scholarly journals Local convergence of a multi-step high order method with divided differences under hypotheses on the first derivative

2017 ◽  
Vol 16 (1) ◽  
pp. 41-50
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Spaces ◽  
Local Convergence ◽  
Sixth Order ◽  
Divided Differences ◽  
Order Method ◽  
Step Method ◽  
Convergence Conditions ◽  
Numerical Examples ◽  
High Order Method ◽  
First Derivative

AbstractThis paper is devoted to the study of a multi-step method with divided differences for solving nonlinear equations in Banach spaces. In earlier studies, hypotheses on the Fréchet derivative up to the sixth order of the operator under consideration is used to prove the convergence of the method. That restricts the applicability of the method. In this paper we extended the applicability of the sixth-order multi-step method by using only hypotheses on the first derivative of the operator involved. Our convergence conditions are weaker than the conditions used in earlier studies. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study.

scholarly journals Local convergence for a family of sixth order methods with parameters

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.

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

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.

scholarly journals On the Local Convergence of an Eighth-order Method for Solving Nonlinear Equations

2016 ◽  
Vol 54 (1) ◽  
pp. 3-16 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Munish Kansal ◽  
V. Kanwar
Keyword(s):  
Linear Equation ◽  
Error Bounds ◽  
Local Convergence ◽  
Order Method ◽  
Eighth Order ◽  
Numerical Examples ◽  
First Derivative ◽  
Non Linear Equation ◽  
Computable Error Bounds ◽  
Local Convergence Analysis

Abstract We present a local convergence analysis of an eighth-order method for approximating a locally unique solution of a non-linear equation. Earlier studies such as have shown convergence of these methods under hypotheses up to the seventh derivative of the function although only the first derivative appears in the method. In this study, we expand the applicability of these methods using only hypotheses up to the first derivative of the function. This way the applicability of these methods is extended under weaker hypotheses. Moreover, the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

scholarly journals Study of Local Convergence and Dynamics of a King-Like Two-Step Method with Applications

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1062
Author(s):  
Ioannis K. Argyros ◽  
Ángel Alberto Magreñán ◽  
Alejandro Moysi ◽  
Íñigo Sarría ◽  
Juan Antonio Sicilia Montalvo
Keyword(s):  
Nonlinear Equations ◽  
Local Convergence ◽  
Step Method ◽  
First Derivative ◽  
Family Behavior ◽  
Local Results

In this paper, we present the local results of the convergence of the two-step King-like method to approximate the solution of nonlinear equations. In this study, we only apply conditions to the first derivative, because we only need this condition to guarantee convergence. As a result, the applicability of the method is expanded. We also use different convergence planes to show family behavior. Finally, the new results are used to solve some applications related to chemistry.

A new efficient implicit four-step method with vanished phase-lag and some of its derivatives for the numerical solution of the radial Schr¨odinger equation

2017 ◽  
Vol 8 (1-2) ◽  
pp. 77 ◽  
Author(s):  
Ali Shokri ◽  
Morteza Tahmourasi
Keyword(s):  
Approximate Solution ◽  
Stability Analysis ◽  
Multistep Method ◽  
Phase Lag ◽  
Order Method ◽  
Step Method ◽  
One Dimensional ◽  
First Derivative ◽  
Algebraic Order ◽  
The One

A new four-step implicit linear sixth algebraic order method with vanished phase-lag and its first derivative is constructed in this paper. The purpose of this paper is to develop an efficient algorithm for the approximate solution of the one-dimensional radial Schr¨odinger equation and related problems. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. An error analysis and a stability analysis is also investigated and a comparison with other methods is also studied. The efficiency of the new methodology is proved via theoretical analysis and numerical applications.

scholarly journals Ball convergence theorem for a Steffensen-type third-order method

2017 ◽  
Vol 51 (1) ◽  
pp. 1-14
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Nonlinear Equation ◽  
Error Bounds ◽  
Order Method ◽  
Third Order ◽  
Numerical Examples ◽  
First Derivative ◽  
Fourth Derivative ◽  
Ball Convergence ◽  
Computable Error Bounds ◽  
Local Convergence Analysis

We present a local convergence analysis for a family of Steffensen-type third-order methods in order to approximate a solution of a nonlinear equation. We use hypothesis up to the first derivative in contrast to earlier studies such as [2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] using hypotheses up to the fourth derivative. This way the applicability of these methods is extended under weaker hypothesis. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

scholarly journals Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative

Algorithms ◽  
2015 ◽  
Vol 8 (4) ◽  
pp. 1076-1087 ◽  
Author(s):  
Ioannis Argyros ◽  
Ramandeep Behl ◽  
S.S. Motsa
Keyword(s):  
Local Convergence ◽  
Convergence Order ◽  
Order Method ◽  
First Derivative

scholarly journals Local Convergence of an Efficient Multipoint Iterative Method in Banach Space

Algorithms ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 25
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Iterative Method ◽  
Local Convergence ◽  
Numerical Experiments ◽  
Order Method ◽  
Eighth Order ◽  
First Derivative ◽  
Taylor Expansions ◽  
Derivative Free ◽  
Class Of Functions

We discuss the local convergence of a derivative-free eighth order method in a Banach space setting. The present study provides the radius of convergence and bounds on errors under the hypothesis based on the first Fréchet-derivative only. The approaches of using Taylor expansions, containing higher order derivatives, do not provide such estimates since the derivatives may be nonexistent or costly to compute. By using only first derivative, the method can be applied to a wider class of functions and hence its applications are expanded. Numerical experiments show that the present results are applicable to the cases wherein previous results cannot be applied.

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