Real dynamics for damped Newton’s method applied to cubic polynomials

Magreñán, Á. Alberto; Jiménez, José Manuel Gutiérrez

doi:10.1016/j.cam.2013.11.019

Cited by 52 publications

(29 citation statements)

References 7 publications

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“…Remark The Scaling Theorem remains true, with almost exactly the same proof, for the damped Newton's function

M_{λ, f}

, see , for arbitrary damping parameter

λ \neq 0

, more precisely, we have

(T \circ M_{λ, f \circ T} \circ T^{- 1}) (x) = M_{λ, f} (x),

for all

x \in R

such that

f^{'} (x) \neq 0

. For the case of the damped Newton's function

N_{λ, f}

, see (3.6), the corresponding generalisation of Lemma ,

(T \circ N_{λ, f \circ T} \circ T^{- 1}) (x) = N_{λ, f} (x),

follows from the proof of Theorem 2.1 in ; although it was stated for analytic functions f , the assumption was not used in that proof.…”

Section: The Scaling Theoremmentioning

confidence: 69%

“…Remark In order to reduce the chaotic behaviour and improve numerical parameters of approximation for lower order polynomials, in and damped Newton's methods have been considered. More precisely, letting λ be the damping parameter, one defines

N_{λ, f}

and

M_{λ, f}

as follows:

N_{λ, f} (x) = x - λ \frac{f (x)}{f^{'} (x)},

and

M_{λ, f} (x) = N_{λ, f} (x) - λ \frac{f (N_{λ, f} (x))}{f^{'} (x)} .

It is easy to observe, by inspection, that Lemma and Theorem remain true if

N_{λ, f}

and

M_{λ, f}

replace

N_{f}

and, respectively,

M_{f}

, for arbitrary damping parameter

λ > 0

, hence the chaotic behaviour characterised by existence of periodic points of any prime period, as well as the uncountability of the set of points of divergence of iteration of

M_{f}

, remain unaltered by damping, for the class of functions considered in Theorem .…”

Section: The Dynamics Of Mfmentioning

confidence: 99%

“…and then, letting x = ay + b, follows from the proof of Theorem 2.1 in [6]; although it was stated for analytic functions f , the assumption was not used in that proof.…”

Section: Lemma 32 Let F Be a Newton Map And Let C 1 And C 2 Be Two Cmentioning

confidence: 99%

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On the dynamics of a third order Newton's approximation method

Gheondea

Şamcı

2016

Mathematische Nachrichten

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We provide an answer to a question raised by S. Amat, S. Busquier, S. Plaza on the qualitative analysis of the dynamics of a certain third order Newton type approximation function Mf, by proving that for functions f twice continuously differentiable and such that both f and its derivative do not have multiple roots, with at least four roots and infinite limits of opposite signs at ±∞, Mf has periodic points of any prime period and that the set of points a at which the approximation sequence (Mfn(a))n∈double-struckN does not converge is uncountable. In addition, we observe that in their Scaling Theorem analyticity can be replaced with differentiability.

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“…Remark The Scaling Theorem remains true, with almost exactly the same proof, for the damped Newton's function

M_{λ, f}

, see , for arbitrary damping parameter

λ \neq 0

, more precisely, we have

(T \circ M_{λ, f \circ T} \circ T^{- 1}) (x) = M_{λ, f} (x),

for all

x \in R

such that

f^{'} (x) \neq 0

. For the case of the damped Newton's function

N_{λ, f}

, see (3.6), the corresponding generalisation of Lemma ,

(T \circ N_{λ, f \circ T} \circ T^{- 1}) (x) = N_{λ, f} (x),

follows from the proof of Theorem 2.1 in ; although it was stated for analytic functions f , the assumption was not used in that proof.…”

Section: The Scaling Theoremmentioning

confidence: 69%

N_{λ, f}

and

M_{λ, f}

as follows:

N_{λ, f} (x) = x - λ \frac{f (x)}{f^{'} (x)},

and

M_{λ, f} (x) = N_{λ, f} (x) - λ \frac{f (N_{λ, f} (x))}{f^{'} (x)} .

It is easy to observe, by inspection, that Lemma and Theorem remain true if

N_{λ, f}

and

M_{λ, f}

replace

N_{f}

and, respectively,

M_{f}

, for arbitrary damping parameter

λ > 0

, hence the chaotic behaviour characterised by existence of periodic points of any prime period, as well as the uncountability of the set of points of divergence of iteration of

M_{f}

, remain unaltered by damping, for the class of functions considered in Theorem .…”

Section: The Dynamics Of Mfmentioning

confidence: 99%

See 1 more Smart Citation

On the dynamics of a third order Newton's approximation method

Gheondea

Şamcı

2016

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…Although the aim of many researches in this area is to design optimal high-order methods, it is also known that the higher the order is, the more sensitive the scheme to initial estimations will be [18]. On the other hand, recent studies on damped Newton's procedure show (see, for example [19]) that small damping parameters widen the set of initial guesses that make the method convergent, although the speed of convergence decreases.…”

Section: Introductionmentioning

confidence: 99%

Choosing the most stable members of Kou’s family of iterative methods

Cordero

Guasp

Torregrosa

2018

Journal of Computational and Applied Mathematics

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In this manuscript, we analyze the dynamical anomalies of a parametric family of iterative schemes designed by Kou et al. It is known that its order of convergence is three for any arbitrary value of the parameter, but it has order four (and it is optimal in the sense of Kung-Traub's conjecture) when an specific value is selected. Among all the elements of this family, one can choose this fourth-order element or any of the infinite members of third order of convergence, if only the speed of convergence is considered. However, the stability of the methods plays an important role in their reliability when they are applied on different problems. This is the reason why we analyze in this paper the dynamical behavior on quadratic polynomials of the mentioned family. The study of fixed points and their stability, joint with the critical points and their associated parameter planes, show the richness of the class and allows us to find members of it with excellent numerical properties, as well as other ones with very unstable behavior. Some test functions are analyzed for confirming the theoretical results.

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“…The dynamics of the real cubic polynomials is a little more complicated than that of the quadratic polynomials. The real dynamics of the cubic polynomials are given in [5,6]. Generally, the dynamics of transcendental functions is more complicated than polynomials.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation and Chaos in Real Dynamics of a Two-Parameter Family Arising from Generating Function of Generalized Apostol-Type Polynomials

Sajid

2018

MCA

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Abstract:The aim of this paper is to investigate the bifurcation and chaotic behaviour in the two-parameter family of transcendental functions f λ,n (x) = λ x (e x +1) n , λ > 0, x ∈ R, n ∈ N\{1} which arises from the generating function of the generalized Apostol-type polynomials. The existence of the real fixed points of f λ,n (x) and their stability are studied analytically and the periodic points of f λ,n (x) are computed numerically. The bifurcation diagrams and Lyapunov exponents are simulated; these demonstrate chaotic behaviour in the dynamical system of the function f λ,n (x) for certain ranges of parameter λ.

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Real dynamics for damped Newton’s method applied to cubic polynomials

Cited by 52 publications

References 7 publications

On the dynamics of a third order Newton's approximation method

On the dynamics of a third order Newton's approximation method

Choosing the most stable members of Kou’s family of iterative methods

Bifurcation and Chaos in Real Dynamics of a Two-Parameter Family Arising from Generating Function of Generalized Apostol-Type Polynomials

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