scholarly journals SOME OPTIMAL METHODS WITH EIGHTH-ORDER CONVERGENCE FOR THE SOLUTION OF NONLINEAR EQUATIONS

2016 ◽  
Vol 29 (4) ◽  
pp. 663-676
Author(s):  
Weonbae Kim ◽  
Changbum Chun
Keyword(s):  
Nonlinear Equations ◽  
Order Convergence ◽  
Eighth Order ◽  
Optimal Methods

scholarly journals On Generalization Based on Bi et al. Iterative Methods with Eighth-Order Convergence for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Taher Lotfi ◽  
Alicia Cordero ◽  
Juan R. Torregrosa ◽  
Morteza Amir Abadi ◽  
Maryam Mohammadi Zadeh
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Convergence ◽  
Eighth Order ◽  
Optimal Methods ◽  
Kung And Traub’S Conjecture

The primary goal of this work is to provide a general optimal three-step class of iterative methods based on the schemes designed by Bi et al. (2009). Accordingly, it requires four functional evaluations per iteration with eighth-order convergence. Consequently, it satisfies Kung and Traub’s conjecture relevant to construction optimal methods without memory. Moreover, some concrete methods of this class are shown and implemented numerically, showing their applicability and efficiency.

scholarly journals On Some Iterative Methods with Memory and High Efficiency Index for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Tahereh Eftekhari
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Efficiency ◽  
Efficiency Index ◽  
Order Convergence ◽  
Eighth Order ◽  
Additional Function ◽  
Numerical Comparisons ◽  
With Memory

Based on iterative methods without memory of eighth-order convergence proposed by Thukral (2012), some iterative methods with memory and high efficiency index are presented. We show that the order of convergence is increased without any additional function evaluations. Numerical comparisons are made to show the performance of the presented 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.

An optimal scheme for multiple roots of nonlinear equations with eighth-order convergence

2018 ◽  
Vol 56 (7) ◽  
pp. 2069-2084 ◽  
Author(s):  
Ramandeep Behl ◽  
Ali Saleh Alshomrani ◽  
S. S. Motsa
Keyword(s):  
Nonlinear Equations ◽  
Multiple Roots ◽  
Order Convergence ◽  
Eighth Order ◽  
Optimal Scheme

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 Family of Iterative Methods with Accelerated Eighth-Order Convergence

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Alicia Cordero ◽  
Mojtaba Fardi ◽  
Mehdi Ghasemi ◽  
Juan R. Torregrosa
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Function Method ◽  
Order Convergence ◽  
Eighth Order ◽  
Weight Function Method ◽  
Theoretical Results

We propose a family of eighth-order iterative methods without memory for solving nonlinear equations. The new iterative methods are developed by using weight function method and using an approximation for the last derivative, which reduces the required number of functional evaluations per step. Their efficiency indices are all found to be 1.682. Several examples allow us to compare our algorithms with known ones and confirm the theoretical results.

scholarly journals New modification of Maheshwari’s method with optimal eighth order convergence for solving nonlinear equations

2016 ◽  
Vol 14 (1) ◽  
pp. 443-451 ◽  
Author(s):  
Somayeh Sharifi ◽  
Massimiliano Ferrara ◽  
Mehdi Salimi ◽  
Stefan Siegmund
Keyword(s):  
Weight Function ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Efficiency Index ◽  
The Other ◽  
Order Convergence ◽  
Eighth Order ◽  
First Derivative ◽  
Attraction Basins

AbstractIn this paper, we present a family of three-point with eight-order convergence methods for finding the simple roots of nonlinear equations by suitable approximations and weight function based on Maheshwari’s method. Per iteration this method requires three evaluations of the function and one evaluation of its first derivative. These class of methods have the efficiency index equal to ${8^{{\textstyle{1 \over 4}}}} \approx 1.682$. We describe the analysis of the proposed methods along with numerical experiments including comparison with the existing methods. Moreover, the attraction basins of the proposed methods are shown with some comparisons to the other existing methods.

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