scholarly journals Family of Optimal Fourth Order Methods for Multiple Roots and their Dynamics

2019 ◽  
Vol 2 (2) ◽  
pp. 1-10
Author(s):  
Prem Bahadur Chand ◽  
Kriti Sethi
Keyword(s):  
Fourth Order ◽  
Linear Equations ◽  
Basins Of Attraction ◽  
Dynamical Behaviour ◽  
Arithmetic Mean ◽  
Multiple Roots ◽  
Chebyshev Method ◽  
Fourth Order Method ◽  
Non Linear Equations ◽  
Theoretical Results

In this paper, we propose a family of fourth order method for solving non-linear equations with multiple roots. The method is based on the arithmetic mean of Weerakoon method and Chebyshev method for multiple roots. Some numerical examples are provided in support of the theoretical results. The numerical results obtained by the method for different values of the parameter are compared with some known methods. The dynamical behaviour of methods is discussed and basins of attraction around the multiple roots for some polynomial is shown at the end of the work.

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 optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior

2019 ◽  
Vol 17 (1) ◽  
pp. 1567-1598
Author(s):  
Tianbao Liu ◽  
Xiwen Qin ◽  
Qiuyue Li
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Second Derivative ◽  
Third Order ◽  
Numerical Tests ◽  
The Family ◽  
Theoretical Results ◽  
High Computational Efficiency

Abstract In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.

The Arithmetic of Field Equations

1953 ◽  
Vol 4 (2) ◽  
pp. 205-230 ◽  
Author(s):  
A. Thom
Keyword(s):  
Fourth Order ◽  
Linear Equations ◽  
Field Equations ◽  
Solution Of Equations ◽  
Non Linear ◽  
Propagation Of Errors ◽  
Fourth Order Equations ◽  
Non Linear Equations ◽  
Poisson Type

SummaryThe paper describes in detail an older method than Relaxation of approximating to the solution of equations of the Laplace and Poisson type. The corresponding fourth order equations are discussed briefly, and a method of dealing with certain non-linear equations is indicated. A description is also given of the propagation of errors in the fields due to various causes.

A family of fourth-order methods for solving non-linear equations

2007 ◽  
Vol 188 (1) ◽  
pp. 1031-1036 ◽  
Author(s):  
Jisheng Kou ◽  
Yitian Li ◽  
Xiuhua Wang
Keyword(s):  
Fourth Order ◽  
Linear Equations ◽  
Non Linear ◽  
Non Linear Equations

scholarly journals Design, Convergence and Stability of a Fourth-Order Class of Iterative Methods for Solving Nonlinear Vectorial Problems

2021 ◽  
Vol 5 (3) ◽  
pp. 125
Author(s):  
Alicia Cordero ◽  
Cristina Jordán ◽  
Esther Sanabria-Codesal ◽  
Juan R. Torregrosa
Keyword(s):  
Fourth Order ◽  
Chaotic Behavior ◽  
Basins Of Attraction ◽  
Order Convergence ◽  
Iterative Schemes ◽  
Numerical Tests ◽  
New Methods ◽  
Stable Elements ◽  
Parameter Values ◽  
Theoretical Results

A new parametric family of iterative schemes for solving nonlinear systems is presented. Fourth-order convergence is demonstrated and its stability is analyzed as a function of the parameter values. This study allows us to detect the most stable elements of the class, to find the fractals in the boundary of the basins of attraction and to reject those with chaotic behavior. Some numerical tests show the performance of the new methods, confirm the theoretical results and allow to compare the proposed schemes with other known ones.

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 Comparison of Proposed and Existing Fourth Order Schemes for Solving Non-linear Equations

2019 ◽  
pp. 1-7
Author(s):  
Khushbu Rajput ◽  
Asif Ali Shaikh ◽  
Sania Qureshi
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Numerical Approximation ◽  
Real Root ◽  
Graphical Representation ◽  
Linear Equations ◽  
Order Convergence ◽  
Derivative Free ◽  
Non Linear Equations ◽  
Number Of Iterations

This paper, investigates the comparison of the convergence behavior of the proposed scheme and existing schemes in literature. While all schemes having fourth-order convergence and derivative-free nature. Numerical approximation demonstrates that the proposed schemes are able to attain up to better accuracy than some classical methods, while still significantly reducing the total number of iterations. This study has considered some nonlinear equations (transcendental, algebraic and exponential) along with two complex mathematical models. For better analysis graphical representation of numerical methods for finding the real root of nonlinear equations with varying parameters has been included. The proposed scheme is better in reducing error rapidly, hence converges faster as compared to the existing schemes.

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