An efficient three-step method to solve system of nonlinear equations

Esmaeili, Hamid Reza; Ahmadi, Mehdi

doi:10.1016/j.amc.2015.05.076

Cited by 14 publications

(9 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…so z 0 , x 1 ∈ U(x * , ρ) and (15), (16) hold for n = 0. Hence, the induction for (14)-(16) is shown for n = 0.…”

Section: Convergencementioning

confidence: 99%

Extended Local Convergence for High Order Schemes Under ω-Continuity Conditions

Argyros

et al. 2020

Contemporary Mathematics

View full text Add to dashboard Cite

There is a plethora of schemes of the same convergence order for generating a sequence approximating a solution of an equation involving Banach space operators. But the set of convergence criteria is not the same in general. This makes the comparison between them challenging and only numerically. Moreover, the convergence is established using Taylor series and by assuming the existence of high order derivatives that do not even appear on these schemes. Furthermore, no computable error estimates, uniqueness for the solution results or a ball of convergence is given. We address all these problems by using only the first derivative that actually appears on these schemes and under the same set of convergence conditions. Our technique is so general that it can be used to extend the applicability of other schemes along the same lines.

show abstract

“…so z 0 , x 1 ∈ U(x * , ρ) and (15), (16) hold for n = 0. Hence, the induction for (14)-(16) is shown for n = 0.…”

Section: Convergencementioning

confidence: 99%

Extended Local Convergence for High Order Schemes Under ω-Continuity Conditions

Argyros

et al. 2020

Contemporary Mathematics

View full text Add to dashboard Cite

show abstract

“…But this can rarely be achieved, so most researchers and practitioners develop iterative schemes which converge to x * . In this paper we extend the convergence ball of a class of an ecient sixth order-scheme studied in [18]. Precisely, we consider the sixth order method dened in [18] for n = 1, 2, .…”

Section: Introductionmentioning

confidence: 99%

“…where A n = F (x n ) + F (y n ). The analysis in [18] uses assumptions on the sixth order derivatives of F and when B 1 = B 2 = R m . The assumptions on higher order derivatives reduce the applicability of method (2).…”

Section: Introductionmentioning

confidence: 99%

Ball analysis for an efficient sixth convergence order scheme under weaker conditions

Argyros

George

2021

Advances in the Theory of Nonlinear Analysis and Its Application

View full text Add to dashboard Cite

In this study we consider an ecient sixth order-scheme for solving Banach space valued equations. The convergence criteria in earlier studies involve higher order derivatives limiting applicability of these methods. In this study we use the rst derivative only in our analysis to expand the usage of these schemes. The technique we use can be used on other schemes to obtain the same advantages. Numerical experiments compare favorably our results to earlier ones.

show abstract

“…These methods use two operators, two Fréchet derivative evaluations, and two linear operator inversions. The sixth convergence order of methods was given in Cordero et al [6], Soleymani et al [7], and Esmaeili and Ahmadi [8], respectively. The conclusions were obtained for the special case when X = Y = R i , using Taylor series with hypotheses up to the seventh derivative even though it does not appear in the methods.…”

Section: Introductionmentioning

confidence: 99%

Ball Comparison between Three Sixth Order Methods for Banach Space Valued Operators

2020

View full text Add to dashboard Cite

Three methods of sixth order convergence are tackled for approximating the solution of an equation defined on the finitely dimensional Euclidean space. This convergence requires the existence of derivatives of, at least, order seven. However, only derivatives of order one are involved in such methods. Moreover, we have no estimates on the error distances, conclusions about the uniqueness of the solution in any domain, and the convergence domain is not sufficiently large. Hence, these methods have limited usage. This paper introduces a new technique on a general Banach space setting based only the first derivative and Lipschitz type conditions that allow the study of the convergence. In addition, we find usable error distances as well as uniqueness of the solution. A comparison between the convergence balls of three methods, not possible to drive with the previous approaches, is also given. The technique is possible to use with methods available in literature improving, consequently, their applicability. Several numerical examples compare these methods and illustrate the convergence criteria.

show abstract

An efficient three-step method to solve system of nonlinear equations

Cited by 14 publications

References 15 publications

Extended Local Convergence for High Order Schemes Under ω-Continuity Conditions

Extended Local Convergence for High Order Schemes Under ω-Continuity Conditions

Ball analysis for an efficient sixth convergence order scheme under weaker conditions

Ball Comparison between Three Sixth Order Methods for Banach Space Valued Operators

Contact Info

Product

Resources

About