Convergence of the family of the deformed Euler–Halley iterations under the Hölder condition of the second derivative

Ye, Xintao; Li, Chong

doi:10.1016/j.cam.2005.07.019

Cited by 40 publications

(23 citation statements)

References 18 publications

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“…Basic results concerning the convergence of the process, existence and uniqueness regions of solutions are given by references [23][24][25][26][27][28][29][30][31][32][33][34]. The results concerning convergence have been published under assumptions of continuity conditions on the second derivative: the Kantorovich-type condition, the γ -condition, or the Lipschitz-condition, or the Hölder-condition.…”

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confidence: 99%

Convergence of the modified Halley’s method for multiple zeros under Hölder continuous derivative

Ren

2011

Numer Algor

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confidence: 99%

Convergence of the modified Halley’s method for multiple zeros under Hölder continuous derivative

Ren

2011

Numer Algor

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“…That is why many authors have used higher order multi-point methods (Ahmad et al, 2009;Amat et al, 2008;Argyros, 2008;Argyros and Hilout, 2010;Bruns and Bailey, 1977;Marquina, 1990a, 1990b;Chun, 1990;Ezquerro and Hernández, 2000, 2005Gutiérrez and Hernández, 1998;Ganesh and Joshi, 1991;Hernández, 2001;Hernández and Salanova, 1999;Kantorovich and Akilov, 1982;Gupta, 2007, 2010;Parida and Gupta, 2007;Ren et al, 2009;Rheinboldt, 1977;Traub, 1964;Wang et al, 2009Wang et al, , 2011Ye and Li, 2006;Ye et al, 2007;Zhao and Wu, 2008;Kou, 2012a, 2012b;Zhu and Wu, 2003). In this paper, we present the local convergence of the derivative free method defined for each n = 0, 1, 2, … by…”

Section: Introductionmentioning

confidence: 97%

“…Newton-like methods are widely used for finding solution of equation (1), these methods are usually studied based on: semi-local and local convergence. The semi-local convergence method is based on the information around an initial point, to give conditions ensuring the convergence of the iterative procedure; while the local one is, based on the information around a solution, to find estimates of the radii of convergence balls (Argyros, 2008;Argyros and Hilout, 2010;Ren et al, 2009;Rheinboldt, 1977;Traub, 1964;Ye and Li, 2006;Zhao and Wu, 2008).…”

Section: Introductionmentioning

confidence: 99%

“…Third order methods such as Euler's, Halley's, super Halley's, Chebyshev's (Ahmad et al, 2009;Amat et al, 2008;Argyros, 2008;Argyros and Hilout, 2010;Bruns and Bailey, 1977;Marquina, 1990a, 1990b;Chun, 1990;Ezquerro and Hernández, 2000, 2005Gutiérrez and Hernández, 1998;Ganesh and Joshi, 1991;Hernández, 2001;Hernández and Salanova, 1999;Kantorovich and Akilov, 1982;Gupta, 2007, 2010;Parida and Gupta, 2007;Ren et al, 2009;Rheinboldt, 1977;Traub, 1964;Wang et al, 2009Wang et al, , 2011Ye and Li, 2006;Ye et al, 2007;Zhao and Wu, 2008;Kou, 2012a, 2012b;Zhu and Wu, 2003) require the evaluation of the second derivative F″ at each step, which in general is very expensive. That is why many authors have used higher order multi-point methods (Ahmad et al, 2009;Amat et al, 2008;Argyros, 2008;Argyros and Hilout, 2010;Bruns and Bailey, 1977;Marquina, 1990a, 1990b;Chun, 1990;Ezquerro and Hernández, 2000, 2005Gutiérrez and Hernández, 1998;Ganesh and Joshi, 1991;Hernández, 2001;Hernández and Salanova, 1999;Kantorovich and Akilov...…”

Section: Introductionmentioning

confidence: 99%

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Local convergence for a derivative free method of order three under weak conditions

Argyros¹,

George²

2016

IJCONVC

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Abstract:A local convergence analysis for a family of a third-order method in order to approximate a solution of a nonlinear equation is presented in this paper. We use hypotheses only on the first derivative in contrast to earlier studies such as Gupta (2007, 2010) and Zhu and Wu (2003) using hypotheses only on the first derivative. 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.

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“…With the special choices of α, we can get some famous methods such as the Halley method (α = 1 2 ) [5, 8-10, 17, 18], the Chebyshev-Euler method (α = 0) [3,6,19] and the SuperHalley method (α = 1) [7,12]. The family (2) has been generalized in latest years too [2,23,24].…”

Section: Introductionmentioning

confidence: 99%

On the local convergence of a family of Euler-Halley type iterations with a parameter

Huang

Guochun

2009

Numer Algor

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This paper aims to study the local convergence of a family of EulerHalley type methods with a parameter α for solving nonlinear operator equations under the second-order generalized Lipschitz assumption. The radius r α of the optimal convergence ball and the error estimation of the method corresponding to α are estimated for each α ∈ (−∞, +∞). For each α > 0, we get r α ≥ r −α and the upper bound of the error estimation of the method with α > 0 is not larger than the one with α < 0. For each α ≤ 0, we get the precise value of r α , which is closely linked to the dynamical property of the method applied to a real or a complex function, and the optimal error estimation, which decreases when α → 0 − . Results show that the method corresponding to α is better than the one corresponding to −α for each α > 0 and the ChebyshevEuler method is the best among all methods in the family with α ∈ (−∞, 0] from the view of both safe choice of the initial point and error estimation.

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Convergence of the family of the deformed Euler–Halley iterations under the Hölder condition of the second derivative

Cited by 40 publications

References 18 publications

Convergence of the modified Halley’s method for multiple zeros under Hölder continuous derivative

Convergence of the modified Halley’s method for multiple zeros under Hölder continuous derivative

Local convergence for a derivative free method of order three under weak conditions

On the local convergence of a family of Euler-Halley type iterations with a parameter

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