scholarly journals A New Family of Iterative Methods Based on an Exponential Model for Solving Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Tianbao Liu ◽  
Hengyan Li ◽  
Zaixiang Pang
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Exponential Model ◽  
New Family

scholarly journals Iteration Methods with an Auxiliary Function for Nonlinear Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Faisal Ali ◽  
Waqas Aslam ◽  
Imran Khalid ◽  
Akbar Nadeem
Keyword(s):  
Nonlinear Equation ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Auxiliary Function ◽  
New Family ◽  
Special Cases ◽  
Engineering Models ◽  
Iteration Methods ◽  
Scientific Engineering ◽  
Taylor’S Series

Various iterative methods have been introduced by involving Taylor’s series on the auxiliary function g x to solve the nonlinear equation f x = 0 . In this paper, we introduce the expansion of g x with the inclusion of weights w i such that ∑ i = 1 p w i = 1 and knots τ i ∈ 0,1 in order to develop a new family of iterative methods. The methods proposed in the present paper are applicable for different choices of auxiliary function g x , and some already known methods can be viewed as the special cases of these methods. We consider the diverse scientific/engineering models to demonstrate the efficiency of the proposed methods.

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 Decomposition Technique and a Family of Efficient Schemes for Nonlinear Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Farooq Ahmed Shah ◽  
Maslina Darus ◽  
Imran Faisal ◽  
Muhammad Arsalan Shafiq
Keyword(s):  
Applied Sciences ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Decomposition Technique ◽  
Unified Framework ◽  
New Methods ◽  
New Family ◽  
And Performance

Various problems of pure and applied sciences can be studied in the unified framework of nonlinear equations. In this paper, a new family of iterative methods for solving nonlinear equations is developed by using a new decomposition technique. The convergence of the new methods is proven. For the implementation and performance of the new methods, some examples are solved and the results are compared with some existing methods.

scholarly journals A New Optimal Eighth-Order Ostrowski-Type Family of Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Taher Lotfi ◽  
Tahereh Eftekhari
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Efficiency Index ◽  
Eighth Order ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
New Family ◽  
Ostrowski’S Method

Based on Ostrowski's method, a new family of eighth-order iterative methods for solving nonlinear equations by using weight function methods is presented. Per iteration the new methods require three evaluations of the function and one evaluation of its first derivative. Therefore, this family of methods has the efficiency index which equals 1.682. Kung and Traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2n−1. Thus, we provide a new class which agrees with the conjecture of Kung-Traub for n=4. Numerical comparisons are made to show the performance of the presented methods.

scholarly journals An Efficient Family of Optimal Eighth-Order Iterative Methods for Solving Nonlinear Equations and Its Dynamics

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Anuradha Singh ◽  
J. P. Jaiswal
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Basins Of Attraction ◽  
Order Method ◽  
Eighth Order ◽  
New Family ◽  
Seventh Order ◽  
Prime Objective ◽  
And Performance

The prime objective of this paper is to design a new family of optimal eighth-order iterative methods by accelerating the order of convergence of the existing seventh-order method without using more evaluations for finding simple root of nonlinear equations. Numerical comparisons have been carried out to demonstrate the efficiency and performance of the proposed method. Finally, we have compared new method with some existing eighth-order methods by basins of attraction and observed that the proposed scheme is more efficient.

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