# A new approach for finding weaker conditions for the convergence of Newton's method

Applicationes Mathematicae (2005)

- Volume: 32, Issue: 4, page 465-475
- ISSN: 1233-7234

## Access Full Article

top## Abstract

top## How to cite

topIoannis K. Argyros. "A new approach for finding weaker conditions for the convergence of Newton's method." Applicationes Mathematicae 32.4 (2005): 465-475. <http://eudml.org/doc/279725>.

@article{IoannisK2005,

abstract = {The Newton-Kantorovich hypothesis (15) has been used for a long time as a sufficient condition for convergence of Newton's method to a locally unique solution of a nonlinear equation in a Banach space setting. Recently in [3], [4] we showed that this hypothesis can always be replaced by a condition weaker in general (see (18), (19) or (20)) whose verification requires the same computational cost. Moreover, finer error bounds and at least as precise information on the location of the solution can be obtained this way. Here we show that we can further weaken conditions (18)-(20) and still improve on the error bounds given in [3], [4] (see Remark 1(c)).},

author = {Ioannis K. Argyros},

journal = {Applicationes Mathematicae},

keywords = {Newton's method; Banach space; majorant principle; Newton-Kantorovich hypothesis/theorem; Fréchet derivative; center-Lipschitz condition; numerical examples; convergence; error bounds},

language = {eng},

number = {4},

pages = {465-475},

title = {A new approach for finding weaker conditions for the convergence of Newton's method},

url = {http://eudml.org/doc/279725},

volume = {32},

year = {2005},

}

TY - JOUR

AU - Ioannis K. Argyros

TI - A new approach for finding weaker conditions for the convergence of Newton's method

JO - Applicationes Mathematicae

PY - 2005

VL - 32

IS - 4

SP - 465

EP - 475

AB - The Newton-Kantorovich hypothesis (15) has been used for a long time as a sufficient condition for convergence of Newton's method to a locally unique solution of a nonlinear equation in a Banach space setting. Recently in [3], [4] we showed that this hypothesis can always be replaced by a condition weaker in general (see (18), (19) or (20)) whose verification requires the same computational cost. Moreover, finer error bounds and at least as precise information on the location of the solution can be obtained this way. Here we show that we can further weaken conditions (18)-(20) and still improve on the error bounds given in [3], [4] (see Remark 1(c)).

LA - eng

KW - Newton's method; Banach space; majorant principle; Newton-Kantorovich hypothesis/theorem; Fréchet derivative; center-Lipschitz condition; numerical examples; convergence; error bounds

UR - http://eudml.org/doc/279725

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.