On two families of high order Newton type methods

Amat, Sergio; Busquier, Sonia; Bermúdez, Concepción; Plaza, Sergio

doi:10.1016/j.aml.2012.06.004

Cited by 35 publications

(56 citation statements)

References 5 publications

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“…Then, obviously function F does not have bounded third derivative in D. Therefore, the results in [1,21,22] cannot be used to show convergence of the preceding methods, since they require the existence of derivatives up to the third order or higher. (1.5).…”

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confidence: 97%

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Local convergence for some high convergence order Newton-like methods with frozen derivatives

Argyros

George

2015

SeMA

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We present a local convergence analysis of some families of Newton-like methods with frozen derivatives in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as Amat et al. (Appl Math Lett. 25:2209-2217, 2012), Petkovic (Multipoint methods for solving nonlinear equations, Elsevier, Amsterdam, 2013), Traub (Iterative methods for the solution of equations, AMS Chelsea Publishing, Providence, 1982) and Xiao and Yin (Appl Math Comput, 2015) the local convergence was proved based on hypotheses on the derivative of order higher than two although only the first derivative appears in these methods. In this paper we expand the applicability of these methods using only hypotheses on the first derivative and Lipschitz constants. Numerical examples are also presented in this study.

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confidence: 97%

“…Notice that operator ϕ defines a sequence {ϕ(a, x n , y n )} of linear operators from Y into X depending on the parameter a and previous iterates x n and y n at each step. Some choices for λ and ϕ can be [1]. Again, if X = Y = R and F is sufficiently many times differentiable, then the convergence order of method (1.3) is m + 1.…”

mentioning

confidence: 98%

Local convergence for some high convergence order Newton-like methods with frozen derivatives

Argyros

George

2015

SeMA

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“…In most real problems, the computational cost of solving a linear system is more expensive than some extra evaluations of the operator. Moreover, from a dynamical point of view [6] the method seem better than the classical two-point Newton method, that is…”

Section: Introductionmentioning

confidence: 99%

Expanding the Applicability of a Third Order Newton-Type Method Free of Bilinear Operators

et al. 2015

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This paper is devoted to the semilocal convergence, using centered hypotheses, of a third order Newton-type method in a Banach space setting. The method is free of bilinear operators and then interesting for the solution of systems of equations. Without imposing any type of Fréchet differentiability on the operator, a variant using divided differences is also analyzed. A variant of the method using only divided differences is also presented.

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“…The dynamical planes of the iterative schemes supply this information graphically, and are developed in Section 3. Several authors have studied and compared the stability of different known iterative methods by means of their dynamical planes, firstly in the work of Varona [3] and later on developed by Amat et al [4], Neta et al [5,6], among others.…”

Section: Introductionmentioning

confidence: 99%

Dynamics and Fractal Dimension of Steffensen-Type Methods

2015

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In this paper, the dynamical behavior of different optimal iterative schemes for solving nonlinear equations with increasing order, is studied. The tendency of the complexity of the Julia set is analyzed and referred to the fractal dimension. In fact, this fractal dimension can be shown to be a powerful tool to compare iterative schemes that estimate the solution of a nonlinear equation. Based on the box-counting algorithm, several iterative derivative-free methods of different convergence orders are compared.

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On two families of high order Newton type methods

Cited by 35 publications

References 5 publications

Local convergence for some high convergence order Newton-like methods with frozen derivatives

Local convergence for some high convergence order Newton-like methods with frozen derivatives

Expanding the Applicability of a Third Order Newton-Type Method Free of Bilinear Operators

Dynamics and Fractal Dimension of Steffensen-Type Methods

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