We provide a local as well as a semilocal convergence analysis for Newton's method to approximate a locally unique solution of an equation in a Banach space setting. Using a combination of center-gamma with a gamma-condition, we obtain an upper bound on the inverses of the operators involved which can be more precise than those given in the elegant works by Smale, Wang, and Zhao and Wang. This observation leads (under the same or less computational cost) to a convergence analysis with the following advantages: local case: larger radius of convergence and finer error estimates on the distances involved; semilocal case: larger domain of convergence, finer error bounds on the distances involved, and at least as precise information on the location of the solution.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-am36-2-9,
author = {Ioannis K. Argyros},
title = {A convergence analysis of Newton's method under the gamma-condition in Banach spaces},
journal = {Applicationes Mathematicae},
volume = {36},
year = {2009},
pages = {225-239},
zbl = {1170.65036},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am36-2-9}
}
Ioannis K. Argyros. A convergence analysis of Newton's method under the gamma-condition in Banach spaces. Applicationes Mathematicae, Tome 36 (2009) pp. 225-239. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am36-2-9/