scholarly journals Another Simple Way of Deriving Several Iterative Functions to Solve Nonlinear Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Ramandeep Behl ◽  
V. Kanwar ◽  
Kapil K. Sharma
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Third Order ◽  
Chebyshev’S Method ◽  
Straight Line ◽  
Special Cases ◽  
Multipoint Iterative Methods ◽  
Order Four ◽  
Square Root Method ◽  
Iterative Functions

We present another simple way of deriving several iterative methods for solving nonlinear equations numerically. The presented approach of deriving these methods is based on exponentially fitted osculating straight line. These methods are the modifications of Newton's method. Also, we obtain well-known methods as special cases, for example, Halley's method, super-Halley method, Ostrowski's square-root method, Chebyshev's method, and so forth. Further, new classes of third-order multipoint iterative methods free from a second-order derivative are derived by semidiscrete modifications of cubically convergent iterative methods. Furthermore, a simple linear combination of two third-order multipoint iterative methods is used for designing new optimal methods of order four.

scholarly journals On the dynamics of a family of third-order iterative functions

2007 ◽  
Vol 48 (3) ◽  
pp. 343-359 ◽  
Author(s):  
Sergio Amat ◽  
Sonia Busquier ◽  
Sergio Plaza
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Conjugacy Classes ◽  
Complex Polynomials ◽  
Third Order ◽  
Iterative Functions

AbstractWe study the dynamics of a family of third-order iterative methods that are used to find roots of nonlinear equations applied to complex polynomials of degrees three and four. This family includes, as particular cases, the Chebyshev, the Halley and the super-Halleyroot-finding algorithms, as well as the so-called c-methods. The conjugacy classes of theseiterative methods are found explicitly.

On some third-order iterative methods for solving nonlinear equations

2005 ◽  
Vol 171 (1) ◽  
pp. 272-280 ◽  
Author(s):  
Mamta ◽  
V. Kanwar ◽  
V.K. Kukreja ◽  
Sukhjit Singh
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Third Order

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 Dynamical Comparison of Several Third-Order Iterative Methods for Nonlinear Equations

2021 ◽  
Vol 67 (2) ◽  
pp. 1951-1962
Author(s):  
Obadah Said Solaiman ◽  
Samsul Ariffin Abdul Karim ◽  
Ishak Hashim
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Third Order

scholarly journals Some Class of Third- and Fourth-Order Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
J. P. Jaiswal
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Basins Of Attraction ◽  
Order Method ◽  
Third Order ◽  
Numerical Examples ◽  
Multivariate Extension ◽  
New Class ◽  
Fourth Order Method

The object of the present work is to give the new class of third- and fourth-order iterative methods for solving nonlinear equations. Our proposed third-order method includes methods of Weerakoon and Fernando (2000), Homeier (2005), and Chun and Kim (2010) as particular cases. The multivariate extension of some of these methods has been also deliberated. Finally, some numerical examples are given to illustrate the performances of our proposed methods by comparing them with some well existing third- and fourth-order methods. The efficiency of our proposed fourth-order method over some fourth-order methods is also confirmed by basins of attraction.

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