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.

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 Higher-Order Root-Finding Algorithms and Their Basins of Attraction

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Amir Naseem ◽  
M. A. Rehman ◽  
Thabet Abdeljawad
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Complex Polynomials ◽  
Convergence Criteria ◽  
Root Finding ◽  
Variational Iteration ◽  
Order Root ◽  
Iteration Technique

In this paper, we proposed and analyzed three new root-finding algorithms for solving nonlinear equations in one variable. We derive these algorithms with the help of variational iteration technique. We discuss the convergence criteria of these newly developed algorithms. The dominance of the proposed algorithms is illustrated by solving several test examples and comparing them with other well-known existing iterative methods in the literature. In the end, we present the basins of attraction using some complex polynomials of different degrees to observe the fractal behavior and dynamical aspects of the proposed algorithms.

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 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.

scholarly journals Several New Third-Order and Fourth-Order Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Anuradha Singh ◽  
J. P. Jaiswal
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Numerical Results ◽  
Function Concept ◽  
Third Order ◽  
Multivariate Case

In order to find the zeros of nonlinear equations, in this paper, we propose a family of third-order and optimal fourth-order iterative methods. We have also obtained some particular cases of these methods. These methods are constructed through weight function concept. The multivariate case of these methods has also been discussed. The numerical results show that the proposed methods are more efficient than some existing third- and fourth-order methods.

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