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.

scholarly journals Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

2015 ◽  
Vol 2015 ◽  
pp. 1-19 ◽  
Author(s):  
Krzysztof Gdawiec ◽  
Wiesław Kotarski ◽  
Agnieszka Lisowska
Keyword(s):  
Iteration Process ◽  
Higher Order ◽  
Point Of View ◽  
Complex Polynomials ◽  
Root Finding ◽  
Points Of View ◽  
Iteration Processes

A survey of some modifications based on the classic Newton’s and the higher order Newton-like root finding methods for complex polynomials is presented. Instead of the standard Picard’s iteration several different iteration processes, described in the literature, which we call nonstandard ones, are used. Kalantari’s visualizations of root finding process are interesting from at least three points of view: scientific, educational, and artistic. By combining different kinds of iterations, different convergence tests, and different colouring we obtain a great variety of polynomiographs. We also check experimentally that using complex parameters instead of real ones in multiparameter iterations do not destabilize the iteration process. Moreover, we obtain nice looking polynomiographs that are interesting from the artistic point of view. Real parts of the parameters alter symmetry, whereas imaginary ones cause asymmetric twisting of polynomiographs.

A family of fifth-order convergent methods for solving nonlinear equations using variational iteration technique

2018 ◽  
Vol 39 (3) ◽  
pp. 673-694 ◽  
Author(s):  
Xiujun Zhang ◽  
Farooq Ahmed Shah ◽  
Yingfang Li ◽  
Li Yan ◽  
Abdul Qudair Baig ◽  
...  
Keyword(s):  
Nonlinear Equations ◽  
Variational Iteration ◽  
Fifth Order ◽  
Iteration Technique

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.

A family of higher order multi-point iterative methods based on power mean for solving nonlinear equations

2015 ◽  
Vol 27 (5-6) ◽  
pp. 865-876 ◽  
Author(s):  
Diyashvir Kreetee Rajiv Babajee ◽  
Kalyanasundaram Madhu ◽  
Jayakumar Jayaraman
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Higher Order ◽  
Power Mean

Higher-order iterative methods by using Householder's method for solving certain nonlinear equations

2013 ◽  
Vol 2 (2) ◽  
pp. 107-120
Author(s):  
Waseem Asghar Khan ◽  
Muhammad Aslam Noor ◽  
Adnan Rauf
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Higher Order

Higher order multi-step interval iterative methods for solving nonlinear equations in $$R^n$$ R n

SeMA Journal ◽  
2016 ◽  
Vol 74 (2) ◽  
pp. 133-146
Author(s):  
Sukhjit Singh ◽  
D. K. Gupta ◽  
Falguni Roy
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Higher Order ◽  
Step Interval

scholarly journals Two Higher Order Iterative Methods for Solving Nonlinear Equations

2018 ◽  
Vol 14 (1) ◽  
pp. 179-187
Author(s):  
Jivandhar Jnawali ◽  
Chet Raj Bhatta
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Higher Order ◽  
Secant Method ◽  
Subject Classification ◽  
Numerical Examples ◽  
Ams Subject Classification

 The main purpose of this paper is to derive two higher order iterative methods for solving nonlinear equations as variants of Mir, Ayub and Rafiq method. These methods are free from higher order derivatives. We obtain these methods by amalgamating Mir, Ayub and Rafiq method with standard secant method and modified secant method given by Amat and Busquier. The order of convergence of new variants are four and six. Also, numerical examples are given to compare the performance of newly introduced methods with the similar existing methods. 2010 AMS Subject Classification: 65H05 Journal of the Institute of Engineering, 2018, 14(1): 179-187

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