scholarly journals Some modifications of Newton’s method with higher-order convergence for solving nonlinear equations

2009 ◽  
Vol 228 (1) ◽  
pp. 296-303 ◽  
Author(s):  
Liang Fang ◽  
Guoping He
Keyword(s):  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Higher Order ◽  
Order Convergence ◽  
Higher Order Convergence

scholarly journals Why using symbolic computation for proving the convergence of iterative methods?

2013 ◽  
Vol 22 (2) ◽  
pp. 127-134
Author(s):  
GHEORGHE ARDELEAN ◽  
◽  
LASZLO BALOG ◽  
Keyword(s):  
Nonlinear Equations ◽  
Symbolic Computation ◽  
Newton’S Method ◽  
Newton's Method ◽  
Order Of Convergence ◽  
Higher Order ◽  
Order Convergence ◽  
Convergence Error ◽  
Higher Order Convergence ◽  
Appl Math

In [YoonMe Ham et al., Some higher-order modifications of Newton’s method for solving nonlinear equations, J. Comput. Appl. Math., 222 (2008) 477–486], some higher-order modifications of Newton’s method for solving nonlinear equations are presented. In [Liang Fang et al., Some modifications of Newton’s method with higher-order convergence for solving nonlinear equations, J. Comput. Appl. Math., 228 (2009) 296–303], the authors point out some flaws in the results of YoonMe Ham et al. and present some modified variants of the method. In this paper we point out that the paper of Liang Fang et al. itself contains some flaw results and we correct them by using symbolic computation in Mathematica. Moreover, we show that the main result in Theorem 3 of Liang Fang et al. is wrong. The order of convergence of the method is’nt 3m+2, but is 2m+4. We give the general expression of convergence error too.

scholarly journals Three New Optimal Fourth-Order Iterative Methods to Solve Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Gustavo Fernández-Torres ◽  
Juan Vásquez-Aquino
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
Classical Method ◽  
Multiple Roots ◽  
Order Convergence ◽  
Numerical Examples ◽  
Optimal Methods

We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published.

Conversion between Cartesian and geodetic coordinates on a rotational ellipsoid by solving a system of nonlinear equations

2011 ◽  
Vol 60 (2) ◽  
pp. 145-159 ◽  
Author(s):  
Marcin Ligas ◽  
Piotr Banasik
Keyword(s):  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
System Of Nonlinear Equations ◽  
Order Convergence ◽  
Ellipsoidal Height ◽  
Third Order ◽  
The Third ◽  
Geodetic Coordinates ◽  
Modified Newton’S Method

Conversion between Cartesian and geodetic coordinates on a rotational ellipsoid by solving a system of nonlinear equationsA new method to transform from Cartesian to geodetic coordinates is presented. It is based on the solution of a system of nonlinear equations with respect to the coordinates of the point projected onto the ellipsoid along the normal. Newton's method and a modification of Newton's method were applied to give third-order convergence. The method developed was compared to some well known iterative techniques. All methods were tested on three ellipsoidal height ranges: namely, (-10 - 10 km) (terrestrial), (20 - 1000 km), and (1000 - 36000 km) (satellite). One iteration of the presented method, implemented with the third-order convergence modified Newton's method, is necessary to obtain a satisfactory level of accuracy for the geodetic latitude (σφ < 0.0004") and height (σh< 10-6km, i.e. less than a millimetre) for all the heights tested. The method is slightly slower than the method of Fukushima (2006) and Fukushima's (1999) fast implementation of Bowring's (1976) method.

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

A New Iterative Method with Sixth-Order Convergence for Solving Nonlinear Equations

2012 ◽  
Vol 542-543 ◽  
pp. 1019-1022
Author(s):  
Han Li
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Numerical Experiments ◽  
Efficiency Index ◽  
Order Convergence ◽  
Second Derivatives ◽  
New Iterative Method ◽  
Better Than

In this paper, we present and analyze a new iterative method for solving nonlinear equations. It is proved that the method is six-order convergent. The algorithm is free from second derivatives, and it requires three evaluations of the functions and two evaluations of derivatives in each iteration. The efficiency index of the presented method is 1.431 which is better than that of classical Newton’s method 1.414. Some numerical experiments illustrate that the proposed method is more efficient and performs better than classical Newton's method and some other methods.

scholarly journals Eighteenth Order Convergent Method for Solving Non-Linear Equations

2017 ◽  
Vol 10 (1) ◽  
pp. 144-150 ◽  
Author(s):  
V.B Vatti ◽  
Ramadevi Sri ◽  
M.S Mylapalli
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Linear Equations ◽  
Subject Classification ◽  
Order Convergence ◽  
Numerical Examples ◽  
Convergent Method ◽  
Non Linear Equations

In this paper, we suggest and discuss an iterative method for solving nonlinear equations of the type f(x)=0 having eighteenth order convergence. This new technique based on Newton’s method and extrapolated Newton’s method. This method is compared with the existing ones through some numerical examples to exhibit its superiority. AMS Subject Classification: 41A25, 65K05, 65H05.

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