scholarly journals Steffensen-Type Method of Super Third-Order Convergence for Solving Nonlinear Equations

2014 ◽  
Vol 02 (07) ◽  
pp. 581-586 ◽  
Author(s):  
Zhongli Liu ◽  
Hong Zhang
Keyword(s):  
Nonlinear Equations ◽  
Order Convergence ◽  
Third Order ◽  
Type Method

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

scholarly journals Newton-Kantorovich and Smale Uniform Type Convergence Theorem for a Deformed Newton Method in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Rongfei Lin ◽  
Yueqing Zhao ◽  
Zdeněk Šmarda ◽  
Yasir Khan ◽  
Qingbiao Wu
Keyword(s):  
Banach Space ◽  
Error Estimate ◽  
Banach Spaces ◽  
Nonlinear Equations ◽  
Newton Method ◽  
Convergence Theorem ◽  
Order Convergence ◽  
Third Order ◽  
The Third

Newton-Kantorovich and Smale uniform type of convergence theorem of a deformed Newton method having the third-order convergence is established in a Banach space for solving nonlinear equations. The error estimate is determined to demonstrate the efficiency of our approach. The obtained results are illustrated with three examples.

scholarly journals A modified Newton-type method with sixth-order convergence for solving nonlinear equations

2011 ◽  
Vol 15 ◽  
pp. 3124-3128 ◽  
Author(s):  
Liang Fang ◽  
Tao Chen ◽  
Li Tian ◽  
Li Sun ◽  
Bin Chen
Keyword(s):  
Nonlinear Equations ◽  
Sixth Order ◽  
Order Convergence ◽  
Type Method

scholarly journals A New Class of Halley’s Method with Third-Order Convergence for Solving Nonlinear Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Mohammed Barrada ◽  
Mariya Ouaissa ◽  
Yassine Rhazali ◽  
Mariyam Ouaissa
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Order Convergence ◽  
Eighth Order ◽  
Third Order ◽  
Numerical Examples ◽  
New Class ◽  
New Methods ◽  
New Family ◽  
Number Of Iterations

In this paper, we present a new family of methods for finding simple roots of nonlinear equations. The convergence analysis shows that the order of convergence of all these methods is three. The originality of this family lies in the fact that these sequences are defined by an explicit expression which depends on a parameter p where p is a nonnegative integer. A first study on the global convergence of these methods is performed. The power of this family is illustrated analytically by justifying that, under certain conditions, the method convergence’s speed increases with the parameter p. This family’s efficiency is tested on a number of numerical examples. It is observed that our new methods take less number of iterations than many other third-order methods. In comparison with the methods of the sixth and eighth order, the new ones behave similarly in the examples considered.

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