Convergence conditions for Secant-type methods

Ioannis K. Argyros; Said Hilout

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 1, page 253-272
  • ISSN: 0011-4642

Abstract

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We provide new sufficient convergence conditions for the convergence of the secant-type methods to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, and Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions are violated. Numerical examples are also provided.

How to cite

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Argyros, Ioannis K., and Hilout, Said. "Convergence conditions for Secant-type methods." Czechoslovak Mathematical Journal 60.1 (2010): 253-272. <http://eudml.org/doc/38005>.

@article{Argyros2010,
abstract = {We provide new sufficient convergence conditions for the convergence of the secant-type methods to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, and Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions are violated. Numerical examples are also provided.},
author = {Argyros, Ioannis K., Hilout, Said},
journal = {Czechoslovak Mathematical Journal},
keywords = {secant method; Banach space; majorizing sequence; divided difference; Fréchet-derivative; secant method; Banach space; majorizing sequence; divided difference; Fréchet-derivative; nonlinear operator equation; convergence; error bounds; numerical examples; Lipschitz-type conditions},
language = {eng},
number = {1},
pages = {253-272},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Convergence conditions for Secant-type methods},
url = {http://eudml.org/doc/38005},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Argyros, Ioannis K.
AU - Hilout, Said
TI - Convergence conditions for Secant-type methods
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 1
SP - 253
EP - 272
AB - We provide new sufficient convergence conditions for the convergence of the secant-type methods to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, and Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions are violated. Numerical examples are also provided.
LA - eng
KW - secant method; Banach space; majorizing sequence; divided difference; Fréchet-derivative; secant method; Banach space; majorizing sequence; divided difference; Fréchet-derivative; nonlinear operator equation; convergence; error bounds; numerical examples; Lipschitz-type conditions
UR - http://eudml.org/doc/38005
ER -

References

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