scholarly journals ON CONSTRUCTING A HIGHER-ORDER EXTENSION OF DOUBLE NEWTON'S METHOD USING A SIMPLE BIVARIATE POLYNOMIAL WEIGHT FUNCTION

2015 ◽  
Vol 28 (3) ◽  
pp. 491-497
Author(s):  
SEON YEONG LEE ◽  
YOUNG IK KIM
Keyword(s):  
Weight Function ◽  
Newton’S Method ◽  
Newton's Method ◽  
Higher Order ◽  
Bivariate Polynomial ◽  
Order Extension

scholarly journals Some New Higher Order Weighted Newton Methods for Solving Nonlinear Equation with Applications

2019 ◽  
Vol 24 (2) ◽  
pp. 59
Author(s):  
Parimala Sivakumar ◽  
Jayakumar Jayaraman
Keyword(s):  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
Efficiency Index ◽  
Higher Order ◽  
Function Technique ◽  
Function Evaluation ◽  
Step Method ◽  
Eighth Order ◽  
Additional Function

Three new iterative methods for solving scalar nonlinear equations using weight function technique are presented. The first one is a two-step fifth order method with four function evaluations which is improved from a two-step Newton’s method having same number of function evaluations. By this, the efficiency index of the new method is improved from 1.414 to 1.495. The second one is a three step method with one additional function evaluation producing eighth order accuracy with efficiency index 1.516. The last one is a new fourth order optimal two-step method with efficiency index 1.587. All these three methods are better than Newton’s method and many other equivalent higher order methods. Convergence analyses are established so that these methods have fifth, eighth and fourth order respectively. Numerical examples ascertain that the proposed methods are efficient and demonstrate better performance when compared to some equivalent and optimal methods. Seven application problems are solved to illustrate the efficiency and performance of the proposed methods.

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 After notes on Chebyshev’s iterative method

2017 ◽  
Vol 2 (1) ◽  
pp. 1-12 ◽  
Author(s):  
S. Amat ◽  
S. Busquier
Keyword(s):  
Global Convergence ◽  
Iterative Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Geometric Interpretation ◽  
Higher Order ◽  
Order Method ◽  
Chebyshev’S Method ◽  
Higher Order Method ◽  
Dynamical Properties

AbstractThis paper is a small review of Chebyshev’s method. The geometric interpretation as a generalization of Newton’s method is derived. Using this interpretation its global convergence is proved. Some dynamical properties are studied. As a higher order method, they are more complicated than in Newton’s method. Finally, some applications are revisited pointing out the advantages of Chebyshev’s method with respect Newton’s method.

Optimal Power Flow Using Newton’s Method

2012 ◽  
Vol 3 (2) ◽  
pp. 167-169
Author(s):  
F.M.PATEL F.M.PATEL ◽  
◽  
N. B. PANCHAL N. B. PANCHAL
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Power Flow ◽  
Optimal Power Flow ◽  
Optimal Power

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.

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