Numerische Mathematik

, Volume 33, Issue 4, pp 391–396

Unified error analysis for Newton-type methods

  • George J. Miel
Article

DOI: 10.1007/BF01399322

Cite this article as:
Miel, G.J. Numer. Math. (1979) 33: 391. doi:10.1007/BF01399322

Summary

Under proper hypotheses, Rheinboldt has shown that Newtonrelated iterates\(x_{n + 1} = x_n - {\cal D}\left( {x_n } \right)^{ - 1} Fx_n \), where some\({\cal D}\left( x \right)\) approximates the Fréchet derivative of an operatorF, converge to a rootx- ofF. Under these hypotheses, this paper establishes error bounds
$$\left\| {x^* - x_n } \right\|B_n \left\| {x_n - x_{n - 1} } \right\|C_n \left\| {x_1 - x_0 } \right\|, \left\| {x_n - \xi _n } \right\|s_n ,$$
whereBn,Cn,sn are constants, and where ξn; are perturbed iterates which take into account rounding errors occuring during actual computations.

Subject Classifications

AMS(MOS): 65J0565H10CR: 5.15

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • George J. Miel
    • 1
  1. 1.Department of Mathematical SciencesUniversity of Nevada, Las VegasLas VegasUSA