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 A Two-Parameter Family of Fourth-Order Iterative Methods with Optimal Convergence for Multiple Zeros

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Young Ik Kim ◽  
Young Hee Geum
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Harmonic Mean ◽  
Optimal Order ◽  
Multiple Roots ◽  
Multiple Zeros ◽  
Asymptotic Error ◽  
Two Parameter

We develop a family of fourth-order iterative methods using the weighted harmonic mean of two derivative functions to compute approximate multiple roots of nonlinear equations. They are proved to be optimally convergent in the sense of Kung-Traub’s optimal order. Numerical experiments for various test equations confirm well the validity of convergence and asymptotic error constants for the developed methods.

A General Way to Construct a New Optimal Scheme with Eighth-Order Convergence for Nonlinear Equations

2019 ◽  
Vol 17 (01) ◽  
pp. 1843011
Author(s):  
Ramandeep Behl ◽  
Changbum Chun ◽  
Ali Saleh Alshormani ◽  
S. S. Motsa
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Functional Approach ◽  
Eighth Order ◽  
Optimal Scheme ◽  
Asymptotic Error ◽  
Error Constant ◽  
Order Scheme ◽  
Asymptotic Error Constant ◽  
Computational Properties

In this paper, we present a new and interesting optimal scheme of order eight in a general way for solving nonlinear equations, numerically. The beauty of our scheme is that it is capable of producing further new and interesting optimal schemes of order eight from every existing optimal fourth-order scheme whose first substep employs Newton’s method. The construction of this scheme is based on rational functional approach. The theoretical and computational properties of the proposed scheme are fully investigated along with a main theorem which establishes the order of convergence and asymptotic error constant. Several numerical examples are given and analyzed in detail to demonstrate faster convergence and higher computational efficiency of our methods.

scholarly journals Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1242
Author(s):  
Ramandeep Behl ◽  
Sonia Bhalla ◽  
Eulalia Martínez ◽  
Majed Aali Alsulami
Keyword(s):  
Iterative Methods ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Real Life ◽  
Multiple Roots ◽  
Computational Results ◽  
Nonlinear Functions ◽  
First Order ◽  
Life Problems ◽  
Derivative Free

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King’s family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots (m≥2). In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung–Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing methods.

scholarly journals Three New Optimal Fourth-Order Iterative Methods to Solve Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Gustavo Fernández-Torres ◽  
Juan Vásquez-Aquino
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
Classical Method ◽  
Multiple Roots ◽  
Order Convergence ◽  
Numerical Examples ◽  
Optimal Methods

We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published.

scholarly journals Novel Computational Iterative Methods with Optimal Order for Nonlinear Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
F. Soleymani
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
General Class ◽  
Analytical Study ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Numerical Examples ◽  
First Order

This paper contributes a very general class of two-point iterative methods without memory for solving nonlinear equations. The class of methods is developed using weight function approach. Per iteration, each method of the class includes two evaluations of the function and one of its first-order derivative. The analytical study of the main theorem is presented in detail to show the fourth order of convergence. Furthermore, it is discussed that many of the existing fourth-order methods without memory are members from this developed class. Finally, numerical examples are taken into account to manifest the accuracy of the derived 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 A new family of fourth-order methods for multiple roots of nonlinear equations

2013 ◽  
Vol 18 (2) ◽  
pp. 143-152 ◽  
Author(s):  
Baoqing Liu ◽  
Xiaojian Zhou
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
Optimal Order ◽  
Multiple Roots ◽  
First Derivative ◽  
New Family ◽  
Modified Newton’S Method

Recently, some optimal fourth-order iterative methods for multiple roots of nonlinear equations are presented when the multiplicity m of the root is known. Different from these optimal iterative methods known already, this paper presents a new family of iterative methods using the modified Newton’s method as its first step. The new family, requiring one evaluation of the function and two evaluations of its first derivative, is of optimal order. Numerical examples are given to suggest that the new family can be competitive with other fourth-order methods and the modified Newton’s method for multiple roots.

scholarly journals An Optimal Derivative Free Family of Chebyshev–Halley’s Method for Multiple Zeros

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 546
Author(s):  
Ramandeep Behl ◽  
Sonia Bhalla ◽  
Ángel Alberto Magreñán ◽  
Alejandro Moysi
Keyword(s):  
Nonlinear Equation ◽  
Weight Function ◽  
Convergence Analysis ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Higher Order ◽  
Multiple Roots ◽  
Iterative Technique ◽  
Derivative Free

In this manuscript, we introduce the higher-order optimal derivative-free family of Chebyshev–Halley’s iterative technique to solve the nonlinear equation having the multiple roots. The designed scheme makes use of the weight function and one parameter α to achieve the fourth-order of convergence. Initially, the convergence analysis is performed for particular values of multiple roots. Afterward, it concludes in general. Moreover, the effectiveness of the presented methods are certified on some applications of nonlinear equations and compared with the earlier derivative and derivative-free schemes. The obtained results depict better performance than the existing methods.

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