scholarly journals An Efficient Family of Root-Finding Methods with Optimal Eighth-Order Convergence

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Rajni Sharma ◽  
Janak Raj Sharma
Keyword(s):  
Computational Cost ◽  
Efficiency Index ◽  
Optimal Order ◽  
Computational Results ◽  
Order Convergence ◽  
Eighth Order ◽  
Root Finding ◽  
First Derivative ◽  
The Family ◽  
Iteration Methods

We derive a family of eighth-order multipoint methods for the solution of nonlinear equations. In terms of computational cost, the family requires evaluations of only three functions and one first derivative per iteration. This implies that the efficiency index of the present methods is 1.682. Kung and Traub (1974) conjectured that multipoint iteration methods without memory based on n evaluations have optimal order . Thus, the family agrees with Kung-Traub conjecture for the case . Computational results demonstrate that the developed methods are efficient and robust as compared with many well-known methods.

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.

scholarly journals New modification of Maheshwari’s method with optimal eighth order convergence for solving nonlinear equations

2016 ◽  
Vol 14 (1) ◽  
pp. 443-451 ◽  
Author(s):  
Somayeh Sharifi ◽  
Massimiliano Ferrara ◽  
Mehdi Salimi ◽  
Stefan Siegmund
Keyword(s):  
Weight Function ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Efficiency Index ◽  
The Other ◽  
Order Convergence ◽  
Eighth Order ◽  
First Derivative ◽  
Attraction Basins

AbstractIn this paper, we present a family of three-point with eight-order convergence methods for finding the simple roots of nonlinear equations by suitable approximations and weight function based on Maheshwari’s method. Per iteration this method requires three evaluations of the function and one evaluation of its first derivative. These class of methods have the efficiency index equal to ${8^{{\textstyle{1 \over 4}}}} \approx 1.682$. We describe the analysis of the proposed methods along with numerical experiments including comparison with the existing methods. Moreover, the attraction basins of the proposed methods are shown with some comparisons to the other existing methods.

scholarly journals A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 339
Author(s):  
Ramandeep Behl ◽  
Eulalia Martínez ◽  
Fabricio Cevallos ◽  
Diego Alarcón
Keyword(s):  
Iterative Methods ◽  
Computational Cost ◽  
Real Life ◽  
Efficiency Index ◽  
Maximum Efficiency ◽  
Multiple Roots ◽  
Order Convergence ◽  
Eighth Order ◽  
Optimal Method ◽  
Computational Mathematics

The aim of this paper is to introduce new high order iterative methods for multiple roots of the nonlinear scalar equation; this is a demanding task in the area of computational mathematics and numerical analysis. Specifically, we present a new Chebyshev–Halley-type iteration function having at least sixth-order convergence and eighth-order convergence for a particular value in the case of multiple roots. With regard to computational cost, each member of our scheme needs four functional evaluations each step. Therefore, the maximum efficiency index of our scheme is 1.6818 for α = 2 , which corresponds to an optimal method in the sense of Kung and Traub’s conjecture. We obtain the theoretical convergence order by using Taylor developments. Finally, we consider some real-life situations for establishing some numerical experiments to corroborate the theoretical results.

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 Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their Basins of Attraction and Its Application

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 322 ◽  
Author(s):  
Yanlin Tao ◽  
Kalyanasundaram Madhu
Keyword(s):  
Dynamical Behavior ◽  
Computational Cost ◽  
Basins Of Attraction ◽  
Order Method ◽  
Eighth Order ◽  
Projectile Motion ◽  
First Derivative ◽  
Iteration Functions ◽  
Optimal Launch Angle ◽  
Fourth Order Method

The principal objective of this work is to propose a fourth, eighth and sixteenth order scheme for solving a nonlinear equation. In terms of computational cost, per iteration, the fourth order method uses two evaluations of the function and one evaluation of the first derivative; the eighth order method uses three evaluations of the function and one evaluation of the first derivative; and sixteenth order method uses four evaluations of the function and one evaluation of the first derivative. So these all the methods have satisfied the Kung-Traub optimality conjecture. In addition, the theoretical convergence properties of our schemes are fully explored with the help of the main theorem that demonstrates the convergence order. The performance and effectiveness of our optimal iteration functions are compared with the existing competitors on some standard academic problems. The conjugacy maps of the presented method and other existing eighth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane. We apply the new scheme to find the optimal launch angle in a projectile motion problem and Planck’s radiation law problem as an application.

scholarly journals An Optimal Eighth-Order Family of Iterative Methods for Multiple Roots

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 672 ◽  
Author(s):  
Saima Akram ◽  
Fiza Zafar ◽  
Nusrat Yasmin
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Multiple Root ◽  
Multiple Roots ◽  
Order Convergence ◽  
Eighth Order ◽  
Root Finding ◽  
New Family ◽  
Two Parameters ◽  
Theoretical Results

In this paper, we introduce a new family of efficient and optimal iterative methods for finding multiple roots of nonlinear equations with known multiplicity ( m ≥ 1 ) . We use the weight function approach involving one and two parameters to develop the new family. A comprehensive convergence analysis is studied to demonstrate the optimal eighth-order convergence of the suggested scheme. Finally, numerical and dynamical tests are presented, which validates the theoretical results formulated in this paper and illustrates that the suggested family is efficient among the domain of multiple root finding methods.

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 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.

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