scholarly journals Ball convergence for Traub-Steffensen like methods in Banach space

2015 ◽  
Vol 53 (2) ◽  
pp. 3-16
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Fréchet Derivative ◽  
Not Given ◽  
Frechet Derivative ◽  
The Third ◽  
Taylor Expansions ◽  
Lipschitz Constants ◽  
Ball Convergence ◽  
Computable Error Bounds ◽  
Local Convergence Analysis

Abstract We present a local convergence analysis for two Traub-Steffensen-like methods in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as [16, 23] Taylor expansions and hypotheses up to the third Fréchet-derivative are used. We expand the applicability of these methods using only hypotheses on the first Fréchet derivative. Moreover, we obtain a radius of convergence and computable error bounds using Lipschitz constants not given before. Numerical examples are also presented in this study.

Ball Convergence for a Family of Quadrature-Based Methods for Solving Equations in Banach Space

2017 ◽  
Vol 14 (02) ◽  
pp. 1750017 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Ramandeep Behl ◽  
S. S. Motsa
Keyword(s):  
Banach Space ◽  
Taylor Series Expansion ◽  
Fréchet Derivative ◽  
Dimensional Euclidean Space ◽  
Frechet Derivative ◽  
Space Setting ◽  
Lipschitz Constants ◽  
Solving Equations ◽  
Ball Convergence ◽  
Predictor Corrector

We present a local convergence analysis for a family of quadrature-based predictor–corrector methods in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as Howk [2016] [Howk, C. L. [2016] “A classs of efficient quadrature-based predictor–corrector methods for solving nonlinear systems, Appl. Math. Comput. 276, 394–406] the [Formula: see text] order of convergence was shown on the [Formula: see text]-dimensional Euclidean space using Taylor series expansion and hypotheses reaching up to the third-order Fréchet-derivative of the operator involved although only the first-order Fréchet-derivative appears in these methods, which restrict the applicability of these methods. In this paper, we expand the applicability of these methods in a Banach space setting and using hypotheses only on the first Fréchet-derivative. Moreover, we provide computable radii of convergence as well as error bounds on the distances involved using Lipschitz constants. Numerical examples are also presented to solve equations in cases where earlier results cannot apply.

Local convergence of deformed Jarratt-type methods in Banach space without inverses

2016 ◽  
Vol 09 (01) ◽  
pp. 1650015
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Computational Cost ◽  
Fréchet Derivative ◽  
Convergence Conditions ◽  
Type Method ◽  
Frechet Derivative ◽  
The Third ◽  
Local Convergence Analysis

We present a local convergence analysis for the Jarratt-type method of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies use hypotheses up to the third Fréchet-derivative. Hence, the applicability of these methods is expanded under weaker hypotheses and less computational cost for the constants involved in the convergence analysis. Numerical examples are also provided in this study.

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 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.

Local convergence of deformed Euler–Halley-type methods in Banach space under weak conditions

2017 ◽  
Vol 10 (02) ◽  
pp. 1750086
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Nonlinear Equation ◽  
Convergence Analysis ◽  
Error Estimates ◽  
Local Convergence ◽  
Fréchet Derivative ◽  
Convergence Ball ◽  
Numerical Examples ◽  
Frechet Derivative ◽  
Local Convergence Analysis

We present a unified local convergence analysis for deformed Euler–Halley-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Euler, Halley and other high order methods. The convergence ball and error estimates are given for these methods under hypotheses up to the first Fréchet derivative in contrast to earlier studies using hypotheses up to the second Fréchet derivative. Numerical examples are also provided in this study.

On a sixth-order Jarratt-type method in Banach spaces

2015 ◽  
Vol 08 (04) ◽  
pp. 1550065 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Spaces ◽  
Semilocal Convergence ◽  
Fréchet Derivative ◽  
Sixth Order ◽  
Convergence Conditions ◽  
Type Method ◽  
Frechet Derivative ◽  
The Third ◽  
Local Convergence Analysis ◽  
Jarratt Method

We present a local convergence analysis of a sixth-order Jarratt-type method in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies such as [X. Wang, J. Kou and C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces, Numer. Algorithms 57 (2011) 441–456.] require hypotheses up to the third Fréchet-derivative. Numerical examples are also provided in this study.

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.

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 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.

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