Drawing Dynamical and Parameters Planes of Iterative Families and Methods

Motsa

et al. 2015

Self Cite

In this paper, we propose a general class of fourth-order optimal multi-point methods without memory for obtaining simple roots. This class requires only three functional evaluations (viz. two evaluations of function f (x n ), f (y n ) and one of its first-order derivative f (x n )) per iteration. Further, we show that the well-known Ostrowski's method and King's family of fourth-order procedures are special cases of our proposed schemes. One of the new particular subclasses is a biparametric family of iterative methods. By using complex dynamics tools, its stability is analyzed, showing stable members of the family. Further on, one of the parameters is fixed and the stability of the resulting class is studied. On the other hand, the accuracy and validity of new schemes is tested by a number of numerical examples by comparing them with recent and classical optimal fourth-order methods available in the literature. It is found that they are very useful in high precision computations.

Section: The Parametric Familymentioning

confidence: 99%

Section: The Parametric Familymentioning

confidence: 99%

Construction of fourth-order optimal families of iterative methods and their dynamics

Behl

Motsa

et al. 2015

Self Cite

“…The graphical tools used to obtain the parameter planes and the different dynamical planes have been designed by Chicharro et al in [22] and are implemented in Matlab language.…”

Section: Dynamical Analysismentioning

confidence: 99%

A stable family with high order of convergence for solving nonlinear equations

Lotfi

Mahdiani

et al. 2015

Self Cite

have published a new family of iterative methods, without memory, with order of convergence five or six, which are not optimal in the sense of Kung and Traub's conjecture. Therefore, we attempt to modify this suggested family in such a way that it becomes optimal. To this end, we consider the same two first steps of the mentioned family, and furthermore, we introduce a better approximation for f ′ (z) in the third step based on interpolation idea as opposed to the Taylor's series used in the work of Li et al. Theoretical, dynamical and numerical aspects of the new family are described and investigated in details.

“…The parameter plane is obtained by iterating one critical (free) point; each point of the parameter plane is associated with a complex value of a, i.e., with an element of the family. To build this parameter plane we use the algorithms designed in [7], with MatLab software. The following figures are made by using these algorithms, by using points, they also are critical points and give rise to their respective Fatou components.…”

mentioning

confidence: 99%

“…In this section, we consider those methods with a small number of critical points and show the dynamical planes for given values of the parameter. The dynamical planes are built by using the algorithms designed in [7], with MatLab software, by using a mesh of 800 × 800 points, a maximum of 80 iterations and a tolerance of 10 −3 .…”

mentioning

confidence: 99%

Dynamics of a multipoint variant of Chebyshev–Halley’s family

Campos

Torregrosa

et al. 2016

Self Cite

In this paper, a complex dynamical study of a parametric Chebyshev-Halley type family of iterative methods on quadratic polynomial is presented. The stability of the fixed points is analyzed in terms of the parameter of the family. We also calculate the critical points building their corresponding parameter planes which allow us to analyze the qualitative behaviour of this family. Moreover, we locate some dynamical planes showing different pathological aspects of this family.