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

Published online by Cambridge University Press:  31 July 2012

José Antonio Ezquerro,
Daniel González and
Miguel Ángel Hernández
José Antonio Ezquerro
Affiliation:
University of La Rioja, Department of Mathematics and Computation, C/ Luis de Ulloa s/n, 26004 Logroño, Spain. @unirioja.es
Daniel González
Affiliation:
University of La Rioja, Department of Mathematics and Computation, C/ Luis de Ulloa s/n, 26004 Logroño, Spain. @unirioja.es
Miguel Ángel Hernández
Affiliation:
University of La Rioja, Department of Mathematics and Computation, C/ Luis de Ulloa s/n, 26004 Logroño, Spain. @unirioja.es

Abstract

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

Keywords

Newton’s methodthe Newton–Kantorovich theoremsemilocal convergencemajorizing sequencea priori error estimatesHammerstein’s integral equation

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 1 , January 2013, pp. 149 - 167
Copyright
© EDP Sciences, SMAI, 2012

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