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

José Antonio EzquerroDaniel GonzálezMiguel Ángel Hernández — 2013

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

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 type.

Multi-Harnack smoothings of real plane branches

Pedro Daniel González PérezJean-Jacques Risler — 2010

Annales scientifiques de l'École Normale Supérieure

Let Δ 𝐑 2 be an integral convex polygon. G. Mikhalkin introduced the notion of, a class of real algebraic curves, defined by polynomials supported on Δ and contained in the corresponding toric surface. He proved their existence, viamethod, and that the topological type of their real parts is unique (and determined by Δ ). This paper is concerned with the description of the analogous statement in the case of a smoothing of a real plane branch ( C , 0 ) . We introduce the class ofsmoothings of ( C , 0 ) by passing through...

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

José Antonio EzquerroDaniel GonzálezMiguel Ángel Hernández — 2012

ESAIM: Mathematical Modelling and Numerical Analysis

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...

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