scholarly journals A New Sixth-Order Steffensen-Type Iterative Method for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Tahereh Eftekhari
Keyword(s):  
Nonlinear Equation ◽  
Iterative Method ◽  
Nonlinear Equations ◽  
Efficiency Index ◽  
Sixth Order ◽  
New Method ◽  
Derivative Free ◽  
Better Than

Based on iterative method proposed by Basto et al. (2006), we present a new derivative-free iterative method for solving nonlinear equations. The aim of this paper is to develop a new method to find the approximation of the root α of the nonlinear equation f(x)=0. This method has the efficiency index which equals 61/4=1.5651. The benefit of this method is that this method does not need to calculate any derivative. Several examples illustrate that the efficiency of the new method is better than that of previous methods.

An Efficient Sixth-Order Convergent Newton-Type Iterative Method for Nonlinear Equations

2012 ◽  
Vol 220-223 ◽  
pp. 2585-2588
Author(s):  
Zhong Yong Hu ◽  
Fang Liang ◽  
Lian Zhong Li ◽  
Rui Chen
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Efficiency Index ◽  
Sixth Order ◽  
Type Method ◽  
Second Derivatives ◽  
Better Than

In this paper, we present a modified sixth order convergent Newton-type method for solving nonlinear equations. It is free from second derivatives, and requires three evaluations of the functions and two evaluations of derivatives per iteration. Hence the efficiency index of the presented method is 1.43097 which is better than that of classical Newton’s method 1.41421. Several results are given to illustrate the advantage and efficiency the algorithm.

Three-Step Method with Fifth-Order Convergence for Nonlinear Equation

2013 ◽  
Vol 846-847 ◽  
pp. 1274-1277
Author(s):  
Ying Peng Zhang ◽  
Li Sun
Keyword(s):  
Nonlinear Equation ◽  
Iterative Method ◽  
Efficiency Index ◽  
Newton Methods ◽  
Order Convergence ◽  
Step Method ◽  
Second Derivatives ◽  
Fifth Order ◽  
Modified Newton Methods ◽  
Better Than

We present a fifth-order iterative method for the solution of nonlinear equation. The new method is based on two ordinary methods, which are modified Newton methods without second derivatives. Its efficiency index is 1.37973 which is better than that of Newton's method. Numerical results show the efficiency of the proposed method.

scholarly journals Further Acceleration of the Simpson Method for Solving Nonlinear Equations

2018 ◽  
Vol 14 (2) ◽  
pp. 7631-7639
Author(s):  
Rajinder Thukral
Keyword(s):  
Nonlinear Equation ◽  
Fourth Order ◽  
Efficiency Index ◽  
New Method ◽  
Order Method ◽  
Third Order ◽  
Type Method ◽  
New Formula ◽  
Order Four ◽  
Better Than

There are two aims of this paper, firstly, we present an improvement of the classical Simpson third-order method for finding zeros a nonlinear equation and secondly, we introduce a new formula for approximating second-order derivative. The new Simpson-type method is shown to converge of the order four.  Per iteration the new method requires same amount of evaluations of the function and therefore the new method has an efficiency index better than the classical Simpson method.  We examine the effectiveness of the new fourth-order Simpson-type method by approximating the simple root of a given nonlinear equation. Numerical comparisons is made with classical Simpson method to show the performance of the presented method.

scholarly journals Some novel Newton-type methods for solving nonlinear equations

2019 ◽  
Vol 38 (3) ◽  
pp. 111-123
Author(s):  
Morteza Bisheh-Niasar ◽  
Abbas Saadatmandi
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Basins Of Attraction ◽  
Sixth Order ◽  
New Method ◽  
Order Convergence ◽  
Cubic Convergence ◽  
Numerical Examples ◽  
Newton Iterative Method

The aim of this paper is to present a new nonstandard Newton iterative method for solving nonlinear equations. The convergence of the proposed method is proved and it is shown that the new method has cubic convergence. Furthermore, two new multi-point methods with sixth-order convergence, based on the introduced method, are presented. Also, we describe the basins of attraction for these methods. Finally, some numerical examples are given to show the performance of our methods by comparing with some other methods available in the literature

A Modification of Newton-Type Method with Sixth-Order Convergence for Solving Nonlinear Equations

2012 ◽  
Vol 490-495 ◽  
pp. 1839-1843
Author(s):  
Rui Chen ◽  
Liang Fang
Keyword(s):  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Efficiency Index ◽  
Numerical Results ◽  
Sixth Order ◽  
Order Convergence ◽  
Type Method ◽  
Second Derivatives ◽  
Better Than

In this paper, we present and analyze a modified Newton-type method with oder of convergence six for solving nonlinear equations. The method is free from second derivatives. It requires three evaluations of the functions and two evaluations of derivatives in each step. Therefore the efficiency index of the presented method is 1.431 which is better than that of classical Newton’s method 1.414. Some numerical results illustrate that the proposed method is more efficient and performs better than classical Newton's method

scholarly journals Modified Newton method to determine multiple zeros of nonlinear equations

2016 ◽  
Vol 11 (10) ◽  
pp. 5774-5780
Author(s):  
Rajinder Thukral
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton Method ◽  
New Method ◽  
Convergence Order ◽  
Multiple Zeros ◽  
Numerical Comparisons ◽  
Modified Newton Method ◽  
Iteration Methods ◽  
Better Than

New one-point iterative method for solving nonlinear equations is constructed.  It is proved that the new method has the convergence order of three. Per iteration the new method requires two evaluations of the function.  Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve maximum convergence order2n-1  but, the new method produces convergence order of three, which is better than expected maximum convergence order of two.  Hence, we demonstrate that the conjecture fails for a particular set of nonlinear equations. Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed method using only a few function evaluations.

scholarly journals New Iterative Method for Solving Nonlinear Equations

2014 ◽  
Vol 11 (4) ◽  
pp. 1649-1654 ◽  
Author(s):  
Baghdad Science Journal
Keyword(s):  
Nonlinear Equation ◽  
Iterative Method ◽  
Convergence Analysis ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
New Method ◽  
New Iterative Method

The aim of this paper is to propose an efficient three steps iterative method for finding the zeros of the nonlinear equation f(x)=0 . Starting with a suitably chosen , the method generates a sequence of iterates converging to the root. The convergence analysis is proved to establish its five order of convergence. Several examples are given to illustrate the efficiency of the proposed new method and its comparison with other methods.

scholarly journals Modifikasi Metode Householder Tiga Parameter yang Bebas Turunan Kedua dengan Orde Konvergensi Optimal

2021 ◽  
Vol 18 (1) ◽  
pp. 62-74
Author(s):  
Wartono ◽  
M Zulianti ◽  
Rahmawati
Keyword(s):  
Numerical Simulation ◽  
Iterative Method ◽  
Iterative Methods ◽  
Efficiency Index ◽  
New Method ◽  
Third Order ◽  
The Third ◽  
New Iterative Method ◽  
Better Than ◽  
Taylor’S Series

The Householder’s method is one of the iterative methods with a third-order convergence that used to solve a nonlinear equation. In this paper, the authors modified the iterative method using the expansion of second order Taylor’s series and approximated its second derivative using equality of two the third-order iterative methods. Based on the results of the study, it was found that the new iterative method has a fourth-order of convergence and requires three evaluations of function with an efficiency index of 1,587401. Numerical simulation is given by using several functions to compare the performance between the new method with other iterative methods. The results of numerical simulation show that the performance of the new method is better than other iterative methods.

Three-Step Method with Fifth-Order Convergence for Nonlinear Equation

2012 ◽  
Vol 524-527 ◽  
pp. 3824-3827 ◽  
Author(s):  
Li Sun ◽  
Liang Fang ◽  
Yun Wang
Keyword(s):  
Nonlinear Equation ◽  
Iterative Method ◽  
Efficiency Index ◽  
Order Method ◽  
Order Convergence ◽  
Step Method ◽  
Third Order ◽  
Second Derivatives ◽  
Fifth Order ◽  
Better Than

We present a fifth-order iterative method for the solution of nonlinear equation. The new method is based on the Noor's third-order method, which is a modified Householder method without second derivatives. Its efficiency index is 1.4953 which is better than that of Newton's method and Noor's method. Numerical results show the efficiency of the proposed method.

An Efficient Iterative Method with Order of Convergence Seven for Nonlinear Equations

2012 ◽  
Vol 220-223 ◽  
pp. 2574-2577
Author(s):  
Yun Hong Hu ◽  
Fang Liang ◽  
Li Fang Guo ◽  
Zhong Yong Hu
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Numerical Results ◽  
Type Method ◽  
Second Derivatives ◽  
Seventh Order ◽  
And Performance ◽  
Better Than

In this paper, we present a modified seventh-order convergent Newton-type method for solving nonlinear equations. It is free from second derivatives, and requires three evaluations of the functions and two evaluations of derivatives at each step. Therefore the efficiency index of the presented method is 1.47577 which is better than that of classical Newton’s method 1.41421. Some numerical results demonstrate the efficiency and performance of the presented method.

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