Skip to main content
SpringerLink
Log in

Über mehrstufige Iterationsverfahren und die Lösung der Hammersteinschen Gleichung

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

The numerical solution of integral equations of Hammerstein type results in solving systems of equationsx+Kf(x)=0 with constant matrixK and diagonal mappingf. It is shown that two-step iterative methods are asymptotically optimal ifK is positive semi-definite andf is isotone and continuously differentiable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literatur

  1. Amann, H.: Über die Existenz und iterative Berechnung einer Lösung der Hammersteinschen Gleichung. Aequ. math.1, 242–266 (1968).

    Google Scholar 

  2. Bittner, L.: Über ein mehrstufiges Iterationsverfahren zur Lösung von linearen Gleichungen. Numer. Math.6, 161–180 (1964).

    Google Scholar 

  3. Hammerstein, A.: Nichtlineare Integralgleichungen nebst Anwendungen. Acta Math.54, 117–176 (1930).

    Google Scholar 

  4. Hyers, D. H.: Integral equations for nonlinear problems in partial differential equations. Eschienen in W. F. Ames (Hrsg.): Nonlinear partial differential New York-London: Academic Press 1967.

    Google Scholar 

  5. Krasnoselskii, M. A., Stecenko, V. A.: Some nonlinear problems with many solutions [Russ.]. Sibirski Mat.4, 120–137 (1963).

    Google Scholar 

  6. Meis, Th., Törnig, W.: Iterative Lösung nichtlinearer Gleichungssysteme und Diskretisierungsverfahren bei elliptischen Differentialgleichungen. Computing5, 281–294 (1970).

    Google Scholar 

  7. Minty, G. J.: On a “Monotonicity” method for the solution of nonlinear equations in Banach spaces. Proc. Nat. Acad. Sci. USA50, 1038–1041 (1963).

    Google Scholar 

  8. Niethammer, W.: Iterationsverfahren und allgemeine Euler-Verfahren. Math. Z.102, 288–317 (1967).

    Google Scholar 

  9. Niethammer, W.: Konvergenzbeschleunigung bei einstufigen Iterationsverfahren durch Summierungsmethoden. Erschienen in L. Collatz, H. Unger (Hrsg.): Iterationsverfahren, Numerische Mathematik, Approximationstheorie. ISNM Bd. 15. Basel-Stuttgart: Birkhäuser Verlag 1970.

    Google Scholar 

  10. Ortega, J. M., Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. New York-London: Academic Press 1970.

    Google Scholar 

  11. Ostrowski, A. M.: Solution of equations and systems of equations. New York-London: Academic Press 1966.

    Google Scholar 

  12. Polyak, B. T.: Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. and Math. Physics4, Nr. 5, 1–17 (1964).

    Google Scholar 

  13. Robert, F.: Normes vectorielles de vecteurs et de matrices. Rev. Franç. Traitement Information Chiffres7, 261–299 (1964).

    Google Scholar 

  14. Schempp, W.: Iterative solution of linear operator equations in Hilbert space and optimal Euler methods. Archiv d. Math.21, 390–395 (1970).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität Mannheim (WH), Postfach 2428, D-6800, Mannheim, L 5, 4, Deutschland

    Eckart Gekeler

Authors
  1. Eckart Gekeler
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gekeler, E. Über mehrstufige Iterationsverfahren und die Lösung der Hammersteinschen Gleichung. Numer. Math. 19, 351–360 (1972). https://doi.org/10.1007/BF01404882

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01404882

Search

Navigation