Advertisement

Computing

, Volume 33, Issue 1, pp 51–64 | Cite as

Difference Newton-like methods under weak continuity conditions

  • T. J. Ypma
Contributed Papers

Abstract

We give a local convergence analysis of difference Newton-like methods for solving the nonlinear equationF(x)=0, without assuming Lipschitz continuity of the derivativeF′. The results are obtained by regarding difference Newton-like methods as inexact Newton methods.

AMS Subject Classification

65H10 

Key words

Nonlinear equations difference approximation Newton-like methods 

Newton-ähnliche Verfahren vom Differenztyp unter schwachen Stetigkeitsvoraussetzungen

Zusammenfassung

Wir befassen uns mit einer Konvergenzanalyse für Newton-ähnliche Verfahren vom Differenztyp zur Lösung der nichtlinearen GleichungF(x)=0, ohne die Voraussetzung, daß die AbleitungF′ Lipschitz-stetig ist. Die Resultate entstehen daraus, daß wir Newton-ähnliche Verfahren vom Differenztyp als approximäre (inexact) Newton-Verfahren ansehen.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bus, J. C. P.: Numerical solution of systems of nonlinear equations. (Mathematical Centre Tract 122.) Amsterdam: Mathematisch Centrum 1980.Google Scholar
  2. [2]
    Dembo, R. S., Eisenstat, S. C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal.19, 400–408 (1982).Google Scholar
  3. [3]
    Dennis, J. E., Moré, J. J.: A characterization of superlinear convergence and its application to quasi-Newton methods. Math. Comput.28, 549–560 (1974).Google Scholar
  4. [4]
    Deuflhard, P., Heindl, G.: Affine invariant convergence theorems for Newton's method and extensions to related methods. SIAM J. Numer. Anal16, 1–10 (1979).Google Scholar
  5. [5]
    Gill, P. E., Murray, W., Saunders, M. A., Wright, M. H.: Computing forward-difference intervals for numerical optimization. SIAM J. Sci. Stat. Comp.4, 310–321 (1983).Google Scholar
  6. [6]
    Griewank, A., Toint, P.: Local convergence analysis for partitioned quasi-Newton updates. Numer. Math.39, 429–448 (1982).Google Scholar
  7. [7]
    Jankowska, J.: Theory of multivariate secant methods. SIAM J. Numer. Anal.16, 547–562 (1979).Google Scholar
  8. [8]
    Mönch, W.: Secant methods for sparse systems of nonlinear equations with a special structure. Computing30, 212–223 (1983).Google Scholar
  9. [9]
    Ortega, J. M., Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. New York: Academic Press 1970.Google Scholar
  10. [10]
    Rokne, J.: Newton's method under mild differentiability conditions with error analysis. Numer. Math.18, 401–412 (1972).Google Scholar
  11. [11]
    Schwetlick, H.: Numerische Lösung nichtlinearer Gleichungen. München: Oldenbourg 1979.Google Scholar
  12. [12]
    Ypma, T. J.: Local convergence of difference Newton-like methods. Math. Comput.41, 527–536 (1983).Google Scholar
  13. [13]
    Ypma, T. J.: Local convergence of inexact Newton methods. SIAM J. Numer. Anal. (to appear, 1984).Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • T. J. Ypma
    • 1
  1. 1.Department of Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa

Personalised recommendations