Displaying similar documents to “Newton and Schinzel sequences in quadratic fields”

A new Kantorovich-type theorem for Newton's method

Ioannis Argyros (1999)

Applicationes Mathematicae

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A new Kantorovich-type convergence theorem for Newton's method is established for approximating a locally unique solution of an equation F(x)=0 defined on a Banach space. It is assumed that the operator F is twice Fréchet differentiable, and that F', F'' satisfy Lipschitz conditions. Our convergence condition differs from earlier ones and therefore it has theoretical and practical value.

A general semilocal convergence result for Newton’s method under centered conditions for the second derivative

José Antonio Ezquerro, Daniel González, Miguel Ángel Hernández (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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From Kantorovich’s theory we present a semilocal convergence result for Newton’s method which is based mainly on a modification of the condition required to the second derivative of the operator involved. In particular, instead of requiring that the second derivative is bounded, we demand that it is centered. As a consequence, we obtain a modification of the starting points for Newton’s method. We illustrate this study with applications to nonlinear integral equations of mixed Hammerstein...

A convergence analysis of Newton-like methods for singular equations using outer or generalized inverses

Ioannis K. Argyros (2005)

Applicationes Mathematicae

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The Newton-Kantorovich approach and the majorant principle are used to provide new local and semilocal convergence results for Newton-like methods using outer or generalized inverses in a Banach space setting. Using the same conditions as before, we provide more precise information on the location of the solution and on the error bounds on the distances involved. Moreover since our Newton-Kantorovich-type hypothesis is weaker than before, we can cover cases where the original Newton-Kantorovich...

On the convergence of Newton's method under ω*-conditioned second derivative

Ioannis K. Argyros, Saïd Hilout (2011)

Applicationes Mathematicae

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We provide a new semilocal result for the quadratic convergence of Newton's method under ω*-conditioned second Fréchet derivative on a Banach space. This way we can handle equations where the usual Lipschitz-type conditions are not verifiable. An application involving nonlinear integral equations and two boundary value problems is provided. It turns out that a similar result using ω-conditioned hypotheses can provide usable error estimates indicating only linear convergence for Newton's...

Note on the Newton number.

Gwoździewicz, Janusz (2008)

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

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Local convergence of inexact Newton methods under affine invariant conditions and hypotheses on the second Fréchet derivative

Ioannis Argyros (1999)

Applicationes Mathematicae

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We use inexact Newton iterates to approximate a solution of a nonlinear equation in a Banach space. Solving a nonlinear equation using Newton iterates at each stage is very expensive in general. That is why we consider inexact Newton methods, where the Newton equations are solved only approximately, and in some unspecified manner. In earlier works [2], [3], natural assumptions under which the forcing sequences are uniformly less than one were given based on the second Fréchet derivative...

A general semilocal convergence result for Newton’s method under centered conditions for the second derivative

José Antonio Ezquerro, Daniel González, Miguel Ángel Hernández (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

Similarity:

From Kantorovich’s theory we present a semilocal convergence result for Newton’s method which is based mainly on a modification of the condition required to the second derivative of the operator involved. In particular, instead of requiring that the second derivative is bounded, we demand that it is centered. As a consequence, we obtain a modification of the starting points for Newton’s method. We illustrate this study with applications to ...

Uniform Convergence of the Newton Method for Aubin Continuous Maps

Dontchev, Asen (1996)

Serdica Mathematical Journal

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* This work was supported by National Science Foundation grant DMS 9404431. In this paper we prove that the Newton method applied to the generalized equation y ∈ f(x) + F(x) with a C^1 function f and a set-valued map F acting in Banach spaces, is locally convergent uniformly in the parameter y if and only if the map (f +F)^(−1) is Aubin continuous at the reference point. We also show that the Aubin continuity actually implies uniform Q-quadratic convergence provided that...