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 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 Stability Anomalies of Some Jacobian-Free Iterative Methods of High Order of Convergence

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 51
Author(s):  
Alicia Cordero ◽  
Javier G. Maimó ◽  
Juan R. Torregrosa ◽  
María P. Vassileva
Keyword(s):  
Difference Operator ◽  
Divided Difference ◽  
Polynomial System ◽  
Nonlinear Problems ◽  
Coefficient Matrix ◽  
Test Problems ◽  
Stability Performance ◽  
Need To Evaluate ◽  
The Stability ◽  
Theoretical Results

In this manuscript, we design two classes of parametric iterative schemes to solve nonlinear problems that do not need to evaluate Jacobian matrices and need to solve three linear systems per iteration with the same divided difference operator as the coefficient matrix. The stability performance of the classes is analyzed on a quadratic polynomial system, and it is shown that for many values of the parameter, only convergence to the roots of the problem exists. Finally, we check the performance of these methods on some test problems to confirm the theoretical results.

scholarly journals An Optimal Eighth-Order Family of Iterative Methods for Multiple Roots

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 672 ◽  
Author(s):  
Saima Akram ◽  
Fiza Zafar ◽  
Nusrat Yasmin
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Multiple Root ◽  
Multiple Roots ◽  
Order Convergence ◽  
Eighth Order ◽  
Root Finding ◽  
New Family ◽  
Two Parameters ◽  
Theoretical Results

In this paper, we introduce a new family of efficient and optimal iterative methods for finding multiple roots of nonlinear equations with known multiplicity ( m ≥ 1 ) . We use the weight function approach involving one and two parameters to develop the new family. A comprehensive convergence analysis is studied to demonstrate the optimal eighth-order convergence of the suggested scheme. Finally, numerical and dynamical tests are presented, which validates the theoretical results formulated in this paper and illustrates that the suggested family is efficient among the domain of multiple root finding methods.

scholarly journals On a Bi-Parametric Family of Fourth Order Composite Newton–Jarratt Methods for Nonlinear Systems

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 492 ◽  
Author(s):  
Janak Raj Sharma ◽  
Deepak Kumar ◽  
Ioannis K. Argyros ◽  
Ángel Alberto Magreñán
Keyword(s):  
Nonlinear Systems ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Systems Of Nonlinear Equations ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Numerical Experimentation ◽  
Theoretical Results ◽  
Two Parameter

We present a new two-parameter family of fourth-order iterative methods for solving systems of nonlinear equations. The scheme is composed of two Newton–Jarratt steps and requires the evaluation of one function and two first derivatives in each iteration. Convergence including the order of convergence, the radius of convergence, and error bounds is presented. Theoretical results are verified through numerical experimentation. Stability of the proposed class is analyzed and presented by means of using new dynamics tool, namely, the convergence plane. Performance is exhibited by implementing the methods on nonlinear systems of equations, including those resulting from the discretization of the boundary value problem. In addition, numerical comparisons are made with the existing techniques of the same order. Results show the better performance of the proposed techniques than the existing ones.

scholarly journals Semilocal Convergence of the Extension of Chun’s Method

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 161
Author(s):  
Alicia Cordero ◽  
Javier G. Maimó ◽  
Eulalia Martínez ◽  
Juan R. Torregrosa ◽  
María P. Vassileva
Keyword(s):  
Iterative Method ◽  
Existence And Uniqueness ◽  
Recurrence Relations ◽  
Nonlinear Integral Equation ◽  
Difference Operator ◽  
Divided Difference ◽  
Semilocal Convergence ◽  
Starting Point ◽  
Domain Of Existence ◽  
Theoretical Results

In this work, we use the technique of recurrence relations to prove the semilocal convergence in Banach spaces of the multidimensional extension of Chun’s iterative method. This is an iterative method of fourth order, that can be transferred to the multivariable case by using the divided difference operator. We obtain the domain of existence and uniqueness by taking a suitable starting point and imposing a Lipschitz condition to the first Fréchet derivative in the whole domain. Moreover, we apply the theoretical results obtained to a nonlinear integral equation of Hammerstein type, showing the applicability of our results.

scholarly journals Generalized High-Order Classes for Solving Nonlinear Systems and Their Applications

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1194 ◽  
Author(s):  
Francisco I. Chicharro ◽  
Alicia Cordero ◽  
Neus Garrido ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Systems ◽  
Iterative Method ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Divided Difference ◽  
High Order ◽  
Weight Functions ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
New Methods

A generalized high-order class for approximating the solution of nonlinear systems of equations is introduced. First, from a fourth-order iterative family for solving nonlinear equations, we propose an extension to nonlinear systems of equations holding the same order of convergence but replacing the Jacobian by a divided difference in the weight functions for systems. The proposed GH family of methods is designed from this fourth-order family using both the composition and the weight functions technique. The resulting family has order of convergence 9. The performance of a particular iterative method of both families is analyzed for solving different test systems and also for the Fisher’s problem, showing the good performance of the new methods.

scholarly journals A New Class of Halley’s Method with Third-Order Convergence for Solving Nonlinear Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Mohammed Barrada ◽  
Mariya Ouaissa ◽  
Yassine Rhazali ◽  
Mariyam Ouaissa
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Order Convergence ◽  
Eighth Order ◽  
Third Order ◽  
Numerical Examples ◽  
New Class ◽  
New Methods ◽  
New Family ◽  
Number Of Iterations

In this paper, we present a new family of methods for finding simple roots of nonlinear equations. The convergence analysis shows that the order of convergence of all these methods is three. The originality of this family lies in the fact that these sequences are defined by an explicit expression which depends on a parameter p where p is a nonnegative integer. A first study on the global convergence of these methods is performed. The power of this family is illustrated analytically by justifying that, under certain conditions, the method convergence’s speed increases with the parameter p. This family’s efficiency is tested on a number of numerical examples. It is observed that our new methods take less number of iterations than many other third-order methods. In comparison with the methods of the sixth and eighth order, the new ones behave similarly in the examples considered.

scholarly journals A New High-Order Jacobian-Free Iterative Method with Memory for Solving Nonlinear Systems

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2122
Author(s):  
Ramandeep Behl ◽  
Alicia Cordero ◽  
Juan R. Torregrosa ◽  
Sonia Bhalla
Keyword(s):  
Nonlinear Systems ◽  
Iterative Method ◽  
Order Of Convergence ◽  
Recent Literature ◽  
Basin Of Attraction ◽  
Qualitative Behavior ◽  
Order Convergence ◽  
Numerical Tests ◽  
Initial Scheme ◽  
With Memory

We used a Kurchatov-type accelerator to construct an iterative method with memory for solving nonlinear systems, with sixth-order convergence. It was developed from an initial scheme without memory, with order of convergence four. There exist few multidimensional schemes using more than one previous iterate in the very recent literature, mostly with low orders of convergence. The proposed scheme showed its efficiency and robustness in several numerical tests, where it was also compared with the existing procedures with high orders of convergence. These numerical tests included large nonlinear systems. In addition, we show that the proposed scheme has very stable qualitative behavior, by means of the analysis of an associated multidimensional, real rational function and also by means of a comparison of its basin of attraction with those of comparison methods.

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.

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