On convergence of the modified Newton’s method under Hölder continuous Fréchet derivative

2009 ◽  
Vol 213 (2) ◽  
pp. 440-448 ◽  
Author(s):  
Hongmin Ren ◽  
Ioannis K. Argyros
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Fréchet Derivative ◽  
Hölder Continuous ◽  
Frechet Derivative ◽  
Modified Newton’S Method

scholarly journals On Newton's method and nondiscrete mathematical induction

1988 ◽  
Vol 38 (1) ◽  
pp. 131-140 ◽  
Author(s):  
Ioannis K. Argyros
Keyword(s):  
Lipschitz Condition ◽  
Error Bounds ◽  
Newton’S Method ◽  
Newton's Method ◽  
Mathematical Induction ◽  
Fréchet Derivative ◽  
Hölder Continuous ◽  
Frechet Derivative ◽  
Nondiscrete Mathematical Induction ◽  
Sharp Error

The method of nondiscrete mathematical induction is used to find sharp error bounds for Newton's method. We assume only that the operator has Hölder continuous derivatives. In the case when the Fréchet-derivative of the operator satisfies a Lipschitz condition, our results reduce to the ones obtained by Ptak and Potra in 1972.

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.

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