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.

scholarly journals On Some Iterative Methods with Memory and High Efficiency Index for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Tahereh Eftekhari
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Efficiency ◽  
Efficiency Index ◽  
Order Convergence ◽  
Eighth Order ◽  
Additional Function ◽  
Numerical Comparisons ◽  
With Memory

Based on iterative methods without memory of eighth-order convergence proposed by Thukral (2012), some iterative methods with memory and high efficiency index are presented. We show that the order of convergence is increased without any additional function evaluations. Numerical comparisons are made to show the performance of the presented methods.

scholarly journals Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 239 ◽  
Author(s):  
Ramandeep Behl ◽  
M. Salimi ◽  
M. Ferrara ◽  
S. Sharifi ◽  
Samaher Alharbi
Keyword(s):  
Iterative Methods ◽  
Chemical Engineering ◽  
Flame Temperature ◽  
Real Life ◽  
Chemical Reactor ◽  
Basins Of Attraction ◽  
Eighth Order ◽  
Present Scheme ◽  
New Family ◽  
Derivative Free

In this study, we present a new higher-order scheme without memory for simple zeros which has two major advantages. The first one is that each member of our scheme is derivative free and the second one is that the present scheme is capable of producing many new optimal family of eighth-order methods from every 4-order optimal derivative free scheme (available in the literature) whose first substep employs a Steffensen or a Steffensen-like method. In addition, the theoretical and computational properties of the present scheme are fully investigated along with the main theorem, which demonstrates the convergence order and asymptotic error constant. Moreover, the effectiveness of our scheme is tested on several real-life problems like Van der Waal’s, fractional transformation in a chemical reactor, chemical engineering, adiabatic flame temperature, etc. In comparison with the existing robust techniques, the iterative methods in the new family perform better in the considered test examples. The study of dynamics on the proposed iterative methods also confirms this fact via basins of attraction applied to a number of test functions.

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 Generating Function Approach to the Derivation of Higher-Order Iterative Methods for Solving Nonlinear Equations

2018 ◽  
Vol 173 ◽  
pp. 03024
Author(s):  
Tugal Zhanlav ◽  
Ochbadrakh Chuluunbaatar ◽  
Vandandoo Ulziibayar
Keyword(s):  
Iterative Methods ◽  
Generating Function ◽  
Generating Functions ◽  
Order Of Convergence ◽  
Sufficient Conditions ◽  
Optimal Order ◽  
Eighth Order ◽  
New Family ◽  
Special Cases ◽  
Necessary And Sufficient

In this paper we propose a generating function method for constructing new two and three-point iterations withp(p= 4, 8) order of convergence. This approach allows us to derive a new family of optimal order iterative methods that include well known methods as special cases. Necessary and sufficient conditions forp-th (p= 4, 8) order convergence of the proposed iterations are given in terms of parameters τnand αn. We also propose some generating functions for τnand αn. We develop a unified representation of all optimal eighth-order methods. The order of convergence of the proposed methods is confirmed by numerical experiments.

scholarly journals A New Class of Halley’s Method with Third-Order Convergence for Solving Nonlinear Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Mohammed Barrada ◽  
Mariya Ouaissa ◽  
Yassine Rhazali ◽  
Mariyam Ouaissa
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Order Convergence ◽  
Eighth Order ◽  
Third Order ◽  
Numerical Examples ◽  
New Class ◽  
New Methods ◽  
New Family ◽  
Number Of Iterations

In this paper, we present a new family of methods for finding simple roots of nonlinear equations. The convergence analysis shows that the order of convergence of all these methods is three. The originality of this family lies in the fact that these sequences are defined by an explicit expression which depends on a parameter p where p is a nonnegative integer. A first study on the global convergence of these methods is performed. The power of this family is illustrated analytically by justifying that, under certain conditions, the method convergence’s speed increases with the parameter p. This family’s efficiency is tested on a number of numerical examples. It is observed that our new methods take less number of iterations than many other third-order methods. In comparison with the methods of the sixth and eighth order, the new ones behave similarly in the examples considered.

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.

Sign in / Sign up

Export Citation Format

Share Document