Superattracting cycles for some Newton type iterative methods

Amat, Sergio; Busquier, Sonia; Navarro, É.; Plaza, Sergio

doi:10.1007/s11425-010-4141-1

Cited by 15 publications

(27 citation statements)

References 11 publications

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“…It is well known that if z 0 ∈ C is chosen close enough to one of the solutions of the equation g(z) = 0, say α, then the sequence {z n = g n (z 0 )} n≥0 has the limit α when n tends to ∞. Moreover the speed of (local) convergence is generically quadratic (see, for instance, [2]). It was Cayley (see [10]) the first to consider Newton's method as a (holomorphic) dynamical system, that is studying the convergence of these sequences for all possible seeds x 0 ∈ C at once, under the assumption that g was a degree 2 or 3 polynomial.…”

Section: Introductionmentioning

confidence: 99%

“…Many authors have studied alternative iterative methods having, for instance, a better local speed of convergence. Two of the best known rootfinding algorithms of order of convergence 3 are Chebyshev's method and Halley's method (see [2]). They are included in the Chebyshev-Halley family of root-finding algorithms, which was introduced in [11] (see also [1]), and is defined as follows.…”

Section: Introductionmentioning

confidence: 99%

“…where α ∈ [0, 1] and L g (z) = g(z)g (z) (g (z)) 2 . Notice that in a real setting, it suffices for g to be a doubly differentiable function such that g (x) is continuous.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Newton-Like Components in the Chebyshev–Halley Family of Degree n Polynomials

Paraschiv

2023

Mediterr. J. Math.

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We study the Chebyshev–Halley methods applied to the family of polynomials $$f_{n,c}(z)=z^n+c$$ f n , c ( z ) = z n + c , for $$n\ge 2$$ n ≥ 2 and $$c\in \mathbb {C}^{*}$$ c ∈ C ∗ . We prove the existence of parameters such that the immediate basins of attraction corresponding to the roots of unity are infinitely connected. We also prove that, for $$n \ge 2$$ n ≥ 2 , the corresponding dynamical plane contains a connected component of the Julia set, which is a quasiconformal deformation of the Julia set of the map obtained by applying Newton’s method to $$f_{n,-1}$$ f n , - 1 .

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Newton-Like Components in the Chebyshev–Halley Family of Degree n Polynomials

Paraschiv

2023

Mediterr. J. Math.

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“…Its is easy to prove that the proposed class of iterative methods LCT satisfies the Scaling Theorem (see for example [1]). So, the dynamical performance of the rational function obtained by applying the class of methods on quadratic polynomials is equivalent by conjugation.…”

Section: Introductionmentioning

confidence: 99%

Design and Dynamical behavior of a fourth order family of iterative methods for solving nonlinear equations

Cordero

Ledesma

Maimó

et al. 2023

Preprint

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In this paper, we propose a weight function to construct a fourth order family of iterative schemes for solving nonlinear equations. This class is parameter-dependent and its numerical performance depends on the value of this free parameter. We analyze the rational function resulting from the fixed point operator applied to a nonlinear polynomial. The dynamics of this rational function allows us to better understand the performance of the iterative methods of the class. In addition, we calculate the critical points and present the parameter spaces dynamical planes, in order to determine the regions with stable and unstable behavior. Finally, parameter values within and outside the stability region are chosen and, with them, numerical tests that confirm the scheme's theoretical convergence and stability are performed, as well as comparisons with other existing methods of the same order of convergence.

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“…It is widely accepted that the dynamical behavior of the rational function related to an iterative scheme provides us with important information about its stability and reliability [11]. In these terms, Amat et al in [12] described the dynamical performance of some known families of iterative methods. More recently, in [9,[13][14][15][16][17], different authors analyze the qualitative behavior of several known methods or classes of iterative schemes.…”

Section: Introductionmentioning

confidence: 99%

Parametric Family of Root-Finding Iterative Methods: Fractals of the Basins of Attraction

et al. 2022

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Research interest in iterative multipoint schemes to solve nonlinear problems has increased recently because of the drawbacks of point-to-point methods, which need high-order derivatives to increase the order of convergence. However, this order is not the only key element to classify the iterative schemes. We aim to design new multipoint fixed point classes without memory, that improve or bring together the existing ones in different areas such as computational efficiency, stability and also convergence order. In this manuscript, we present a family of parametric iterative methods, whose order of convergence is four, that has been designed by using composition and weight function techniques. A qualitative analysis is made, based on complex discrete dynamics, to select those elements of the class with best stability properties on low-degree polynomials. This stable behavior is directly related with the simplicity of the fractals defined by the basins of attraction. In the opposite, particular methods with unstable performance present high-complexity in the fractals of their basins. The stable members are demonstrated also be the best ones in terms of numerical performance of non-polynomial functions, with special emphasis on Colebrook-White equation, with wide applications in Engineering.

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Superattracting cycles for some Newton type iterative methods

Cited by 15 publications

References 11 publications

Newton-Like Components in the Chebyshev–Halley Family of Degree n Polynomials

Newton-Like Components in the Chebyshev–Halley Family of Degree n Polynomials

Design and Dynamical behavior of a fourth order family of iterative methods for solving nonlinear equations

Parametric Family of Root-Finding Iterative Methods: Fractals of the Basins of Attraction

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