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New sufficient convergence conditions for the secant method

Ioannis K. Argyros

Czechoslovak Mathematical Journal (2005)

  • Volume: 55, Issue: 1, page 175-187
  • ISSN: 0011-4642

Abstract

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We provide new sufficient conditions for the convergence of the secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses “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 the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated.

How to cite

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Argyros, Ioannis K.. "New sufficient convergence conditions for the secant method." Czechoslovak Mathematical Journal 55.1 (2005): 175-187. <http://eudml.org/doc/30936>.

@article{Argyros2005,
abstract = {We provide new sufficient conditions for the convergence of the secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses “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 the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated.},
author = {Argyros, Ioannis K.},
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},
language = {eng},
number = {1},
pages = {175-187},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {New sufficient convergence conditions for the secant method},
url = {http://eudml.org/doc/30936},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Argyros, Ioannis K.
TI - New sufficient convergence conditions for the secant method
JO - Czechoslovak Mathematical Journal
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 1
SP - 175
EP - 187
AB - We provide new sufficient conditions for the convergence of the secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses “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 the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated.
LA - eng
KW - secant method; Banach space; majorizing sequence; divided difference; Fréchet-derivative; secant method; Banach space; majorizing sequence; divided difference; Fréchet-derivative
UR - http://eudml.org/doc/30936
ER -

References

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  8. 10.1016/S0377-0427(99)00116-8, J.  Comput. Appl. Math. 115 (2000), 245–254. (2000) MR1747223DOI10.1016/S0377-0427(99)00116-8
  9. Functional Analysis in Normed Spaces, Pergamon Press, Oxford Press, , 1982. (1982) MR0664597
  10. Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. (1970) MR0273810
  11. Sharp error bounds for a class of Newton-like methods, Libertas Mathematica 5 (1985), 71–84. (1985) Zbl0581.47050MR0816258
  12. 10.1007/BF01400355, Numer. Math. 51 (1987), 545–557. (1987) Zbl0633.65049MR0910864DOI10.1007/BF01400355

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