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

Three methods of sixth order convergence are tackled for approximating the solution of an equation defined on the finitely dimensional Euclidean space. This convergence requires the existence of derivatives of, at least, order seven. However, only derivatives of order one are involved in such methods. Moreover, we have no estimates on the error distances, conclusions about the uniqueness of the solution in any domain, and the convergence domain is not sufficiently large. Hence, these methods have limited usage. This paper introduces a new technique on a general Banach space setting based only the first derivative and Lipschitz type conditions that allow the study of the convergence. In addition, we find usable error distances as well as uniqueness of the solution. A comparison between the convergence balls of three methods, not possible to drive with the previous approaches, is also given. The technique is possible to use with methods available in literature improving, consequently, their applicability. Several numerical examples compare these methods and illustrate the convergence criteria.

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Local convergence comparison between two novel sixth order methods for solving equations

Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica ◽

10.2478/aupcsm-2019-0001 ◽

2019 ◽

Vol 18 (1) ◽

pp. 5-19

Author(s):

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Banach Space ◽

Convergence Analysis ◽

Local Convergence ◽

Sixth Order ◽

Convergence Order ◽

Convergence Criteria ◽

Numerical Examples ◽

First Derivative ◽

Solving Equations ◽

Local Convergence Analysis

Abstract The aim of this article is to provide the local convergence analysis of two novel competing sixth convergence order methods for solving equations involving Banach space valued operators. Earlier studies have used hypotheses reaching up to the sixth derivative but only the first derivative appears in these methods. These hypotheses limit the applicability of the methods. That is why we are motivated to present convergence analysis based only on the first derivative. Numerical examples where the convergence criteria are tested are provided. It turns out that in these examples the criteria in the earlier works are not satisfied, so these results cannot be used to solve equations but our results can be used.

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Extended convergence of a sixth order scheme for solving equations under ω–continuity conditions

Moroccan Journal of Pure and Applied Analysis ◽

10.2478/mjpaa-2022-0008 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Samundra Regmi ◽

Christopher I. Argyros ◽

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Banach Space ◽

Sixth Order ◽

Convergence Order ◽

Not Given ◽

Numerical Examples ◽

First Derivative ◽

Uniqueness Of The Solution ◽

Order Scheme ◽

Solving Equations

Abstract The applicability of an efficient sixth convergence order scheme is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the scheme. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided (not given in earlier works) based on ω–continuity conditions. Numerical examples complete this article.

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Extending the Convergence of Two Similar Sixth Order Schemes for Solving Equations under Generalized Conditions

Contemporary Mathematics ◽

10.37256/cm.242021991 ◽

2021 ◽

pp. 246-257

Author(s):

Ioannis K. Argyros ◽

Santhosh George ◽

Christopher I. Argyros

Keyword(s):

Banach Space ◽

Sixth Order ◽

Convergence Order ◽

Not Given ◽

Numerical Examples ◽

First Derivative ◽

Uniqueness Of The Solution ◽

Solving Equations

The applicability of two competing efficient sixth convergence order schemes is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the schemes. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided not given in the earlier works based on ω-continuity conditions. Our technique extends other schemes analogously, since it is so general. Numerical examples complete this work.

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A comparison between two competing sixth convergence order algorithms under the same set of conditions

Creative Mathematics and Informatics ◽

10.37193/cmi.2021.01.03 ◽

2021 ◽

Vol 30 (1) ◽

pp. 19-28

Author(s):

GUS ARGYROS ◽

MICHAEL ARGYROS ◽

IOANNIS K. ARGYROS ◽

GEORGE SANTHOSH

Keyword(s):

Banach Space ◽

Error Estimates ◽

Taylor Series ◽

High Order ◽

Convergence Order ◽

Convergence Criteria ◽

Convergence Conditions ◽

First Derivative ◽

Banach Space Operators

There is a plethora of algorithms of the same convergence order for generating a sequence approximating a solution of an equation involving Banach space operators. But the set of convergence criteria is not the same in general. This makes the comparison between them hard and only numerically. Moreover, the convergence is established using Taylor series and by assuming the existence of high order derivatives not even appearing on these algorithms. Furthermore, no computable error estimates, uniqueness for the solution results or a ball of convergence is given. We address all these problems by using only the first derivative that actually appears on these algorithms and under the same set of convergence conditions. Our technique is so general that it can be used to extend the applicability of other algorithms along the same lines.

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Some Imbedding Theorems for Sobolev Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-055-3 ◽

1971 ◽

Vol 23 (3) ◽

pp. 517-530 ◽

Cited By ~ 16

Author(s):

R. A. Adams ◽

John Fournier

Keyword(s):

Banach Space ◽

Positive Integer ◽

Euclidean Space ◽

Sobolev Space ◽

Dimensional Euclidean Space ◽

Dense Subspace ◽

Bounded Subset ◽

Partial Derivatives ◽

Image Position ◽

Derivatives Of

We shall be concerned throughout this paper with the Sobolev space Wm,p(G) and the existence and compactness (or lack of it) of its imbeddings (i.e. continuous inclusions) into various LP spaces over G, where G is an open, not necessarily bounded subset of n-dimensional Euclidean space En. For each positive integer m and each real p ≧ 1 the space Wm,p(G) consists of all u in LP(G) whose distributional partial derivatives of all orders up to and including m are also in LP(G). With respect to the norm1.1Wm,p(G) is a Banach space. It has been shown by Meyers and Serrin [9] that the set of functions in Cm(G) which, together with their partial derivatives of orders up to and including m, are in LP(G) forms a dense subspace of Wm,p(G).

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Ball Convergence of a Parametric Efficient Family of Iterative Methods for Solving Nonlinear Equations

Foundations ◽

10.3390/foundations1010004 ◽

2021 ◽

Vol 1 (1) ◽

pp. 23-31

Author(s):

Samundra Regmi ◽

Christopher I. Argyros ◽

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Linear Convergence ◽

Error Tolerance ◽

First Derivative ◽

Uniqueness Of The Solution ◽

Ball Convergence ◽

Space Case ◽

Local Results ◽

Lipschitz Conditions ◽

Theoretical Results ◽

General Banach Space

The goal is to extend the applicability of Newton-Traub-like methods in cases not covered in earlier articles requiring the usage of derivatives up to order seven that do not appear in the methods. The price we pay by using conditions on the first derivative that actually appear in the method is that we show only linear convergence. To find the convergence order is not our intention, however, since this is already known in the case where the spaces coincide with the multidimensional Euclidean space. Note that the order is rediscovered by using ACOC or COC, which require only the first derivative. Moreover, in earlier studies using Taylor series, no computable error distances were available based on generalized Lipschitz conditions. Therefore, we do not know, for example, in advance, how many iterates are needed to achieve a predetermined error tolerance. Furthermore, no uniqueness of the solution results is available in the aforementioned studies, but we also provide such results. Our technique can be used to extend the applicability of other methods in an analogous way, since it is so general. Finally note that local results of this type are important, since they demonstrate the difficulty in choosing initial points. Our approach also extends the applicability of this family of methods from the multi-dimensional Euclidean to the more general Banach space case. Numerical examples complement the theoretical results.

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Expanding the applicability of secant-like methods for solving nonlinear equations

Carpathian Journal of Mathematics ◽

10.37193/cjm.2015.01.02 ◽

2015 ◽

Vol 31 (1) ◽

pp. 11-30

Author(s):

I. K. ARGYROS ◽

◽

J. A. EZQUERRO ◽

M. A. HERNANDEZ ◽

S. HILOUT ◽

...

Keyword(s):

Banach Space ◽

Integral Equation ◽

Nonlinear Equation ◽

Nonlinear Integral Equation ◽

Semilocal Convergence ◽

Convergence Criteria ◽

Convergence Domain ◽

Nonlinear Integral Equations ◽

Recurrent Functions ◽

New Approach

We use the method of recurrent functions to provide a new semilocal convergence analysis for secant-like methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our sufficient convergence criteria are weaker than in earlier studies such as [18, 19, 20, 21, 25, 26]. Therefore, the new approach has a larger convergence domain and uses the same constants. A numerical example involving a nonlinear integral equation of mixed Hammerstein type is given to illustrate the advantages of the new approach. Another example of nonlinear integral equations is presented to show that the old convergence criteria are not satisfied but the new convergence are satisfied.

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Local convergence for a family of sixth order methods with parameters

Open Journal of Mathematical Sciences ◽

10.30538/oms2021.0166 ◽

2021 ◽

Vol 5 (1) ◽

pp. 300-305

Author(s):

Christopher I. Argyros ◽

◽

Michael Argyros ◽

Ioannis K. Argyros ◽

Santhosh George ◽

...

Keyword(s):

Banach Space ◽

Convergence Analysis ◽

Real Line ◽

Local Convergence ◽

Sixth Order ◽

Numerical Examples ◽

The Real ◽

First Derivative ◽

Local Convergence Analysis ◽

Methods Numerical

Local convergence of a family of sixth order methods for solving Banach space valued equations is considered in this article. The local convergence analysis is provided using only the first derivative in contrast to earlier works on the real line using the seventh derivative. This way the applicability is expanded for these methods. Numerical examples complete the article.

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Convergence Criteria of Three Step Schemes for Solving Equations

Mathematics ◽

10.3390/math9233106 ◽

2021 ◽

Vol 9 (23) ◽

pp. 3106

Author(s):

Samundra Regmi ◽

Christopher I. Argyros ◽

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Banach Space ◽

Euclidean Space ◽

Convergence Analysis ◽

Local Convergence ◽

Dimensional Euclidean Space ◽

Convergence Criteria ◽

Iterative Schemes ◽

Finite Dimensional ◽

Taylor Expansions ◽

Solving Equations

We develop a unified convergence analysis of three-step iterative schemes for solving nonlinear Banach space valued equations. The local convergence order has been shown before to be five on the finite dimensional Euclidean space assuming Taylor expansions and the existence of the sixth derivative not on these schemes. So, the usage of them is restricted six or higher differentiable mappings. But in our paper only the first Frèchet derivative is utilized to show convergence. Consequently, the scheme is expanded. Numerical applications are also given to test convergence.

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Local Comparison between Two Ninth Convergence Order Algorithms for Equations

Algorithms ◽

10.3390/a13060147 ◽

2020 ◽

Vol 13 (6) ◽

pp. 147

Author(s):

Samundra Regmi ◽

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Error Estimates ◽

Nonlinear Equations ◽

Local Convergence ◽

Numerical Experiments ◽

Radius Of Convergence ◽

Convergence Order ◽

Convergence Criteria ◽

First Derivative ◽

Local Comparison ◽

Uniqueness Of The Solution

A local convergence comparison is presented between two ninth order algorithms for solving nonlinear equations. In earlier studies derivatives not appearing on the algorithms up to the 10th order were utilized to show convergence. Moreover, no error estimates, radius of convergence or results on the uniqueness of the solution that can be computed were given. The novelty of our study is that we address all these concerns by using only the first derivative which actually appears on these algorithms. That is how to extend the applicability of these algorithms. Our technique provides a direct comparison between these algorithms under the same set of convergence criteria. This technique can be used on other algorithms. Numerical experiments are utilized to test the convergence criteria.

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