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.

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 A New Class of Iterative Processes for Solving Nonlinear Systems by Using One Divided Differences Operator

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 776 ◽  
Author(s):  
Alicia Cordero ◽  
Cristina Jordán ◽  
Esther Sanabria ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Systems ◽  
Boundary Problem ◽  
Fourth Order ◽  
Difference Operator ◽  
Divided Difference ◽  
Order Convergence ◽  
New Class ◽  
Numerical Tests ◽  
New Family ◽  
Theoretical Results

In this manuscript, a new family of Jacobian-free iterative methods for solving nonlinear systems is presented. The fourth-order convergence for all the elements of the class is established, proving, in addition, that one element of this family has order five. The proposed methods have four steps and, in all of them, the same divided difference operator appears. Numerical problems, including systems of academic interest and the system resulting from the discretization of the boundary problem described by Fisher’s equation, are shown to compare the performance of the proposed schemes with other known ones. The numerical tests are in concordance with the theoretical results.

scholarly journals A New Jarratt-Type Fourth-Order Method for Solving System of Nonlinear Equations and Applications

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Moin-ud-Din Junjua ◽  
Saima Akram ◽  
Nusrat Yasmin ◽  
Fiza Zafar
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Basins Of Attraction ◽  
Systems Of Nonlinear Equations ◽  
System Of Nonlinear Equations ◽  
Numerical Tests ◽  
New Methods ◽  
New Family ◽  
Engineering Problems ◽  
Fourth Order Method

Solving systems of nonlinear equations plays a major role in engineering problems. We present a new family of optimal fourth-order Jarratt-type methods for solving nonlinear equations and extend these methods to solve system of nonlinear equations. Convergence analysis is given for both cases to show that the order of the new methods is four. Cost of computations, numerical tests, and basins of attraction are presented which illustrate the new methods as better alternates to previous methods. We also give an application of the proposed methods to well-known Burger's equation.

scholarly journals Optimal High-Order Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
S. Artidiello ◽  
A. Cordero ◽  
Juan R. Torregrosa ◽  
M. P. Vassileva
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Order ◽  
Function Technique ◽  
Numerical Tests ◽  
New Methods ◽  
Theoretical Results ◽  
Made In

A class of optimal iterative methods for solving nonlinear equations is extended up to sixteenth-order of convergence. We design them by using the weight function technique, with functions of three variables. Some numerical tests are made in order to confirm the theoretical results and to compare the new methods with other known ones.

scholarly journals New Phase-Fitted and Amplification-Fitted Fourth-Order and Fifth-Order Runge-Kutta-Nyström Methods for Oscillatory Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
K. W. Moo ◽  
N. Senu ◽  
F. Ismail ◽  
M. Suleiman
Keyword(s):  
Fourth Order ◽  
Variable Coefficients ◽  
Phase Lag ◽  
Oscillatory Solutions ◽  
Numerical Tests ◽  
New Methods ◽  
Nyström Methods ◽  
Runge Kutta ◽  
Fifth Order ◽  
New Phase

Two new Runge-Kutta-Nyström (RKN) methods are constructed for solving second-order differential equations with oscillatory solutions. These two new methods are constructed based on two existing RKN methods. Firstly, a three-stage fourth-order Garcia’s RKN method. Another method is Hairer’s RKN method of four-stage fifth-order. Both new derived methods have two variable coefficients with phase-lag of order infinity and zero amplification error (zero dissipative). Numerical tests are performed and the results show that the new methods are more accurate than the other methods in the literature.

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 FAMILY OF EIGHTH-ORDER ITERATIVE METHODS WITH AN INVERSE INTERPOLATORY RATIONAL FUNCTION ERROR CORRECTOR FOR NONLINEAR EQUATIONS

2017 ◽  
Vol 22 (3) ◽  
pp. 321-336 ◽  
Author(s):  
Young I. Kim ◽  
Ramandeep Behl ◽  
Sandile S. Motsa
Keyword(s):  
Rational Function ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Iterative Scheme ◽  
Basins Of Attraction ◽  
Theoretical Development ◽  
Eighth Order ◽  
Main Motivation ◽  
Iterative Schemes ◽  
Computational Properties

The main motivation of this study is to propose an optimal scheme with an inverse interpolatory rational function error corrector in a general way that can be applied to any existing optimal multi-point fourth-order iterative scheme whose first sub step employs Newton’s method to further produce optimal eighth-order iterative schemes. In addition, we also discussed the theoretical and computational properties of our scheme. Variety of concrete numerical experiments and basins of attraction are extensively treated to confirm the theoretical development.

scholarly journals On a Class of Optimal Fourth Order Multiple Root Solvers without Using Derivatives

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1452 ◽  
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Lorentz Jäntschi
Keyword(s):  
Fourth Order ◽  
Multiple Root ◽  
Optimal Order ◽  
Test Problems ◽  
Multiple Roots ◽  
Motivational Factor ◽  
New Methods ◽  
Derivative Free ◽  
The Family ◽  
Theoretical Results

Many optimal order multiple root techniques involving derivatives have been proposed in literature. On the contrary, optimal order multiple root techniques without derivatives are almost nonexistent. With this as a motivational factor, here we develop a family of optimal fourth-order derivative-free iterative schemes for computing multiple roots. The procedure is based on two steps of which the first is Traub–Steffensen iteration and second is Traub–Steffensen-like iteration. Theoretical results proved for particular cases of the family are symmetric to each other. This feature leads us to prove the general result that shows the fourth-order convergence. Efficacy is demonstrated on different test problems that verifies the efficient convergent nature of the new methods. Moreover, the comparison of performance has proven the presented derivative-free techniques as good competitors to the existing optimal fourth-order methods that use derivatives.

Local Convergence of a Family of Iterative Methods with Sixth and Seventh Order Convergence Under Weak Conditions

2019 ◽  
Vol 16 (08) ◽  
pp. 1850120 ◽  
Author(s):  
Tianbao liu ◽  
Xiwen Qin ◽  
Peng Wang
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Basins Of Attraction ◽  
Order Convergence ◽  
The Family ◽  
Lipschitz Constants ◽  
Seventh Order ◽  
Parameter Values ◽  
Local Convergence Theorem

In this paper, we study a local convergence analysis of a family of iterative methods with sixth and seventh order convergence for nonlinear equations, which was established by [Cordero et al. [2010] in “A family of iterative methods with sixth and seventh order convergence for nonlinear equations,” Math. Comput. Model. 52, 1190–1496]. Earlier studies have shown convergence using Taylor expansions and hypotheses reaching up to the sixth derivative. In our work, we make an attempt to study and establish a local convergence theorem by using only hypotheses the first derivative of the function and Lipschitz constants. We can also obtain error bounds and radii of convergence based on our results. Hence, the applicability of the methods is expanded. Moreover, we consider some different numerical examples and obtain the radii of convergence centered at the solution for different parameter values [Formula: see text] of the family. Furthermore, the basins of attraction of the family with different parameter values are also studied, which allow us to distinguish between the good and bad members of the family in terms of convergence and stable properties, and help us find the members with better or the best stable behavior.

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.

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