scholarly journals New Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction

2022 ◽  
Vol 21 ◽  
pp. 9-16
Author(s):  
O. Ababneh
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Newton's Method ◽  
Order Of Convergence ◽  
Iterative Algorithms ◽  
Basins Of Attraction ◽  
Second Derivative ◽  
Numerical Examples ◽  
Convergence Results ◽  
Modified Newton’S Method

The purpose of this paper is to propose new modified Newton’s method for solving nonlinear equations and free from second derivative. Convergence results show that the order of convergence is four. Several numerical examples are given to illustrate that the new iterative algorithms are effective.In the end, we present the basins of attraction to observe the fractal behavior and dynamical aspects of the proposed algorithms.

scholarly journals Numerical Algorithms for Finding Zeros of Nonlinear Equations and Their Dynamical Aspects

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Amir Naseem ◽  
M. A. Rehman ◽  
Thabet Abdeljawad ◽  
Francisco Balibrea
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Numerical Algorithms ◽  
One Dimension ◽  
Second Derivative ◽  
Numerical Examples ◽  
Derivative Free ◽  
Interpolation Technique ◽  
Taylor’S Series

In this paper, we developed two new numerical algorithms for finding zeros of nonlinear equations in one dimension and one of them is second derivative free which has been removed using the interpolation technique. We derive these algorithms with the help of Taylor’s series expansion and Golbabai and Javidi’s method. The convergence analysis of these algorithms is discussed. It is established that the newly developed algorithms have sixth order of convergence. Several numerical examples have been solved which prove the better efficiency of these algorithms as compared to other well-known iterative methods of the same kind. Finally, the comparison of polynomiographs generated by other well-known iterative methods with our developed algorithms has been made which reflects their dynamical aspects.

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.

scholarly journals A Family of Fourteenth-Order Convergent Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fiza Zafar ◽  
Gulshan Bibi
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Numerical Examples ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
Specific Step

We present a family of fourteenth-order convergent iterative methods for solving nonlinear equations involving a specific step which when combined with any two-step iterative method raises the convergence order by n+10, if n is the order of convergence of the two-step iterative method. This new class include four evaluations of function and one evaluation of the first derivative per iteration. Therefore, the efficiency index of this family is 141/5 =1.695218203. Several numerical examples are given to show that the new methods of this family are comparable with the existing methods.

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

scholarly journals New Modified Newton Type Iterative Methods

2021 ◽  
Vol 2 (1) ◽  
pp. 17-24
Author(s):  
Jivandhar Jnawali
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Single Variable ◽  
Numerical Examples

In this work, we present two Newton type iterative methods for finding the solution of nonlinear equations of single variable. One is obtained as variant of McDougall and Wotherspoon method, and another is obtained by amalgamation of Potra and Pta’k method and our newly introduced method. The order of convergence of these methods are 1 + √2 and 3.5615. Some numerical examples are given to compare the performance of these methods with some similar existing methods.

scholarly journals Novel Computational Iterative Methods with Optimal Order for Nonlinear Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
F. Soleymani
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
General Class ◽  
Analytical Study ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Numerical Examples ◽  
First Order

This paper contributes a very general class of two-point iterative methods without memory for solving nonlinear equations. The class of methods is developed using weight function approach. Per iteration, each method of the class includes two evaluations of the function and one of its first-order derivative. The analytical study of the main theorem is presented in detail to show the fourth order of convergence. Furthermore, it is discussed that many of the existing fourth-order methods without memory are members from this developed class. Finally, numerical examples are taken into account to manifest the accuracy of the derived methods.

scholarly journals Modification of Newton-Househölder Method for Determining Multiple Roots of Unknown Multiplicity of Nonlinear Equations

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1020
Author(s):  
Syahmi Afandi Sariman ◽  
Ishak Hashim ◽  
Faieza Samat ◽  
Mohammed Alshbool
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Accuracy ◽  
Basins Of Attraction ◽  
Convergence Order ◽  
Multiple Roots ◽  
Eighth Order ◽  
Numerical Examples

In this study, we propose an extension of the modified Newton-Househölder methods to find multiple roots with unknown multiplicity of nonlinear equations. With four functional evaluations per iteration, the proposed method achieves an optimal eighth order of convergence. The higher the convergence order, the quicker we get to the root with a high accuracy. The numerical examples have shown that this scheme can compete with the existing methods. This scheme is also stable across all of the functions tested based on the graphical basins of attraction.

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 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 Using symbolic computation in Mathematica for verifying the convergence of the iterative methods

2013 ◽  
Vol 22 (1) ◽  
pp. 9-13
Author(s):  
GHEORGHE ARDELEAN ◽  
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Symbolic Computation ◽  
Newton’S Method ◽  
Newton's Method ◽  
New Method ◽  
Correct Result ◽  
Modified Newton’S Method ◽  
Second Derivatives ◽  
Appl Math

In [Jisheng Kou, The improvements of modified Newton’s method, Appl. Math. Comput., 189 (2007) 602–609], the improvements of some thirdorder modifications of Newton’s method for solving nonlinear equations are presented. In this paper we point out some flaws in the results of Jisheng Kou and we correct them by using symbolic computation in Mathematica. In [M. A. Noor et al., A new modified Halley method without second derivatives for nonlinear equations, Appl. Math. Comput., 189 (2007) 1268–1273] , the error equation obtained for the new method presented is wrong. We present the correct result by using symbolic computation, too. Finally, we present two examples of very simply proofs for the convergence of iterative methods by using symbolic computation. We consider that the Mathematica programs in this paper are good examples for other authors to prove the convergence of the iterative methods or to verify their results.

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