On a quadratically convergent method using divided differences of order one under the gamma condition

Ioannis Argyros; Hongmin Ren

Open Mathematics (2008)

  • Volume: 6, Issue: 2, page 262-271
  • ISSN: 2391-5455

Abstract

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We re-examine a quadratically convergent method using divided differences of order one in order to approximate a locally unique solution of an equation in a Banach space setting [4, 5, 7]. Recently in [4, 5, 7], using Lipschitz conditions, and a Newton-Kantorovich type approach, we provided a local as well as a semilocal convergence analysis for this method which compares favorably to other methods using two function evaluations such as the Steffensen’s method [1, 3, 13]. Here, we provide an analysis of this method under the gamma condition [6, 7, 19, 20]. In particular, we also show the quadratic convergence of this method. Numerical examples further validating the theoretical results are also provided.

How to cite

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Ioannis Argyros, and Hongmin Ren. "On a quadratically convergent method using divided differences of order one under the gamma condition." Open Mathematics 6.2 (2008): 262-271. <http://eudml.org/doc/269199>.

@article{IoannisArgyros2008,
abstract = {We re-examine a quadratically convergent method using divided differences of order one in order to approximate a locally unique solution of an equation in a Banach space setting [4, 5, 7]. Recently in [4, 5, 7], using Lipschitz conditions, and a Newton-Kantorovich type approach, we provided a local as well as a semilocal convergence analysis for this method which compares favorably to other methods using two function evaluations such as the Steffensen’s method [1, 3, 13]. Here, we provide an analysis of this method under the gamma condition [6, 7, 19, 20]. In particular, we also show the quadratic convergence of this method. Numerical examples further validating the theoretical results are also provided.},
author = {Ioannis Argyros, Hongmin Ren},
journal = {Open Mathematics},
keywords = {Banach space; local/semilocal convergence; majorizing sequence; Fréchet-derivative; two-point iterative method; Steffensen’s method; gamma condition; local and semilocal convergence; Fréchet derivative; Steffensen method; Newton-Kantorovich condition; nonlinear operator equation},
language = {eng},
number = {2},
pages = {262-271},
title = {On a quadratically convergent method using divided differences of order one under the gamma condition},
url = {http://eudml.org/doc/269199},
volume = {6},
year = {2008},
}

TY - JOUR
AU - Ioannis Argyros
AU - Hongmin Ren
TI - On a quadratically convergent method using divided differences of order one under the gamma condition
JO - Open Mathematics
PY - 2008
VL - 6
IS - 2
SP - 262
EP - 271
AB - We re-examine a quadratically convergent method using divided differences of order one in order to approximate a locally unique solution of an equation in a Banach space setting [4, 5, 7]. Recently in [4, 5, 7], using Lipschitz conditions, and a Newton-Kantorovich type approach, we provided a local as well as a semilocal convergence analysis for this method which compares favorably to other methods using two function evaluations such as the Steffensen’s method [1, 3, 13]. Here, we provide an analysis of this method under the gamma condition [6, 7, 19, 20]. In particular, we also show the quadratic convergence of this method. Numerical examples further validating the theoretical results are also provided.
LA - eng
KW - Banach space; local/semilocal convergence; majorizing sequence; Fréchet-derivative; two-point iterative method; Steffensen’s method; gamma condition; local and semilocal convergence; Fréchet derivative; Steffensen method; Newton-Kantorovich condition; nonlinear operator equation
UR - http://eudml.org/doc/269199
ER -

References

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