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 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.

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 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 General Family of Third Order Methods for Multiple Roots of Nonlinear Equations and Basin Attractors for Various Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Rajni Sharma ◽  
Ashu Bahl
Keyword(s):  
Nonlinear Equations ◽  
Complex Plane ◽  
General Scheme ◽  
Basins Of Attraction ◽  
Multiple Roots ◽  
Order Convergence ◽  
Third Order ◽  
Numerical Examples ◽  
General Family ◽  
Theoretical Results

A general scheme of third order convergence is described for finding multiple roots of nonlinear equations. The proposed scheme requires one evaluation of f, f′, and f′′ each per iteration and contains several known one-point third order methods for finding multiple roots, as particular cases. Numerical examples are included to confirm the theoretical results and demonstrate convergence behavior of the proposed methods. In the end, we provide the basins of attraction for some methods to observe their dynamics in the complex plane.

scholarly journals On Constructing Two-Point Optimal Fourth-Order Multiple-Root Finders with a Generic Error Corrector and Illustrating Their Dynamics

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Young Ik Kim ◽  
Young Hee Geum
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Multiple Root ◽  
Optimal Order ◽  
Convergence Behavior ◽  
Principal Branch ◽  
Basins Of Attractions ◽  
Made In

With an error corrector via principal branch of themth root of a function-to-function ratio, we propose optimal quartic-order multiple-root finders for nonlinear equations. The relevant optimal order satisfies Kung-Traub conjecture made in 1974. Numerical experiments performed for various test equations demonstrate convergence behavior agreeing with theory and the basins of attractions for several examples are presented.

scholarly journals Design and Complex Dynamics of Potra–Pták-Type Optimal Methods for Solving Nonlinear Equations and Its Applications

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 942 ◽  
Author(s):  
Prem B. Chand ◽  
Francisco I. Chicharro ◽  
Neus Garrido ◽  
Pankaj Jain
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Complex Dynamics ◽  
Basins Of Attraction ◽  
Weight Functions ◽  
Order Method ◽  
Eighth Order ◽  
Numerical Examples ◽  
Fourth Order Method ◽  
Cubic Polynomials

In this paper, using the idea of weight functions on the Potra–Pták method, an optimal fourth order method, a non optimal sixth order method, and a family of optimal eighth order methods are proposed. These methods are tested on some numerical examples, and the results are compared with some known methods of the corresponding order. It is proved that the results obtained from the proposed methods are compatible with other methods. The proposed methods are tested on some problems related to engineering and science. Furthermore, applying these methods on quadratic and cubic polynomials, their stability is analyzed by means of their basins of attraction.

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.

An Accelerated Iterative Method with Third-Order Convergence

2012 ◽  
Vol 220-223 ◽  
pp. 2658-2661
Author(s):  
Zhong Yong Hu ◽  
Liang Fang ◽  
Lian Zhong Li
Keyword(s):  
Iterative Method ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
New Method ◽  
Order Convergence ◽  
Third Order ◽  
Numerical Examples ◽  
Modified Newton’S Method ◽  
Jarratt Method

We present a new modified Newton's method with third-order convergence and compare it with the Jarratt method, which is of fourth-order. Based on this new method, we obtain a family of Newton-type methods, which converge cubically. Numerical examples show that the presented method can compete with Newton's method and other known third-order modifications of Newton's method.

A Class of Higher-Order Newton-Like Methods for Systems of Nonlinear Equations

2021 ◽  
pp. 2150059
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Ioannis K. Argyros
Keyword(s):  
Nonlinear Equations ◽  
Inverse Operator ◽  
General Setting ◽  
Basins Of Attraction ◽  
Function Evaluation ◽  
Systems Of Nonlinear Equations ◽  
Third Order ◽  
Additional Function ◽  
Numerical Experimentation ◽  
The Cost

In this paper, a class of efficient iterative methods with increasing order of convergence for solving systems of nonlinear equations is developed and analyzed. The methodology uses well-known third-order Potra–Pták iteration in the first step and Newton-like iterations in the subsequent steps. Novelty of the methods is the increase in convergence order by an amount three per step at the cost of only one additional function evaluation. In addition, the algorithm uses a single inverse operator in each iteration, which makes it computationally more efficient and attractive. Local convergence is studied in the more general setting of a Banach space under suitable assumptions. Theoretical results of convergence and computational efficiency are verified through numerical experimentation. Comparison of numerical results indicates that the developed algorithms outperform the other similar algorithms available in the literature, particularly when applied to solve the large systems of equations. The basins of attraction of some of the existing methods along with the proposed method are given to exhibit their performance.

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