The effect of rounding errors on a certain class of iterative methods

Ioannis Argyros

Applicationes Mathematicae (2000)

  • Volume: 27, Issue: 3, page 369-375
  • ISSN: 1233-7234

Abstract

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In this study we are concerned with the problem of approximating a solution of a nonlinear equation in Banach space using Newton-like methods. Due to rounding errors the sequence of iterates generated on a computer differs from the sequence produced in theory. Using Lipschitz-type hypotheses on the mth Fréchet derivative (m ≥ 2 an integer) instead of the first one, we provide sufficient convergence conditions for the inexact Newton-like method that is actually generated on the computer. Moreover, we show that the ratio of convergence improves under our conditions. Furthermore, we provide a wider choice of initial guesses than before. Finally, a numerical example is provided to show that our results compare favorably with earlier ones.

How to cite

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Argyros, Ioannis. "The effect of rounding errors on a certain class of iterative methods." Applicationes Mathematicae 27.3 (2000): 369-375. <http://eudml.org/doc/219279>.

@article{Argyros2000,
abstract = {In this study we are concerned with the problem of approximating a solution of a nonlinear equation in Banach space using Newton-like methods. Due to rounding errors the sequence of iterates generated on a computer differs from the sequence produced in theory. Using Lipschitz-type hypotheses on the mth Fréchet derivative (m ≥ 2 an integer) instead of the first one, we provide sufficient convergence conditions for the inexact Newton-like method that is actually generated on the computer. Moreover, we show that the ratio of convergence improves under our conditions. Furthermore, we provide a wider choice of initial guesses than before. Finally, a numerical example is provided to show that our results compare favorably with earlier ones.},
author = {Argyros, Ioannis},
journal = {Applicationes Mathematicae},
keywords = {Fréchet derivative; Lipschitz conditions; Newton-like method; inexact Newton-like method; Banach space; iterative methods; nonlinear operator equation; Newton method; rounding errors; convergence; numerical examples},
language = {eng},
number = {3},
pages = {369-375},
title = {The effect of rounding errors on a certain class of iterative methods},
url = {http://eudml.org/doc/219279},
volume = {27},
year = {2000},
}

TY - JOUR
AU - Argyros, Ioannis
TI - The effect of rounding errors on a certain class of iterative methods
JO - Applicationes Mathematicae
PY - 2000
VL - 27
IS - 3
SP - 369
EP - 375
AB - In this study we are concerned with the problem of approximating a solution of a nonlinear equation in Banach space using Newton-like methods. Due to rounding errors the sequence of iterates generated on a computer differs from the sequence produced in theory. Using Lipschitz-type hypotheses on the mth Fréchet derivative (m ≥ 2 an integer) instead of the first one, we provide sufficient convergence conditions for the inexact Newton-like method that is actually generated on the computer. Moreover, we show that the ratio of convergence improves under our conditions. Furthermore, we provide a wider choice of initial guesses than before. Finally, a numerical example is provided to show that our results compare favorably with earlier ones.
LA - eng
KW - Fréchet derivative; Lipschitz conditions; Newton-like method; inexact Newton-like method; Banach space; iterative methods; nonlinear operator equation; Newton method; rounding errors; convergence; numerical examples
UR - http://eudml.org/doc/219279
ER -

References

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  1. [1] I. K. Argyros, On the convergence of some projection methods with perturbations, J. Comput. Appl. Math. 36 (1991), 255-258. Zbl0755.65056
  2. [2] I. K. Argyros, Concerning the radius of convergence of Newton's method and applications, Korean J. Comput. Appl. Math. 6 (1999), 451-462. Zbl0937.65065
  3. [3] I. K. Argyros and F. Szidarovszky, The Theory and Application of Iteration Methods, CRC Press, Boca Raton, FL, 1993. Zbl0844.65052
  4. [4] R. S. Dembo, S. C. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal. 19 (1982), 400-408. Zbl0478.65030
  5. [5] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982. Zbl0484.46003
  6. [6] T. J. Ypma, Numerical solution of systems of nonlinear algebraic equations, Ph.D. thesis, Oxford, 1982. 
  7. [7] T. J. Ypma, Affine invariant convergence results for Newton's method, BIT 22 (1982), 108-118. Zbl0481.65027
  8. [8] T. J. Ypma, The effect of rounding errors on Newton-like methods, IMA J. Numer. Anal. 3 (1983), 109-118. Zbl0519.65026

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