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.

scholarly journals Local convergence for an eighth order method for solving equations and systems of equations

2019 ◽  
Vol 8 (1) ◽  
pp. 74-79
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Local Convergence ◽  
Divided Difference ◽  
Dimensional Euclidean Space ◽  
Order Method ◽  
Systems Of Equations ◽  
Eighth Order ◽  
First Derivative ◽  
Solving Equations ◽  
Computable Error Bounds

AbstractThe aim of this study is to extend the applicability of an eighth convergence order method from thek−dimensional Euclidean space to a Banach space setting. We use hypotheses only on the first derivative to show the local convergence of the method. Earlier studies use hypotheses up to the eighth derivative although only the first derivative and a divided difference of order one appear in the method. Moreover, we provide computable error bounds based on Lipschitz-type functions.

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

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1203
Author(s):  
Janak Raj Sharma ◽  
Deepak Kumar ◽  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Local Convergence ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Geometrical Approach ◽  
Stable Convergence ◽  
Numerical Testing ◽  
Taylor Expansions ◽  
Attraction Basins ◽  
Class Of Functions

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.

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 Local Convergence of an Optimal Eighth Order Method under Weak Conditions

Algorithms ◽  
2015 ◽  
Vol 8 (3) ◽  
pp. 645-655 ◽  
Author(s):  
Ioannis Argyros ◽  
Ramandeep Behl ◽  
S.S. Motsa
Keyword(s):  
Local Convergence ◽  
Order Method ◽  
Eighth Order

Convergence of derivative free iterative methods

2019 ◽  
Vol 28 (1) ◽  
pp. 19-26
Author(s):  
IOANNIS K. ARGYROS ◽  
◽  
SANTHOSH GEORGE ◽  
Keyword(s):  
Banach Space ◽  
Iterative Methods ◽  
Local Convergence ◽  
Practical Interest ◽  
Systems Of Equations ◽  
Hölder Continuous ◽  
Numerical Examples ◽  
Lipschitz Continuous ◽  
Derivative Free ◽  
Special Cases

We present the local as well as the semi-local convergence of some iterative methods free of derivatives for Banach space valued operators. These methods contain the secant and the Kurchatov method as special cases. The convergence is based on weak hypotheses specializing to Lipschitz continuous or Holder continuous hypotheses. The results are of theoretical and practical interest. In particular the method is compared favorably ¨ to other methods using concrete numerical examples to solve systems of equations containing a nondifferentiable term.

scholarly journals Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their Basins of Attraction and Its Application

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 322 ◽  
Author(s):  
Yanlin Tao ◽  
Kalyanasundaram Madhu
Keyword(s):  
Dynamical Behavior ◽  
Computational Cost ◽  
Basins Of Attraction ◽  
Order Method ◽  
Eighth Order ◽  
Projectile Motion ◽  
First Derivative ◽  
Iteration Functions ◽  
Optimal Launch Angle ◽  
Fourth Order Method

The principal objective of this work is to propose a fourth, eighth and sixteenth order scheme for solving a nonlinear equation. In terms of computational cost, per iteration, the fourth order method uses two evaluations of the function and one evaluation of the first derivative; the eighth order method uses three evaluations of the function and one evaluation of the first derivative; and sixteenth order method uses four evaluations of the function and one evaluation of the first derivative. So these all the methods have satisfied the Kung-Traub optimality conjecture. In addition, the theoretical convergence properties of our schemes are fully explored with the help of the main theorem that demonstrates the convergence order. The performance and effectiveness of our optimal iteration functions are compared with the existing competitors on some standard academic problems. The conjugacy maps of the presented method and other existing eighth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane. We apply the new scheme to find the optimal launch angle in a projectile motion problem and Planck’s radiation law problem as an application.

scholarly journals Ball Convergence of an Efficient Eighth Order Iterative Method Under Weak Conditions

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 260 ◽  
Author(s):  
Janak Sharma ◽  
Ioannis Argyros ◽  
Sunil Kumar
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Higher Order ◽  
High Order ◽  
Convergence Order ◽  
Eighth Order ◽  
First Derivative ◽  
Ball Convergence ◽  
Derivatives Of

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods.

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 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.

Sign in / Sign up

Export Citation Format

Share Document