scholarly journals Local Convergence for Multi-Step High Order Solvers under Weak Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 179
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Nonlinear Problems ◽  
Fréchet Derivative ◽  
High Order ◽  
Order Derivative ◽  
First Order ◽  
Lipschitz Constants ◽  
Convergence Study

Our aim in this article is to suggest an extended local convergence study for a class of multi-step solvers for nonlinear equations valued in a Banach space. In comparison to previous studies, where they adopt hypotheses up to 7th Fŕechet-derivative, we restrict the hypotheses to only first-order derivative of considered operators and Lipschitz constants. Hence, we enlarge the suitability region of these solvers along with computable radii of convergence. In the end of this study, we choose a variety of numerical problems which illustrate that our works are applicable but not earlier to solve nonlinear problems.

Ball Convergence for a Multi-Step Harmonic Mean Newton-Like Method in Banach Space

2019 ◽  
Vol 17 (05) ◽  
pp. 1940018 ◽  
Author(s):  
Ramandeep Behl ◽  
Ali Saleh Alshormani ◽  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Harmonic Mean ◽  
Order Derivative ◽  
Systems Of Nonlinear Equations ◽  
Numerical Examples ◽  
First Order ◽  
Ball Convergence ◽  
Local Convergence Analysis

In this paper, we present a local convergence analysis of some iterative methods to approximate a locally unique solution of nonlinear equations in a Banach space setting. In the earlier study [Babajee et al. (2015) “On some improved harmonic mean Newton-like methods for solving systems of nonlinear equations,” Algorithms 8(4), 895–909], demonstrate convergence of their methods under hypotheses on the fourth-order derivative or even higher. However, only first-order derivative of the function appears in their proposed scheme. In this study, we have shown that the local convergence of these methods depends on hypotheses only on the first-order derivative and the Lipschitz condition. In this way, we not only expand the applicability of these methods but also proposed the theoretical radius of convergence of these methods. Finally, a variety of concrete numerical examples demonstrate that our results even apply to solve those nonlinear equations where earlier studies cannot apply.

scholarly journals Local Convergence of Solvers with Eighth Order Having Weak Conditions

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 70
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros
Keyword(s):  
Closed Form Solution ◽  
Nonlinear Problems ◽  
Form Solution ◽  
Fréchet Derivative ◽  
Not Given ◽  
Eighth Order ◽  
Iterative Schemes ◽  
Frechet Derivative ◽  
First Order ◽  
Lipschitz Constants

In particular, the problem of approximating a solution of an equation is of extreme importance in many disciplines, since numerous problems from diverse disciplines reduce to solving such equations. The solutions are found using iterative schemes since in general to find closed form solution is not possible. That is why it is important to study convergence order of solvers. We extended the applicability of an eighth-order convergent solver for solving Banach space valued equations. Earlier considerations adopting suppositions up to the ninth Fŕechet-derivative, although higher than one derivatives are not appearing on these solvers. But, we only practiced supposition on Lipschitz constants and the first-order Fŕechet-derivative. Hence, we extended the applicability of these solvers and provided the computable convergence radii of them not given in the earlier works. We only showed improvements for a certain class of solvers. But, our technique can be used to extend the applicability of other solvers in the literature in a similar fashion. We used a variety of numerical problems to show that our results are applicable to solve nonlinear problems but not earlier ones.

scholarly journals On the Semi-Local Convergence of a Traub-Type Method for Solving Equations

Foundations ◽  
2022 ◽  
Vol 2 (1) ◽  
pp. 114-127
Author(s):  
Samundra Regmi ◽  
Christopher I. Argyros ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Local Convergence ◽  
Type Method ◽  
Lipschitz Constants ◽  
Solving Equations ◽  
Original Domain ◽  
Theoretical Results

The celebrated Traub’s method involving Banach space-defined operators is extended. The main feature in this study involves the determination of a subset of the original domain that also contains the Traub iterates. In the smaller domain, the Lipschitz constants are smaller too. Hence, a finer analysis is developed without the usage of additional conditions. This methodology applies to other methods. The examples justify the theoretical results.

scholarly journals On the Solution of Equations by Extended Discretization

Computation ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 69
Author(s):  
Gus I. Argyros ◽  
Michael I. Argyros ◽  
Samundra Regmi ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Nonlinear Equations ◽  
Error Bounds ◽  
Hölder Continuity ◽  
Fréchet Derivative ◽  
Holder Continuity ◽  
Frechet Derivative ◽  
Solution Of Equations ◽  
Discretization Technique

The method of discretization is used to solve nonlinear equations involving Banach space valued operators using Lipschitz or Hölder constants. But these constants cannot always be found. That is why we present results using ω− continuity conditions on the Fréchet derivative of the operator involved. This way, we extend the applicability of the discretization technique. It turns out that if we specialize ω− continuity our new results improve those in the literature too in the case of Lipschitz or Hölder continuity. Our analysis includes tighter upper error bounds on the distances involved.

scholarly journals A New Higher-Order Iterative Scheme for the Solutions of Nonlinear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 271 ◽  
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros
Keyword(s):  
Mathematical Modeling ◽  
Banach Space ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Iterative Scheme ◽  
Real Life ◽  
System Of Nonlinear Equations ◽  
Life Problems ◽  
Lipschitz Constants ◽  
Better Than

Many real-life problems can be reduced to scalar and vectorial nonlinear equations by using mathematical modeling. In this paper, we introduce a new iterative family of the sixth-order for a system of nonlinear equations. In addition, we present analyses of their convergences, as well as the computable radii for the guaranteed convergence of them for Banach space valued operators and error bounds based on the Lipschitz constants. Moreover, we show the applicability of them to some real-life problems, such as kinematic syntheses, Bratu’s, Fisher’s, boundary value, and Hammerstein integral problems. We finally wind up on the ground of achieved numerical experiments, where they perform better than other competing schemes.

A METHOD TO CONSTRUCT WEIERSTRASS ELLIPTIC FUNCTION SOLUTION FOR NONLINEAR EQUATIONS

2011 ◽  
Vol 25 (14) ◽  
pp. 1931-1939 ◽  
Author(s):  
LIANG-MA SHI ◽  
LING-FENG ZHANG ◽  
HAO MENG ◽  
HONG-WEI ZHAO ◽  
SHI-PING ZHOU
Keyword(s):  
Nonlinear Equations ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Order Derivative ◽  
First Order ◽  
Function Solution ◽  
Weierstrass Elliptic Function ◽  
Klein Gordon

A method for constructing the solutions of nonlinear evolution equations by using the Weierstrass elliptic function and its first-order derivative was presented. This technique was then applied to Burgers and Klein–Gordon equations which showed its efficiency and validality for exactly some solving nonlinear evolution equations.

On the Ostrowski Method for Solving Equations

2021 ◽  
Vol 2 ◽  
pp. 3
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George ◽  
Christopher I. Argyros
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Numerical Experiments ◽  
Convergence Domain ◽  
Lipschitz Constants ◽  
Solving Equations ◽  
Ostrowski’S Method

In this paper, we revisited the Ostrowski's method for solving Banach space valued equations. We developed a technique  to determine a subset of the original convergence domain and using this new Lipschitz constants derived. These constants are at least as tight as the earlier ones leading to a finer convergence analysis in both the semi-local and the local convergence case. These techniques are very general, so they can be used to extend the applicability of other methods without additional hypotheses. Numerical experiments complete this study.

scholarly journals Convergence Analysis of a Three Step Newton-like Method for Nonlinear Equations in Banach Space under Weak Conditions

2016 ◽  
Vol 54 (2) ◽  
pp. 37-46
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Convergence Order ◽  
Order Method ◽  
Numerical Examples ◽  
Local Convergence Analysis

Abstract In the present paper, we study the local convergence analysis of a fifth convergence order method considered by Sharma and Guha in [15] to solve equations in Banach space. Using our idea of restricted convergence domains we extend the applicability of this method. Numerical examples where earlier results cannot apply to solve equations but our results can apply are also given in this study.

scholarly journals A New Derivative-Free Method to Solve Nonlinear Equations

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 583
Author(s):  
Beny Neta
Keyword(s):  
Nonlinear Equation ◽  
Nonlinear Equations ◽  
High Order ◽  
Order Derivative ◽  
Eighth Order ◽  
High Order Derivative ◽  
Derivative Free ◽  
Derivative Free Method

A new high-order derivative-free method for the solution of a nonlinear equation is developed. The novelty is the use of Traub’s method as a first step. The order is proven and demonstrated. It is also shown that the method has much fewer divergent points and runs faster than an optimal eighth-order derivative-free method.

scholarly journals Extending the Applicability of the Super-Halley-Like Method Using ω-Continuous Derivatives and Restricted Convergence Domains

2019 ◽  
Vol 33 (1) ◽  
pp. 21-40
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Uniqueness Result ◽  
Second Derivative ◽  
Numerical Examples ◽  
The Third ◽  
Lipschitz Constants ◽  
Local Convergence Analysis ◽  
Third Derivative

AbstractWe present a local convergence analysis of the super-Halley-like method in order to approximate a locally unique solution of an equation in a Banach space setting. The convergence analysis in earlier studies was based on hypotheses reaching up to the third derivative of the operator. In the present study we expand the applicability of the super-Halley-like method by using hypotheses up to the second derivative. We also provide: a computable error on the distances involved and a uniqueness result based on Lipschitz constants. Numerical examples are also presented in this study.

Sign in / Sign up

Export Citation Format

Share Document