Expanding the applicability of Newton’s method and of a robust modified Newton’s method

2019 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Modified Newton’S Method

An Accelerated Iterative Method with Third-Order Convergence

2012 ◽  
Vol 220-223 ◽  
pp. 2658-2661
Author(s):  
Zhong Yong Hu ◽  
Liang Fang ◽  
Lian Zhong Li
Keyword(s):  
Iterative Method ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
New Method ◽  
Order Convergence ◽  
Third Order ◽  
Numerical Examples ◽  
Modified Newton’S Method ◽  
Jarratt Method

We present a new modified Newton's method with third-order convergence and compare it with the Jarratt method, which is of fourth-order. Based on this new method, we obtain a family of Newton-type methods, which converge cubically. Numerical examples show that the presented method can compete with Newton's method and other known third-order modifications of Newton's method.

Modified Newton's method applied to potential field based navigation for nonholonomic robots in dynamic environments

Robotica ◽  
2008 ◽  
Vol 26 (3) ◽  
pp. 285-294 ◽  
Author(s):  
Jing Ren ◽  
Kenneth A. McIsaac ◽  
Rajni V. Patel
Keyword(s):  
Newton’S Method ◽  
Potential Field ◽  
Newton's Method ◽  
Dynamic Environment ◽  
Dynamic Environments ◽  
Field Methods ◽  
Control Laws ◽  
Nonholonomic Robots ◽  
Modified Newton’S Method ◽  
Speed Up

SUMMARYThis paper is to investigate inherent oscillations problems of Potential Field Methods (PFMs) for nonholonomic robots in dynamic environments. In prior work, we proposed a modification of Newton's method to eliminate oscillations for omnidirectional robots in static environment. In this paper, we develop control laws for nonholonomic robots in dynamic environment using modifications of Newton's method. We have validated this technique in a multirobot search-and-forage task. We found that the use of the modifications of Newton's method, which applies anywhere C2 continuous navigation functions are defined, can greatly reduce oscillations and speed up robot's movement, when compared to the standard gradient approaches.

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.

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