A Variant of Sharma-Arora's Optimal Eighth-Order Family of Methods for Finding A Simple Root of Nonlinear Equation

2018 ◽  
Author(s):  
Dejan Cebic ◽  
Marija Paunovic ◽  
Nebojsa M. Ralevic
Keyword(s):  
Nonlinear Equation ◽  
Simple Root ◽  
Eighth Order

scholarly journals Polynomial and Rational Approximations and the Link between Schröder’s Processes of the First and Second Kind

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
François Dubeau
Keyword(s):  
Nonlinear Equation ◽  
Simple Root ◽  
Rational Approximations

We show that Schröder’s processes of the first kind and of the second kind to obtain a simple root of a nonlinear equation are related by polynomial and rational approximations.

scholarly journals A New Derivative-Free Method to Solve Nonlinear Equations

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 583
Author(s):  
Beny Neta
Keyword(s):  
Nonlinear Equation ◽  
Nonlinear Equations ◽  
High Order ◽  
Order Derivative ◽  
Eighth Order ◽  
High Order Derivative ◽  
Derivative Free ◽  
Derivative Free Method

A new high-order derivative-free method for the solution of a nonlinear equation is developed. The novelty is the use of Traub’s method as a first step. The order is proven and demonstrated. It is also shown that the method has much fewer divergent points and runs faster than an optimal eighth-order derivative-free method.

scholarly journals New Iterative Method with Application

2021 ◽  
Vol 20 ◽  
pp. 424-430
Author(s):  
O. Ababneh ◽  
N. Zomot
Keyword(s):  
Nonlinear Equation ◽  
Iterative Method ◽  
Iterative Methods ◽  
Simple Root ◽  
Open Interval ◽  
Scalar Function ◽  
New Iterative Method

In this paper, we consider iterative methods to find a simple root of a nonlinear equation f(x) = 0, where f : D∈R→R for an open interval D is a scalar function.

scholarly journals Basin of Attraction from Modification Variant of Chebyshev-Halley Methods

2019 ◽  
Vol 8 (6S3) ◽  
pp. 119-124
Keyword(s):  
Nonlinear Equation ◽  
Convergence Analysis ◽  
Hermite Interpolation ◽  
Basin Of Attraction ◽  
Efficiency Index ◽  
Sixth Order ◽  
Order Convergence ◽  
Eighth Order ◽  
Basin Of Attractions

The development of Chebyshev-Halley Method for solving nonlinear equation is presented in this paper. Varian of Chebyshev-Halley method by Xiaojian (2008) was modified using Hermite Interpolation. The convergence analysis shows that these methods have sixth-order convergence for   0 and   1 eighth-order convergence for   1 2 . The methods are classified by the order and efficiency index. Here, we considered other criteria, the basin of attractions which are presented for several examples.The development of Chebyshev-Halley Method for solving nonlinear equation is presented in this paper. Varian of Chebyshev-Halley method by Xiaojian (2008) was modified using Hermite Interpolation. The convergence analysis shows that these methods have sixth-order convergence for   0 and   1 eighth-order convergence for   1 2 . The methods are classified by the order and efficiency index. Here, we considered other criteria, the basin of attractions which are presented for several examples.

Ball convergence for a three-point method with optimal convergence order eight under weak conditions

2018 ◽  
Vol 13 (02) ◽  
pp. 2050048
Author(s):  
Ioannis K. Argyros ◽  
Munish Kansal ◽  
V. Kanwar
Keyword(s):  
Nonlinear Equation ◽  
Error Bounds ◽  
Eighth Order ◽  
Numerical Examples ◽  
Optimal Convergence ◽  
New Class ◽  
First Derivative ◽  
Ball Convergence ◽  
Computable Error Bounds ◽  
Local Convergence Analysis

We present a local convergence analysis of an optimal eighth-order family of Ostrowski like methods for approximating a locally unique solution of a nonlinear equation. Earlier studies [T. Lotfi, S. Sharifi, M. Salimi and S. Siegmund, A new class of three-point methods with optimal convergence and its dynamics, Numer. Algorithms 68 (2015) 261–288.] have shown convergence of these methods under hypotheses up to the eighth derivative of the function although only the first derivative appears in the method. In this study, we expand the applicability of these methods using only hypotheses up to the first derivative of the function. By this way the applicability of these methods is extended under weaker hypotheses. Moreover, the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

scholarly journals Further development on Traub’s method for solving system of nonlinear equations and ODE’s

2020 ◽  
Author(s):  
Kalyanasundaram Madhu ◽  
Mo'tassem Al-arydah
Keyword(s):  
Nonlinear Equation ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
System Of Nonlinear Equations ◽  
Functional Evaluation ◽  
Order Method ◽  
Eighth Order ◽  
Numerical Examples ◽  
First Derivative ◽  
Iteration Methods

Abstract The foremost objective of this work is to propose a eighth and sixteenth order scheme for handling a nonlinear equation. The eighth order method uses three evaluations of the function and one assessment of the first derivative and sixteenth order method uses four evaluations of the function and one appraisal of the first derivative. Kung-Traub conjecture is satisfied, theoretical analysis of the methods are presented and numerical examples are added to confirm the order of convergence. The performance and efficiency of our iteration methods are compared with the equivalent existing methods on some standard academic problems. We tested projectile motion problem, Planck’s radiation law problem as an application. The basins of attraction are also given to demonstrate their dynamical behavior in the complex plane. Further, we attempt to proposed a sixteenth order iterative method for solving system of nonlinear equation with four functional evaluation, namely two F and two F 0 and only one inverse of Jacobian. The theoretical proof of the method is given and numerical examples are included to confirm the convergence order of the presented methods. We apply the new scheme to find solution on 1-D bratu problem. The performance and efficiency of our iteration methods are compared.

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