Efficient, process-preserving modified Newton's method for solving reactive multicomponent transport problems in porous media

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.

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Modified Newton’s method for systems of nonlinear equations with singular Jacobian

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2008.04.013 ◽

2009 ◽

Vol 224 (1) ◽

pp. 77-83 ◽

Cited By ~ 21

Author(s):

José L. Hueso ◽

Eulalia Martínez ◽

Juan R. Torregrosa

Keyword(s):

Nonlinear Equations ◽

Newton’S Method ◽

Newton's Method ◽

Systems Of Nonlinear Equations ◽

Modified Newton’S Method ◽

Singular Jacobian

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

Robotica ◽

10.1017/s026357470700389x ◽

2008 ◽

Vol 26 (3) ◽

pp. 285-294 ◽

Cited By ~ 6

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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Conversion between Cartesian and geodetic coordinates on a rotational ellipsoid by solving a system of nonlinear equations

Geodesy and Cartography ◽

10.2478/v10277-012-0013-x ◽

2011 ◽

Vol 60 (2) ◽

pp. 145-159 ◽

Cited By ~ 20

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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