scholarly journals Some New Variants of Cauchy's Methods for Solving Nonlinear Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Tianbao Liu ◽  
Hengyan Li
Keyword(s):  
Convergence Analysis ◽  
Nonlinear Equations ◽  
Numerical Results ◽  
Convergence Order ◽  
Second Derivative ◽  
Numerical Examples ◽  
First Derivative ◽  
New Methods

We present and analyze some variants of Cauchy's methods free from second derivative for obtaining simple roots of nonlinear equations. The convergence analysis of the methods is discussed. It is established that the methods have convergence order three. Per iteration the new methods require two function and one first derivative evaluations. Numerical examples show that the new methods are comparable with the well-known existing methods and give better numerical results in many aspects.

scholarly journals A Family of Fourteenth-Order Convergent Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fiza Zafar ◽  
Gulshan Bibi
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Numerical Examples ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
Specific Step

We present a family of fourteenth-order convergent iterative methods for solving nonlinear equations involving a specific step which when combined with any two-step iterative method raises the convergence order by n+10, if n is the order of convergence of the two-step iterative method. This new class include four evaluations of function and one evaluation of the first derivative per iteration. Therefore, the efficiency index of this family is 141/5 =1.695218203. Several numerical examples are given to show that the new methods of this family are comparable with the existing methods.

scholarly journals Convergence Analysis of a Three Step Newton-like Method for Nonlinear Equations in Banach Space under Weak Conditions

2016 ◽  
Vol 54 (2) ◽  
pp. 37-46
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Convergence Order ◽  
Order Method ◽  
Numerical Examples ◽  
Local Convergence Analysis

Abstract In the present paper, we study the local convergence analysis of a fifth convergence order method considered by Sharma and Guha in [15] to solve equations in Banach space. Using our idea of restricted convergence domains we extend the applicability of this method. Numerical examples where earlier results cannot apply to solve equations but our results can apply are also given in this study.

scholarly journals Local convergence comparison between two novel sixth order methods for solving equations

2019 ◽  
Vol 18 (1) ◽  
pp. 5-19
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Sixth Order ◽  
Convergence Order ◽  
Convergence Criteria ◽  
Numerical Examples ◽  
First Derivative ◽  
Solving Equations ◽  
Local Convergence Analysis

Abstract The aim of this article is to provide the local convergence analysis of two novel competing sixth convergence order methods for solving equations involving Banach space valued operators. Earlier studies have used hypotheses reaching up to the sixth derivative but only the first derivative appears in these methods. These hypotheses limit the applicability of the methods. That is why we are motivated to present convergence analysis based only on the first derivative. Numerical examples where the convergence criteria are tested are provided. It turns out that in these examples the criteria in the earlier works are not satisfied, so these results cannot be used to solve equations but our results can be used.

scholarly journals Improved Convergence Analysis of Gauss-Newton-Secant Method for Solving Nonlinear Least Squares Problems

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 99 ◽  
Author(s):  
Ioannis Argyros ◽  
Stepan Shakhno ◽  
Yurii Shunkin
Keyword(s):  
Least Squares ◽  
Convergence Analysis ◽  
Difference Method ◽  
Divided Difference ◽  
Nonlinear Least Squares ◽  
Convergence Order ◽  
Least Squares Problems ◽  
Convergence Radius ◽  
Numerical Examples ◽  
Theoretical Results

We study an iterative differential-difference method for solving nonlinear least squares problems, which uses, instead of the Jacobian, the sum of derivative of differentiable parts of operator and divided difference of nondifferentiable parts. Moreover, we introduce a method that uses the derivative of differentiable parts instead of the Jacobian. Results that establish the conditions of convergence, radius and the convergence order of the proposed methods in earlier work are presented. The numerical examples illustrate the theoretical results.

scholarly journals Some New Iterative Methods for Nonlinear Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Eisa Al-Said ◽  
Muhammad Waseem
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
System Of Equations ◽  
Decomposition Technique ◽  
Methods Comparison ◽  
Numerical Examples ◽  
New Methods ◽  
And Performance

We suggest and analyze some new iterative methods for solving the nonlinear equationsf(x)=0using the decomposition technique coupled with the system of equations. We prove that new methods have convergence of fourth order. Several numerical examples are given to illustrate the efficiency and performance of the new methods. Comparison with other similar methods is given.

scholarly journals A family of two-point methods with memory for solving nonlinear equations

2011 ◽  
Vol 5 (2) ◽  
pp. 298-317 ◽  
Author(s):  
Miodrag Petkovic ◽  
Jovana Dzunic ◽  
Ljiljana Petkovic
Keyword(s):  
Nonlinear Equations ◽  
Free Parameter ◽  
Convergence Order ◽  
Optimal Order ◽  
Additional Function ◽  
Numerical Examples ◽  
Derivative Free ◽  
With Memory ◽  
Theoretical Results ◽  
High Computational Efficiency

An efficient family of two-point derivative free methods with memory for solving nonlinear equations is presented. It is proved that the convergence order of the proposed family is increased from 4 to at least 2 + ?6 ? 4.45, 5, 1/2 (5 + ?33) ? 5.37 and 6, depending on the accelerating technique. The increase of convergence order is attained using a suitable accelerating technique by varying a free parameter in each iteration. The improvement of convergence rate is achieved without any additional function evaluations meaning that the proposed methods with memory are very efficient. Moreover, the presented methods are more efficient than all existing methods known in literature in the class of two-point methods and three-point methods of optimal order eight. Numerical examples and the comparison with the existing two-point methods are included to confirm theoretical results and high computational efficiency. 2010 Mathematics Subject Classification. 65H05

scholarly journals Families of optimal multipoint methods for solving nonlinear equations: A survey

2010 ◽  
Vol 4 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Miodrag Petkovic ◽  
Ljiljana Petkovic
Keyword(s):  
Nonlinear Equations ◽  
Computational Efficiency ◽  
Rapid Development ◽  
Convergence Order ◽  
Important Class ◽  
Iterative Root ◽  
Numerical Examples ◽  
Multipoint Methods ◽  
The 1960S ◽  
Order Four

Multipoint iterative root-solvers belong to the class of the most powerful methods for solving nonlinear equations since they overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency. Although the construction of these methods has occurred in the 1960s, their rapid development have started in the first decade of the 21-st century. The most important class of multipoint methods are optimal methods which attain the convergence order 2n using n + 1 function evaluations per iteration. In this paper we give a review of optimal multipoint methods of the order four (n = 2), eight (n = 3) and higher (n > 3), some of which being proposed by the authors. All of them possess as high as possible computational efficiency in the sense of the Kung-Traub hypothesis (1974). Numerical examples are included to demonstrate a very fast convergence of the presented optimal multipoint methods.

scholarly journals Local convergence for a family of sixth order methods with parameters

2021 ◽  
Vol 5 (1) ◽  
pp. 300-305
Author(s):  
Christopher I. Argyros ◽  
◽  
Michael Argyros ◽  
Ioannis K. Argyros ◽  
Santhosh George ◽  
...  
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Real Line ◽  
Local Convergence ◽  
Sixth Order ◽  
Numerical Examples ◽  
The Real ◽  
First Derivative ◽  
Local Convergence Analysis ◽  
Methods Numerical

Local convergence of a family of sixth order methods for solving Banach space valued equations is considered in this article. The local convergence analysis is provided using only the first derivative in contrast to earlier works on the real line using the seventh derivative. This way the applicability is expanded for these methods. Numerical examples complete the article.

scholarly journals New Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction

2022 ◽  
Vol 21 ◽  
pp. 9-16
Author(s):  
O. Ababneh
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Newton's Method ◽  
Order Of Convergence ◽  
Iterative Algorithms ◽  
Basins Of Attraction ◽  
Second Derivative ◽  
Numerical Examples ◽  
Convergence Results ◽  
Modified Newton’S Method

The purpose of this paper is to propose new modified Newton’s method for solving nonlinear equations and free from second derivative. Convergence results show that the order of convergence is four. Several numerical examples are given to illustrate that the new iterative algorithms are effective.In the end, we present the basins of attraction to observe the fractal behavior and dynamical aspects of the proposed algorithms.

scholarly journals Extending the Applicability of the Super-Halley-Like Method Using ω-Continuous Derivatives and Restricted Convergence Domains

2019 ◽  
Vol 33 (1) ◽  
pp. 21-40
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Uniqueness Result ◽  
Second Derivative ◽  
Numerical Examples ◽  
The Third ◽  
Lipschitz Constants ◽  
Local Convergence Analysis ◽  
Third Derivative

AbstractWe present a local convergence analysis of the super-Halley-like method in order to approximate a locally unique solution of an equation in a Banach space setting. The convergence analysis in earlier studies was based on hypotheses reaching up to the third derivative of the operator. In the present study we expand the applicability of the super-Halley-like method by using hypotheses up to the second derivative. We also provide: a computable error on the distances involved and a uniqueness result based on Lipschitz constants. Numerical examples are also presented in this study.

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