scholarly journals A Novel Family of Efficient Weighted-Newton Multiple Root Iterations

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1494
Author(s):  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Lorentz Jăntschi
Keyword(s):  
Fourth Order ◽  
Nonlinear Function ◽  
Multiple Root ◽  
Efficiency Index ◽  
Basins Of Attraction ◽  
Order Method ◽  
Graphical Tool ◽  
Seventh Order ◽  
Fourth Order Method ◽  
Local Convergence Analysis

We propose a novel family of seventh-order iterative methods for computing multiple zeros of a nonlinear function. The algorithm consists of three steps, of which the first two are the steps of recently developed Liu–Zhou fourth-order method, whereas the third step is based on a Newton-like step. The efficiency index of the proposed scheme is 1.627, which is better than the efficiency index 1.587 of the basic Liu–Zhou fourth-order method. In this sense, the proposed iteration is the modification over the Liu–Zhou iteration. Theoretical results are fully studied including the main theorem of local convergence analysis. Moreover, convergence domains are also assessed using the graphical tool, namely, basins of attraction which are symmetrical through the fractal like boundaries. Accuracy and computational efficiency are demonstrated by implementing the algorithms on different numerical problems. Comparison of numerical experiments shows that the new methods have an edge over the existing methods.

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 An Efficient Class of Weighted-Newton Multiple Root Solvers with Seventh Order Convergence

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1054
Author(s):  
Janak Raj Sharma ◽  
Deepak Kumar ◽  
Carlo Cattani
Keyword(s):  
Nonlinear Function ◽  
Multiple Root ◽  
Basins Of Attraction ◽  
Stability And Convergence ◽  
Good Convergence ◽  
Cpu Time ◽  
Numerical Experimentation ◽  
Seventh Order ◽  
Graphical Technique ◽  
Analysis Stability

In this work, we construct a family of seventh order iterative methods for finding multiple roots of a nonlinear function. The scheme consists of three steps, of which the first is Newton’s step and last two are the weighted-Newton steps. Hence, the name of the scheme is ‘weighted-Newton methods’. Theoretical results are studied exhaustively along with the main theorem describing convergence analysis. Stability and convergence domain of the proposed class are also demonstrated by means of using a graphical technique, namely, basins of attraction. Boundaries of these basins are fractal like shapes through which basins are symmetric. Efficacy is demonstrated through numerical experimentation on variety of different functions that illustrates good convergence behavior. Moreover, the theoretical result concerning computational efficiency is verified by computing the elapsed CPU time. The overall comparison of numerical results including accuracy and CPU-time shows that the new methods are strong competitors for the existing methods.

scholarly journals An Optimal Fourth Order Iterative Method for Solving Nonlinear Equations and Its Dynamics

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Rajni Sharma ◽  
Ashu Bahl
Keyword(s):  
Fourth Order ◽  
Dynamical Behavior ◽  
Computational Cost ◽  
Efficiency Index ◽  
Basins Of Attraction ◽  
Optimal Order ◽  
Extraneous Fixed Points ◽  
Fourth Order Method ◽  
Jarratt Method ◽  
Better Than

We present a new fourth order method for finding simple roots of a nonlinear equation f(x)=0. In terms of computational cost, per iteration the method uses one evaluation of the function and two evaluations of its first derivative. Therefore, the method has optimal order with efficiency index 1.587 which is better than efficiency index 1.414 of Newton method and the same with Jarratt method and King’s family. Numerical examples are given to support that the method thus obtained is competitive with other similar robust methods. The conjugacy maps and extraneous fixed points of the presented method and other existing fourth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane.

scholarly journals Local Convergence Analysis of an Efficient Fourth Order Weighted-Newton Method under Weak Conditions

2018 ◽  
Vol 56 (1) ◽  
pp. 23-34
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Convergence Analysis ◽  
Newton Method ◽  
Fourth Order ◽  
Local Convergence ◽  
Systems Of Nonlinear Equations ◽  
Order Method ◽  
First Derivative ◽  
Fourth Order Method ◽  
Order Four ◽  
Local Convergence Analysis

Abstract Local convergence analysis of a fourth order method considered by Sharma et. al in [19] for solving systems of nonlinear equations. Using conditions on derivatives upto the order five, they proved that the method is of order four. In this study using conditions only on the first derivative, we prove the convergence of the method in [19]. This way we extended the applicability of the method. Numerical example which do not satisfy earlier conditions but satisfy our conditions are presented in this study.

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 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.

Dynamic Analysis of Prey-Predator Model with Harvesting

2020 ◽  
Vol 5 (5) ◽  
pp. 22-27
Author(s):  
Puskar R. Pokhrel ◽  
Bhabani Lamsal
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dynamic Analysis ◽  
Fourth Order ◽  
Model Equation ◽  
Order Method ◽  
Eigen Values ◽  
Predator Model ◽  
The Stability ◽  
Fourth Order Method

Employing the Lotka -Voltera (1926) prey-predator model equation, the system is presented with harvesting effort for both species prey and predator. We analyze the stability of the system of ordinary differential equation after calculating the Eigen values of the system. We include the harvesting term for both species in the model equation, and observe the dynamic analysis of prey-predator populations by including the harvesting efforts on the model equation. We also analyze the population dynamic of the system by varying the harvesting efforts on the system. The model equation are solved numerically by applying Runge - Kutta fourth order method.

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