scholarly journals An Iterative Solver in the Presence and Absence of Multiplicity for Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Fazlollah Soleymani ◽  
Stanford Shateyi ◽  
Gülcan Özkum
Keyword(s):  
Fixed Point ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Chaotic Behavior ◽  
Multiple Root ◽  
Basins Of Attraction ◽  
Iterative Solver ◽  
Type Method ◽  
New Class ◽  
Point Type

We develop a high-order fixed point type method to approximate a multiple root. By using three functional evaluations per full cycle, a new class of fourth-order methods for this purpose is suggested and established. The methods from the class require the knowledge of the multiplicity. We also present a method in the absence of multiplicity for nonlinear equations. In order to attest the efficiency of the obtained methods, we employ numerical comparisons alongside obtaining basins of attraction to compare them in the complex plane according to their convergence speed and chaotic behavior.

scholarly journals Optimal Methods for Finding Simple and Multiple Roots of Nonlinear Equations and their Basins of Attraction

2020 ◽  
Vol 37 (1-2) ◽  
pp. 14-29
Author(s):  
Prem Bahadur Chand
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Multiple Root ◽  
Basins Of Attraction ◽  
Weight Functions ◽  
Multiple Roots ◽  
Order Method ◽  
Numerical Examples ◽  
Fourth Order Method ◽  
Simple And Multiple Roots

In this paper, using the variant of Frontini-Sormani method, some higher order methods for finding the roots (simple and multiple) of nonlinear equations are proposed. In particular, we have constructed an optimal fourth order method and a family of sixth order method for finding a simple root. Further, an optimal fourth order method for finding a multiple root of a nonlinear equation is also proposed. We have used different weight functions to a cubically convergent For ntini-Sormani method for the construction of these methods. The proposed methods are tested on numerical examples and compare the results with some existing methods. Further, we have presented the basins of attraction of these methods to understand their dynamics visually.

scholarly journals On generalized Halley-like methods for solving nonlinear equations

2019 ◽  
Vol 13 (2) ◽  
pp. 399-422
Author(s):  
Miodrag Petkovic ◽  
Ljiljana Petkovic ◽  
Beny Neta
Keyword(s):  
Computer Graphics ◽  
Nonlinear Equation ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Multiple Root ◽  
Basins Of Attraction ◽  
Third Order ◽  
Numerical Examples ◽  
Theoretical Results

Generalized Halley-like one-parameter families of order three and four for finding multiple root of a nonlinear equation are constructed and studied. This presentation is, actually, a mixture of theoretical results, algorithmic aspects, numerical experiments, and computer graphics. Starting from the proposed class of third order methods and using an accelerating procedure, we construct a new fourth order family of Halley's type. To analyze convergence behavior of two presented families, we have used two methodologies: (i) testing by numerical examples and (ii) dynamic study using basins of attraction.

scholarly journals Some Class of Third- and Fourth-Order Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
J. P. Jaiswal
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Basins Of Attraction ◽  
Order Method ◽  
Third Order ◽  
Numerical Examples ◽  
Multivariate Extension ◽  
New Class ◽  
Fourth Order Method

The object of the present work is to give the new class of third- and fourth-order iterative methods for solving nonlinear equations. Our proposed third-order method includes methods of Weerakoon and Fernando (2000), Homeier (2005), and Chun and Kim (2010) as particular cases. The multivariate extension of some of these methods has been also deliberated. Finally, some numerical examples are given to illustrate the performances of our proposed methods by comparing them with some well existing third- and fourth-order methods. The efficiency of our proposed fourth-order method over some fourth-order methods is also confirmed by basins of attraction.

scholarly journals An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior

2019 ◽  
Vol 17 (1) ◽  
pp. 1567-1598
Author(s):  
Tianbao Liu ◽  
Xiwen Qin ◽  
Qiuyue Li
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Second Derivative ◽  
Third Order ◽  
Numerical Tests ◽  
The Family ◽  
Theoretical Results ◽  
High Computational Efficiency

Abstract In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.

A new fourth order Newton-type method for solution of system of nonlinear equations

2015 ◽  
Vol 270 ◽  
pp. 724-730 ◽  
Author(s):  
Waseem Asghar Khan ◽  
Khalida Inayat Noor ◽  
Kaleemulah Bhatti ◽  
Faryal Aijaz Ansari
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
System Of Nonlinear Equations ◽  
Type Method

scholarly journals A New Biparametric Family of Two-Point Optimal Fourth-Order Multiple-Root Finders

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Young Ik Kim ◽  
Young Hee Geum
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Multiple Root ◽  
Multiple Roots ◽  
Asymptotic Error ◽  
Error Constant ◽  
Asymptotic Error Constant

We construct a biparametric family of fourth-order iterative methods to compute multiple roots of nonlinear equations. This method is verified to be optimally convergent. Various nonlinear equations confirm our proposed method with order of convergence of four and show that the computed asymptotic error constant agrees with the theoretical one.

scholarly journals Design, Convergence and Stability of a Fourth-Order Class of Iterative Methods for Solving Nonlinear Vectorial Problems

2021 ◽  
Vol 5 (3) ◽  
pp. 125
Author(s):  
Alicia Cordero ◽  
Cristina Jordán ◽  
Esther Sanabria-Codesal ◽  
Juan R. Torregrosa
Keyword(s):  
Fourth Order ◽  
Chaotic Behavior ◽  
Basins Of Attraction ◽  
Order Convergence ◽  
Iterative Schemes ◽  
Numerical Tests ◽  
New Methods ◽  
Stable Elements ◽  
Parameter Values ◽  
Theoretical Results

A new parametric family of iterative schemes for solving nonlinear systems is presented. Fourth-order convergence is demonstrated and its stability is analyzed as a function of the parameter values. This study allows us to detect the most stable elements of the class, to find the fractals in the boundary of the basins of attraction and to reject those with chaotic behavior. Some numerical tests show the performance of the new methods, confirm the theoretical results and allow to compare the proposed schemes with other known ones.

A New Class of Three-Dimensional Maps with Hidden Chaotic Dynamics

2016 ◽  
Vol 26 (12) ◽  
pp. 1650206 ◽  
Author(s):  
Haibo Jiang ◽  
Yang Liu ◽  
Zhouchao Wei ◽  
Liping Zhang
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Chaotic Dynamics ◽  
Three Dimensional ◽  
Computational Procedure ◽  
Basins Of Attraction ◽  
Chaotic Attractors ◽  
Computer Search ◽  
New Class ◽  
Special Concern

This paper studies a new class of three-dimensional maps in a Jerk-like structure with a special concern of their hidden chaotic dynamics. Our investigation focuses on the hidden chaotic attractors in three typical scenarios of fixed points, namely no fixed point, single fixed point, and two fixed points. A systematic computer search is performed to explore possible hidden chaotic attractors, and a number of examples of the proposed maps are used for demonstration. Numerical results show that the routes to hidden chaotic attractors are complex, and the basins of attraction for the hidden chaotic attractors could be tiny, so that using the standard computational procedure for localization is impossible.

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