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.
@article{bwmeta1.element.bwnjournal-article-zmv27i3p369bwm,
author = {Ioannis Argyros},
title = {The effect of rounding errors on a certain class of iterative methods},
journal = {Applicationes Mathematicae},
volume = {27},
year = {2000},
pages = {369-375},
zbl = {0998.65061},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-zmv27i3p369bwm}
}
Argyros, Ioannis. The effect of rounding errors on a certain class of iterative methods. Applicationes Mathematicae, Tome 27 (2000) pp. 369-375. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-zmv27i3p369bwm/
[000] [1] I. K. Argyros, On the convergence of some projection methods with perturbations, J. Comput. Appl. Math. 36 (1991), 255-258. | Zbl 0755.65056
[001] [2] I. K. Argyros, Concerning the radius of convergence of Newton's method and applications, Korean J. Comput. Appl. Math. 6 (1999), 451-462. | Zbl 0937.65065
[002] [3] I. K. Argyros and F. Szidarovszky, The Theory and Application of Iteration Methods, CRC Press, Boca Raton, FL, 1993. | Zbl 0844.65052
[003] [4] R. S. Dembo, S. C. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal. 19 (1982), 400-408. | Zbl 0478.65030
[004] [5] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982. | Zbl 0484.46003
[005] [6] T. J. Ypma, Numerical solution of systems of nonlinear algebraic equations, Ph.D. thesis, Oxford, 1982.
[006] [7] T. J. Ypma, Affine invariant convergence results for Newton's method, BIT 22 (1982), 108-118. | Zbl 0481.65027
[007] [8] T. J. Ypma, The effect of rounding errors on Newton-like methods, IMA J. Numer. Anal. 3 (1983), 109-118. | Zbl 0519.65026