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

Applicationes Mathematicae (2000)

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

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topArgyros, 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

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