scholarly journals New Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Rajinder Thukral
Keyword(s):  
Nonlinear Equations ◽  
Convergence Order ◽  
Order Derivative ◽  
Optimal Order ◽  
Eighth Order ◽  
Optimal Order Of Convergence ◽  
New Family ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Iteration Methods

A new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based onnevaluations could achieve optimal convergence order of . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for . Numerical comparisons are made to demonstrate the performance of the methods presented.

scholarly journals Eighth-Order Iterative Methods without Derivatives for Solving Nonlinear Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
R. Thukral
Keyword(s):  
Nonlinear Equations ◽  
Convergence Order ◽  
Optimal Order ◽  
Eighth Order ◽  
Optimal Order Of Convergence ◽  
New Methods ◽  
New Family ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Iteration Methods

A new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on evaluations, could achieve optimal convergence order . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for . Numerical comparisons are made to demonstrate the performance of the methods presented.

A general class of optimal eighth-order derivative free methods for nonlinear equations

2020 ◽  
Vol 58 (4) ◽  
pp. 854-867
Author(s):  
Ramandeep Behl ◽  
Ali Saleh Alshomrani ◽  
Changbum Chun
Keyword(s):  
Nonlinear Equations ◽  
General Class ◽  
Order Derivative ◽  
Eighth Order ◽  
Derivative Free ◽  
Derivative Free Methods

scholarly journals A family of two-point methods with memory for solving nonlinear equations

2011 ◽  
Vol 5 (2) ◽  
pp. 298-317 ◽  
Author(s):  
Miodrag Petkovic ◽  
Jovana Dzunic ◽  
Ljiljana Petkovic
Keyword(s):  
Nonlinear Equations ◽  
Free Parameter ◽  
Convergence Order ◽  
Optimal Order ◽  
Additional Function ◽  
Numerical Examples ◽  
Derivative Free ◽  
With Memory ◽  
Theoretical Results ◽  
High Computational Efficiency

An efficient family of two-point derivative free methods with memory for solving nonlinear equations is presented. It is proved that the convergence order of the proposed family is increased from 4 to at least 2 + ?6 ? 4.45, 5, 1/2 (5 + ?33) ? 5.37 and 6, depending on the accelerating technique. The increase of convergence order is attained using a suitable accelerating technique by varying a free parameter in each iteration. The improvement of convergence rate is achieved without any additional function evaluations meaning that the proposed methods with memory are very efficient. Moreover, the presented methods are more efficient than all existing methods known in literature in the class of two-point methods and three-point methods of optimal order eight. Numerical examples and the comparison with the existing two-point methods are included to confirm theoretical results and high computational efficiency. 2010 Mathematics Subject Classification. 65H05

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 On two new families of iterative methods for solving nonlinear equations with optimal order

2011 ◽  
Vol 5 (1) ◽  
pp. 93-109 ◽  
Author(s):  
M. Heydari ◽  
S.M. Hosseini ◽  
G.B. Loghmani
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Efficiency Index ◽  
Convergence Order ◽  
Optimal Order ◽  
Eighth Order ◽  
Optimal Convergence ◽  
Numerical Comparisons ◽  
First Derivative ◽  
The Third

In this paper, two new families of eighth-order iterative methods for solving nonlinear equations is presented. These methods are developed by combining a class of optimal two-point methods and a modified Newton?s method in the third step. Per iteration the presented methods require three evaluations of the function and one evaluation of its first derivative and therefore have the efficiency index equal to 1:682. Kung and Traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2n?1. Thus the new families of eighth-order methods agrees with the conjecture of Kung-Traub for the case n = 4. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.

Design and Analysis of a New Class of Derivative-Free Optimal Order Methods for Nonlinear Equations

2018 ◽  
Vol 15 (03) ◽  
pp. 1850010 ◽  
Author(s):  
Janak Raj Sharma ◽  
Ioannis K. Argyros ◽  
Deepak Kumar
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Optimal Order Of Convergence ◽  
Numerical Examples ◽  
New Class ◽  
New Methods ◽  
Derivative Free ◽  
Local Convergence Analysis ◽  
Theoretical Results

We develop a general class of derivative free iterative methods with optimal order of convergence in the sense of Kung–Traub hypothesis for solving nonlinear equations. The methods possess very simple design, which makes them easy to remember and hence easy to implement. The Methodology is based on quadratically convergent Traub–Steffensen scheme and further developed by using Padé approximation. Local convergence analysis is provided to show that the iterations are locally well defined and convergent. Numerical examples are provided to confirm the theoretical results and to show the good performance of new methods.

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