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 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 Optimal Methods for Finding Simple and Multiple Roots of Nonlinear Equations and their Basins of Attraction

2020 ◽  
Vol 37 (1-2) ◽  
pp. 14-29
Author(s):  
Prem Bahadur Chand
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Multiple Root ◽  
Basins Of Attraction ◽  
Weight Functions ◽  
Multiple Roots ◽  
Order Method ◽  
Numerical Examples ◽  
Fourth Order Method ◽  
Simple And Multiple Roots

In this paper, using the variant of Frontini-Sormani method, some higher order methods for finding the roots (simple and multiple) of nonlinear equations are proposed. In particular, we have constructed an optimal fourth order method and a family of sixth order method for finding a simple root. Further, an optimal fourth order method for finding a multiple root of a nonlinear equation is also proposed. We have used different weight functions to a cubically convergent For ntini-Sormani method for the construction of these methods. The proposed methods are tested on numerical examples and compare the results with some existing methods. Further, we have presented the basins of attraction of these methods to understand their dynamics visually.

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 Design and Complex Dynamics of Potra–Pták-Type Optimal Methods for Solving Nonlinear Equations and Its Applications

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 942 ◽  
Author(s):  
Prem B. Chand ◽  
Francisco I. Chicharro ◽  
Neus Garrido ◽  
Pankaj Jain
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Complex Dynamics ◽  
Basins Of Attraction ◽  
Weight Functions ◽  
Order Method ◽  
Eighth Order ◽  
Numerical Examples ◽  
Fourth Order Method ◽  
Cubic Polynomials

In this paper, using the idea of weight functions on the Potra–Pták method, an optimal fourth order method, a non optimal sixth order method, and a family of optimal eighth order methods are proposed. These methods are tested on some numerical examples, and the results are compared with some known methods of the corresponding order. It is proved that the results obtained from the proposed methods are compatible with other methods. The proposed methods are tested on some problems related to engineering and science. Furthermore, applying these methods on quadratic and cubic polynomials, their stability is analyzed by means of their basins of attraction.

scholarly journals Extended local convergence for Newton-type solver under weak conditions

2021 ◽  
Vol 66 (4) ◽  
pp. 757-768
Author(s):  
Ioannis K. Argyros ◽  
◽  
Santhosh George ◽  
Kedarnath Senapati ◽  
◽  
...  
Keyword(s):  
Local Convergence ◽  
Order Of Convergence ◽  
Basins Of Attraction ◽  
Dimensional Euclidean Space ◽  
Test Problems ◽  
Eighth Order ◽  
Convergence Conditions ◽  
Numerical Examples ◽  
Special Case ◽  
Methods Numerical

We present the local convergence of a Newton-type solver for equations involving Banach space valued operators. The eighth order of convergence was shown earlier in the special case of the k-dimensional Euclidean space, using hypotheses up to the eighth derivative although these derivatives do not appear in the method. We show convergence using only the rst derivative. This way we extend the applicability of the methods. Numerical examples are used to show the convergence conditions. Finally, the basins of attraction of the method, on some test problems are presented.

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 General Family of Third Order Methods for Multiple Roots of Nonlinear Equations and Basin Attractors for Various Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Rajni Sharma ◽  
Ashu Bahl
Keyword(s):  
Nonlinear Equations ◽  
Complex Plane ◽  
General Scheme ◽  
Basins Of Attraction ◽  
Multiple Roots ◽  
Order Convergence ◽  
Third Order ◽  
Numerical Examples ◽  
General Family ◽  
Theoretical Results

A general scheme of third order convergence is described for finding multiple roots of nonlinear equations. The proposed scheme requires one evaluation of f, f′, and f′′ each per iteration and contains several known one-point third order methods for finding multiple roots, as particular cases. Numerical examples are included to confirm the theoretical results and demonstrate convergence behavior of the proposed methods. In the end, we provide the basins of attraction for some methods to observe their dynamics in the complex plane.

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 Metode Iterasi Tiga Langkah untuk Menyelesaikan Persamaan NonLinear dengan Menggunakan Matlab

2019 ◽  
Vol 4 (2) ◽  
pp. 34
Author(s):  
Deasy Wahyuni ◽  
Elisawati Elisawati
Keyword(s):  
Numerical Simulations ◽  
Nonlinear Equations ◽  
Newton Method ◽  
Iteration Method ◽  
Newton's Method ◽  
Order Of Convergence ◽  
Higher Order ◽  
Convergence Order ◽  
Matlab Program ◽  
Analytical Results

Newton method is one of the most frequently used methods to find solutions to the roots of nonlinear equations. Along with the development of science, Newton's method has undergone various modifications. One of them is the hasanov method and the newton method variant (vmn), with a higher order of convergence. In this journal focuses on the three-step iteration method in which the order of convergence is higher than the three methods. To find the convergence order of the three-step iteration method requires a program that can support the analytical results of both methods. One of them using the help of the matlab program. Which will then be compared with numerical simulations also using the matlab program.  Keywords : newton method, newton method variant, Hasanov Method and order of convergence

scholarly journals Generating Optimal Eighth Order Methods for Computing Multiple Roots

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1947
Author(s):  
Deepak Kumar ◽  
Sunil Kumar ◽  
Janak Raj Sharma ◽  
Matteo d’Amore
Keyword(s):  
Real Life ◽  
Nonlinear Function ◽  
Basins Of Attraction ◽  
Computational Time ◽  
Multiple Roots ◽  
Time Stability ◽  
Academic Problems ◽  
Eighth Order ◽  
Multiple Zeros ◽  
New Family

There are a few optimal eighth order methods in literature for computing multiple zeros of a nonlinear function. Therefore, in this work our main focus is on developing a new family of optimal eighth order iterative methods for multiple zeros. The applicability of proposed methods is demonstrated on some real life and academic problems that illustrate the efficient convergence behavior. It is shown that the newly developed schemes are able to compete with other methods in terms of numerical error, convergence and computational time. Stability is also demonstrated by means of a pictorial tool, namely, basins of attraction that have the fractal-like shapes along the borders through which basins are symmetric.

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