Numerische Mathematik

, Volume 33, Issue 4, pp 391–396

Unified error analysis for Newton-type methods

  • George J. Miel
Article

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): 65J05 65H10 CR: 5.15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Gragg, W.B., Tapia, R.A.: Optimal error bounds for the Newton-Kantorovich theorem. SIAM J. Numer. Anal.11, 10–13 (1974)CrossRefGoogle Scholar
  2. 2.
    Lancaster, P.: Error analysis for the Newton-Raphson method. Numer. Math.9, 55–68 (1966)Google Scholar
  3. 3.
    Miel, G.J.: The Kantorovich theorem with optimal error bounds. Amer. Math. Monthly86, 212–215 (1979)Google Scholar
  4. 4.
    Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. New York: Academic Press 1970Google Scholar
  5. 5.
    Rockne, J.: Newton's method under mild differentiability conditions with error analysis. Numer. Math.,18, 401–412 (1972)Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

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

Personalised recommendations