Dynamics and Fractal Dimension of Steffensen-Type Methods

Chicharro, Francisco Israel; Cordero, Alicia; Torregrosa, Juan R.

doi:10.3390/a8020271

Cited by 13 publications

(8 citation statements)

References 14 publications

(19 reference statements)

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“…Further information about this basis can be found in some works. [11][12][13][14] In addition, the complex dynamics has been developed as a method for the stability analysis in many manuscripts such as other works, 9,[15][16][17] amongs others.…”

Section: Basics On Complex Dynamicsmentioning

confidence: 99%

Wide stability in a new family of optimal fourth‐order iterative methods

Chicharro

Cordero

Garrido

et al. 2019

Comp and Math Methods

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A new family of two-steps fourth-order iterative methods for solving nonlinear equations is introduced based on the weight functions procedure. This family is optimal in the sense of Kung-Traub conjecture and it is extended to design a class of iterative schemes with four step and seventh order of convergence. We are interested in analyzing the dynamical behavior of different elements of the fourth-order class. This analysis gives us important information about the stability of these members of the family. The methods are also tested with nonlinear functions and compared with other known schemes. The results show the good features of the introduced class. KEYWORDS complex dynamics, iterative methods, weight functions Comp and Math Methods. 2019;1:e1023.wileyonlinelibrary.com/journal/cmm4

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Section: Basics On Complex Dynamicsmentioning

confidence: 99%

Wide stability in a new family of optimal fourth‐order iterative methods

Chicharro

Cordero

Garrido

et al. 2019

Comp and Math Methods

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“…8 illustrates the surface that is divided into a series of small sub-areas. Curves L a and L b are then divided into N sections [19], [20], and the length of each curve may be described as:…”

Section: Optimization Criterionmentioning

confidence: 99%

Parameter Deduction and Accurac y Analysis of Track Beam Curves in Straddle - type Monorail Systems

Zhao¹,

Wang²,

Xian-feng³

et al. 2015

JESTR

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The accuracy of the bottom curve of a PC track beam is strongly related to the production quality of the entire beam. Many factors may affect the parameters of the bottom curve, such as the superelevation of the curve and the deformation of a PC track beam. At present, no effective method has been developed to determine the bottom curve of a PC track beam; therefore, a new technique is presented in this paper to deduce the parameters of such a curve and to control the accuracy of the computation results. First, the domain of the bottom curve of a PC track beam is assumed to be a spindle plane. Then, the corresponding supposed top curve domain is determined based on a geometrical relationship that is the opposite of that identified by the conventional method. Second, several optimal points are selected from the supposed top curve domain according to the dichotomy algorithm; the supposed top curve is thus generated by connecting these points. Finally, one rigorous criterion is established in the fractal dimension to assess the accuracy of the assumed top curve deduced in the previous step. If this supposed curve coincides completely with the known top curve, then the assumed bottom curve corresponding to the assumed top curve is considered to be the real bottom curve. This technique of determining the bottom curve of a PC track beam is thus proven to be efficient and accurate.

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“…The design of new efficient methods is an ongoing issue for scientists. Recently, this design goes hand to hand with the dynamical analysis [5][6][7][8][9], allowing the knowledge of the stability of the methods involved.…”

Section: Introductionmentioning

confidence: 99%

Impact on Stability by the Use of Memory in Traub-Type Schemes

et al. 2020

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In this work, two Traub-type methods with memory are introduced using accelerating parameters. To obtain schemes with memory, after the inclusion of these parameters in Traub’s method, they have been designed using linear approximations or the Newton’s interpolation polynomials. In both cases, the parameters use information from the current and the previous iterations, so they define a method with memory. Moreover, they achieve higher order of convergence than Traub’s scheme without any additional functional evaluations. The real dynamical analysis verifies that the proposed methods with memory not only converge faster, but they are also more stable than the original scheme. The methods selected by means of this analysis can be applied for solving nonlinear problems with a wider set of initial estimations than their original partners. This fact also involves a lower number of iterations in the process.

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Dynamics and Fractal Dimension of Steffensen-Type Methods

Cited by 13 publications

References 14 publications

Wide stability in a new family of optimal fourth‐order iterative methods

Wide stability in a new family of optimal fourth‐order iterative methods

Parameter Deduction and Accurac y Analysis of Track Beam Curves in Straddle - type Monorail Systems

Impact on Stability by the Use of Memory in Traub-Type Schemes

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