Extending the applicability of Newton's method using nondiscrete induction

Ioannis K. Argyros; Saïd Hilout

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 1, page 115-141
  • ISSN: 0011-4642

Abstract

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We extend the applicability of Newton's method for approximating a solution of a nonlinear operator equation in a Banach space setting using nondiscrete mathematical induction concept introduced by Potra and Pták. We obtain new sufficient convergence conditions for Newton's method using Lipschitz and center-Lipschitz conditions instead of only the Lipschitz condition used in F. A. Potra, V. Pták, Sharp error bounds for Newton's process, Numer. Math., 34 (1980), 63–72, and F. A. Potra, V. Pták, Nondiscrete Induction and Iterative Processes, Research Notes in Mathematics, 103. Pitman Advanced Publishing Program, Boston, 1984. Under the same computational cost as before, we provide: weaker sufficient convergence conditions; tighter error estimates on the distances involved and more precise information on the location of the solution. Numerical examples are also provided in this study.

How to cite

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Argyros, Ioannis K., and Hilout, Saïd. "Extending the applicability of Newton's method using nondiscrete induction." Czechoslovak Mathematical Journal 63.1 (2013): 115-141. <http://eudml.org/doc/252486>.

@article{Argyros2013,
abstract = {We extend the applicability of Newton's method for approximating a solution of a nonlinear operator equation in a Banach space setting using nondiscrete mathematical induction concept introduced by Potra and Pták. We obtain new sufficient convergence conditions for Newton's method using Lipschitz and center-Lipschitz conditions instead of only the Lipschitz condition used in F. A. Potra, V. Pták, Sharp error bounds for Newton's process, Numer. Math., 34 (1980), 63–72, and F. A. Potra, V. Pták, Nondiscrete Induction and Iterative Processes, Research Notes in Mathematics, 103. Pitman Advanced Publishing Program, Boston, 1984. Under the same computational cost as before, we provide: weaker sufficient convergence conditions; tighter error estimates on the distances involved and more precise information on the location of the solution. Numerical examples are also provided in this study.},
author = {Argyros, Ioannis K., Hilout, Saïd},
journal = {Czechoslovak Mathematical Journal},
keywords = {Newton's method; Banach space; rate of convergence; semilocal convergence; nondiscrete mathematical induction; estimate function; Newton's method; Banach space; rate of convergence; semilocal convergence; nondiscrete mathematical induction; estimate function; nonlinear operator equation; center-Lipschitz conditions; error estimates; numerical examples},
language = {eng},
number = {1},
pages = {115-141},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Extending the applicability of Newton's method using nondiscrete induction},
url = {http://eudml.org/doc/252486},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Argyros, Ioannis K.
AU - Hilout, Saïd
TI - Extending the applicability of Newton's method using nondiscrete induction
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 1
SP - 115
EP - 141
AB - We extend the applicability of Newton's method for approximating a solution of a nonlinear operator equation in a Banach space setting using nondiscrete mathematical induction concept introduced by Potra and Pták. We obtain new sufficient convergence conditions for Newton's method using Lipschitz and center-Lipschitz conditions instead of only the Lipschitz condition used in F. A. Potra, V. Pták, Sharp error bounds for Newton's process, Numer. Math., 34 (1980), 63–72, and F. A. Potra, V. Pták, Nondiscrete Induction and Iterative Processes, Research Notes in Mathematics, 103. Pitman Advanced Publishing Program, Boston, 1984. Under the same computational cost as before, we provide: weaker sufficient convergence conditions; tighter error estimates on the distances involved and more precise information on the location of the solution. Numerical examples are also provided in this study.
LA - eng
KW - Newton's method; Banach space; rate of convergence; semilocal convergence; nondiscrete mathematical induction; estimate function; Newton's method; Banach space; rate of convergence; semilocal convergence; nondiscrete mathematical induction; estimate function; nonlinear operator equation; center-Lipschitz conditions; error estimates; numerical examples
UR - http://eudml.org/doc/252486
ER -

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