example is given. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

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