Ball convergence of a sixth-order Newton-like method based on means under weak conditions

Magreñán, Á. Alberto; Argyros, Ioannis K.; Rainer, J. Javier; Sicilia, Juan Antonio

doi:10.1007/s10910-018-0856-y

Cited by 22 publications

(11 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The same benefits can be obtained for the other single-step methods. Moreover, our idea can similarly be extended to multi-step and multi-point methods [4,5,13,[19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].…”

Section: Theorem 5 Under the Preceding Conditions Secant Methods {Xmentioning

confidence: 99%

Unified Convergence Criteria for Iterative Banach Space Valued Methods with Applications

Argyros

2021

Mathematics

View full text Add to dashboard Cite

A plethora of sufficient convergence criteria has been provided for single-step iterative methods to solve Banach space valued operator equations. However, an interesting question remains unanswered: is it possible to provide unified convergence criteria for single-step iterative methods, which are weaker than earlier ones without additional hypotheses? The answer is yes. In particular, we provide only one sufficient convergence criterion suitable for single-step methods. Moreover, we also give a finer convergence analysis. Numerical experiments involving boundary value problems and Hammerstein-like integral equations complete this paper.

show abstract

Section: Theorem 5 Under the Preceding Conditions Secant Methods {Xmentioning

confidence: 99%

Unified Convergence Criteria for Iterative Banach Space Valued Methods with Applications

Argyros

2021

Mathematics

View full text Add to dashboard Cite

show abstract

“…In general a closed form solution for (1.1) is not possible, so iterative methods are used for approximating a solution x * of (1.1). Many iterative methods are studied for approximating x * [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27].…”

Section: Introductionmentioning

confidence: 99%

“…The local convergence of the Homeier method was shown to be of order three using Taylor expansion and assumptions on the fourth order derivative of G, which is not on these methods [13]. So, the assumptions on the fourth derivative reduce the applicability of these methods [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. For example:…”

Section: Introductionmentioning

confidence: 99%

On the semi-local convergence of the Homeier method in Banach space for solving equations

Regmi

Argyros

George

et al. 2022

CMCMA

View full text Add to dashboard Cite

In this paper we consider the semi-local convergence analysis of the Homeier method for solving nonlinear equation in Banach space. As far as we know no semi-local convergence has been given for the Homeier under Lipschitz conditions. Our goal is to extend the applicability of the Homeier method in the semi-local convergence under these conditions. We use majorizing sequences and conditions only on the first derivative which appear on the method for proving our results. Numerical experiments are provided in this study.

show abstract

“…The motivation for writing this paper is the extension of the applicability of method (2), as already illustrated by the example. The novelty of the paper includes the extension of the convergence domain in both the local as well as the semi-local convergence case and the introduction of the recurrent functions proving technique, which can be used in other methods too [14][15][16][17][18][19][20][21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Extended Kung–Traub Methods for Solving Equations with Applications

et al. 2021

View full text Add to dashboard Cite

Kung and Traub (1974) proposed an iterative method for solving equations defined on the real line. The convergence order four was shown using Taylor expansions, requiring the existence of the fifth derivative not in this method. However, these hypotheses limit the utilization of it to functions that are at least five times differentiable, although the methods may converge. As far as we know, no semi-local convergence has been given in this setting. Our goal is to extend the applicability of this method in both the local and semi-local convergence case and in the more general setting of Banach space valued operators. Moreover, we use our idea of recurrent functions and conditions only on the first derivative and divided difference, which appear in the method. This idea can be used to extend other high convergence multipoint and multistep methods. Numerical experiments testing the convergence criteria complement this study.

show abstract

Ball convergence of a sixth-order Newton-like method based on means under weak conditions

Cited by 22 publications

References 23 publications

Unified Convergence Criteria for Iterative Banach Space Valued Methods with Applications

Unified Convergence Criteria for Iterative Banach Space Valued Methods with Applications

On the semi-local convergence of the Homeier method in Banach space for solving equations

Extended Kung–Traub Methods for Solving Equations with Applications

Contact Info

Product

Resources

About