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 Using symbolic computation in Mathematica for verifying the convergence of the iterative methods

2013 ◽  
Vol 22 (1) ◽  
pp. 9-13
Author(s):  
GHEORGHE ARDELEAN ◽  
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Symbolic Computation ◽  
Newton’S Method ◽  
Newton's Method ◽  
New Method ◽  
Correct Result ◽  
Modified Newton’S Method ◽  
Second Derivatives ◽  
Appl Math

In [Jisheng Kou, The improvements of modified Newton’s method, Appl. Math. Comput., 189 (2007) 602–609], the improvements of some thirdorder modifications of Newton’s method for solving nonlinear equations are presented. In this paper we point out some flaws in the results of Jisheng Kou and we correct them by using symbolic computation in Mathematica. In [M. A. Noor et al., A new modified Halley method without second derivatives for nonlinear equations, Appl. Math. Comput., 189 (2007) 1268–1273] , the error equation obtained for the new method presented is wrong. We present the correct result by using symbolic computation, too. Finally, we present two examples of very simply proofs for the convergence of iterative methods by using symbolic computation. We consider that the Mathematica programs in this paper are good examples for other authors to prove the convergence of the iterative methods or to verify their results.

scholarly journals Metode Iterasi Tiga Langkah untuk Menyelesaikan Persamaan NonLinear dengan Menggunakan Matlab

2019 ◽  
Vol 4 (2) ◽  
pp. 34
Author(s):  
Deasy Wahyuni ◽  
Elisawati Elisawati
Keyword(s):  
Numerical Simulations ◽  
Nonlinear Equations ◽  
Newton Method ◽  
Iteration Method ◽  
Newton's Method ◽  
Order Of Convergence ◽  
Higher Order ◽  
Convergence Order ◽  
Matlab Program ◽  
Analytical Results

Newton method is one of the most frequently used methods to find solutions to the roots of nonlinear equations. Along with the development of science, Newton's method has undergone various modifications. One of them is the hasanov method and the newton method variant (vmn), with a higher order of convergence. In this journal focuses on the three-step iteration method in which the order of convergence is higher than the three methods. To find the convergence order of the three-step iteration method requires a program that can support the analytical results of both methods. One of them using the help of the matlab program. Which will then be compared with numerical simulations also using the matlab program.  Keywords : newton method, newton method variant, Hasanov Method and order of convergence

scholarly journals On an new Open type variant of Newton´s method

2014 ◽  
Vol 10 (2) ◽  
pp. 21-31
Author(s):  
Manoj Kumar
Keyword(s):  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Order Of Convergence ◽  
Open Type ◽  
Numerical Tests ◽  
Comparison Of The Results

Abstract The aim of the present paper is to introduce and investigate a new Open type variant of Newton's method for solving nonlinear equations. The order of convergence of the proposed method is three. In addition to numerical tests verifying the theory, a comparison of the results for the proposed method and some of the existing ones have also been given.

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.

scholarly journals Optimal Sixteenth Order Convergent Method Based on Quasi-Hermite Interpolation for Computing Roots

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
Fiza Zafar ◽  
Nawab Hussain ◽  
Zirwah Fatimah ◽  
Athar Kharal
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Order Of Convergence ◽  
Hermite Interpolation ◽  
Convergent Method ◽  
Basins Of Attractions

We have given a four-step, multipoint iterative method without memory for solving nonlinear equations. The method is constructed by using quasi-Hermite interpolation and has order of convergence sixteen. As this method requires four function evaluations and one derivative evaluation at each step, it is optimal in the sense of the Kung and Traub conjecture. The comparisons are given with some other newly developed sixteenth-order methods. Interval Newton’s method is also used for finding the enough accurate initial approximations. Some figures show the enclosure of finitely many zeroes of nonlinear equations in an interval. Basins of attractions show the effectiveness of the 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

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