A new Kantorovich-type theorem for Newton's method

Ioannis Argyros

Applicationes Mathematicae (1999)

  • Volume: 26, Issue: 2, page 151-157
  • ISSN: 1233-7234

Abstract

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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.

How to cite

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Argyros, Ioannis. "A new Kantorovich-type theorem for Newton's method." Applicationes Mathematicae 26.2 (1999): 151-157. <http://eudml.org/doc/219231>.

@article{Argyros1999,
abstract = {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.},
author = {Argyros, Ioannis},
journal = {Applicationes Mathematicae},
keywords = {Newton's method; Lipschitz-Hölder condition; Kantorovich hypothesis; Banach space; Kantorovich theorem; convergence; Newton method},
language = {eng},
number = {2},
pages = {151-157},
title = {A new Kantorovich-type theorem for Newton's method},
url = {http://eudml.org/doc/219231},
volume = {26},
year = {1999},
}

TY - JOUR
AU - Argyros, Ioannis
TI - A new Kantorovich-type theorem for Newton's method
JO - Applicationes Mathematicae
PY - 1999
VL - 26
IS - 2
SP - 151
EP - 157
AB - 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.
LA - eng
KW - Newton's method; Lipschitz-Hölder condition; Kantorovich hypothesis; Banach space; Kantorovich theorem; convergence; Newton method
UR - http://eudml.org/doc/219231
ER -

References

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  1. [1] I. K. Argyros, Newton-like methods under mild differentiability conditions with error analysis, Bull. Austral. Math. Soc. 37 (1988), 131-147. 
  2. [2] I. K. Argyros and F. Szidarovszky, The Theory and Applications of Iteration Methods, C.R.C. Press, Boca Raton, Fla., 1993. Zbl0844.65052
  3. [3] J. M. Gutiérrez, A new semilocal convergence theorem for Newton's method, J. Comput. Appl. Math. 79 (1997), 131-145. Zbl0872.65045
  4. [4] J. M. Gutiérrez, M. A. Hernández, and M. A. Salanova, Accessibility of solutions by Newton's method, Internat. J. Comput. Math. 57 (1995), 239-247. Zbl0844.47035
  5. [5] Z. Huang, A note on the Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993), 211-217. Zbl0782.65071
  6. [6] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982. Zbl0484.46003

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