Book ChapterDOI
Newton's Method under Different Lipschitz Conditions
José M. Gutiérrez,M. A. Hernández +1 more
- pp 368-376
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The classical Kantorovich theorem for Newton's method assumes that the derivative of the involved operator satisfies a Lipschitz condition, with a special emphasis in the center-Lipschitzer condition.Abstract:
The classical Kantorovich theorem for Newton's method assumes that the derivative of the involved operator satisfies a Lipschitz condition ?F? (x0-1 [F? (x) - F? (y)] ? ? L?x - y? In this communication, we analyse the different modifications of this condition, with a special emphasis in the center-Lipschitz condition: ?F? (x0)-1 [F? (x) - F? (x0] ? ? ?(?x - x0?) being ? a positive increasing real function and x0 the starting point for Newton's iterationread more
Citations
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Journal ArticleDOI
On the semilocal convergence of Newton-Kantorovich method under center-Lipschitz conditions
TL;DR: Newton's method for solving nonlinear equations with operators defined between two Banach spaces is studied to obtain a generalization of Kantorovich's theorem that improves the values of the universal constant that appears in it as well as the radius where the solution is located and where it is unique.
Journal ArticleDOI
Enlarging the domain of starting points for Newton's method under center conditions on the first Fréchet-derivative
TL;DR: This work analyzes the semilocal convergence of Newton's method under center conditions on the first Frechet-derivative of the operator involved and illustrates the theoretical results obtained with some mildly nonlinear elliptic equations.
Journal ArticleDOI
On the Domain of Starting Points of Newton’s Method Under Center Lipschitz Conditions
TL;DR: In this article, the authors studied the semilocal convergence of Newton's method in Banach spaces under center Lipschitz conditions, and improved this choice by looking for a domain of initial points (a convergence domain).
Journal ArticleDOI
A Convergence Criterion of Newton’s Method Based on the Heisenberg Uncertainty Principle
Book ChapterDOI
Using Center ω-Lipschitz Conditions for the First Derivative at Auxiliary Points
TL;DR: In this article, it was shown that the semilocal convergence of Newton's method is guaranteed for any starting point x 0 provided the values of x and y are such that the parameters L, β and η are fixed by the following conditions:
References
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Journal ArticleDOI
Convergence of Newton's method and uniqueness of the solution of equations in Banach space
TL;DR: In this article, the convergence of Newton's method in Banach spaces is established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak L average.
Journal ArticleDOI
A new semilocal convergence theorem for Newton's method
TL;DR: In this paper, a new semilocal convergence theorem for Newton's method is established for solving a nonlinear equation F ( x ) = 0, defined in Banach spaces. But it is assumed that the operator F is twice Frechet differentiable, and satisfies a Lipschitz type condition.
Journal ArticleDOI
A note on the Kantorovich theorem for Newton iteration
TL;DR: In this article, a new theorem for the Newton method convergence is obtained, which is different from that of the Kantorovich theorem and therefore has theoretical and practical value, and it has been shown to be correct.
Journal ArticleDOI
Newton's method under mild differentiability conditions with error analysis
TL;DR: In this article, a detailed analysis of Newton's method applied to operators with Holder continuous derivatives is given, and the analysis is shown to reduce the analysis of Lancaster (Num. Math.A.N. 1968) when the operator has a continuous second derivative.
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Characterizations and Extensions of Lipschitz–α Operators
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Daniel Azagra,R. Fry,L. Keener +2 more