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.

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.

Semilocal Convergence Theorem for a Newton-like Method

2017 ◽  
Vol 7 (3) ◽  
pp. 482-494
Author(s):  
Rong-Fei Lin ◽  
Qing-Biao Wu ◽  
Min-Hong Chen ◽  
Lu Liu ◽  
Ping-Fei Dai
Keyword(s):  
Error Estimate ◽  
Nonlinear Equations ◽  
Convergence Theorem ◽  
Nonlinear Operator ◽  
Semilocal Convergence ◽  
Weak Condition ◽  
Third Order ◽  
Convergence Theorems ◽  
Numerical Examples ◽  
Semilocal Convergence Theorem

AbstractThe semilocal convergence of a third-order Newton-like method for solving nonlinear equations is considered. Under a weak condition (the so-called γ-condition) on the derivative of the nonlinear operator, we establish a new semilocal convergence theorem for the Newton-like method and also provide an error estimate. Some numerical examples show the applicability and efficiency of our result, in comparison to other semilocal convergence theorems.

scholarly journals On the canonical projection of the third dual of a Banach space onto the first dual

1976 ◽  
Vol 15 (3) ◽  
pp. 351-354 ◽  
Author(s):  
A.L. Brown
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dual Space ◽  
Canonical Projection ◽  
The Third ◽  
Current Belief ◽  
A Current

For a Banach space B let P denote the canonical projection of the third dual space of B onto the embedding of the first dual into the third. It is shown that if B = l1 then ‖I-P‖ = 2.This fact shows to be mistaken a current belief in a statement which is equivalent to the statement that for all Banach spaces B the operator I – P is of norm one.

scholarly journals On Some Efficient Techniques for Solving Systems of Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Janak Raj Sharma ◽  
Puneet Gupta
Keyword(s):  
Nonlinear Equations ◽  
Computational Efficiency ◽  
Newton Iteration ◽  
Systems Of Nonlinear Equations ◽  
Order Method ◽  
Third Order ◽  
Newton Step ◽  
The Third ◽  
Large Systems ◽  
Theoretical Results

We present iterative methods of convergence order three, five, and six for solving systems of nonlinear equations. Third-order method is composed of two steps, namely, Newton iteration as the first step and weighted-Newton iteration as the second step. Fifth and sixth-order methods are composed of three steps of which the first two steps are same as that of the third-order method whereas the third is again a weighted-Newton step. Computational efficiency in its general form is discussed and a comparison between the efficiencies of proposed techniques with existing ones is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are verified in the examples. It is shown that the present methods have an edge over similar existing methods, particularly when applied to large systems of equations.

On solutions of third order nonlinear differential equations

2006 ◽  
Vol 4 (1) ◽  
pp. 46-63 ◽  
Author(s):  
Ivan Mojsej ◽  
Ján Ohriska
Keyword(s):  
Differential Equations ◽  
Nonlinear Equations ◽  
Nonlinear Differential Equations ◽  
Asymptotic Properties ◽  
Property A ◽  
Comparison Result ◽  
Third Order ◽  
Deviating Arguments ◽  
Deviating Argument ◽  
The Third

AbstractThe aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with deviating argument. In particular, we prove a comparison theorem for properties A and B as well as a comparison result on property A between nonlinear equations with and without deviating arguments. Our assumptions on nonlinearity f are related to its behavior only in a neighbourhood of zero and/or of infinity.

scholarly journals The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius

2019 ◽  
Vol 27 (4) ◽  
pp. 539-557
Author(s):  
Barbara Kaltenbacher ◽  
Andrej Klassen ◽  
Mario Luiz Previatti de Souza
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Newton Method ◽  
Noise Level ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
A Priori ◽  
Discrepancy Principle ◽  
A Posteriori ◽  
Source Condition

Abstract In this paper, we consider the iteratively regularized Gauss–Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton/discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., {L^{\infty}} as a preimage space. The theoretical findings are illustrated by numerical experiments.

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