Numerische Mathematik

, Volume 29, Issue 1, pp 45–58

Eine Methode zur berechnung sämtlicher Lösungen von Polynomgleichungssystemen

  • Franz-Josef Drexler
Article

A method for computing all solutions of systems of polynomial equations

Summary

In this paper a numerical method is given to compute all solutions of systemsT ofn polynomial equations inn unknowns on the only premises that the sets of solutions of these systems are finite. The method employed is that of “embedding”, i.e. the systemT is embedded in a set of systems which are successively solved, starting with one having solutions easily to compute and proceding toT in a finite series of steps. An estimation of the number of steps necessary is given. The practicability of the method is proved for all systemsT. Numerical examples and results are contained.

Subject Classifications

AMS (MOS): 65H10 CR: 5.15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literatur

  1. 1.
    Brown, K.M., Gearhart, W.B.: Deflation techniques for the calculation of further solutions of a nonlinear system. Numer. Math.16, 334–342 (1970/71)Google Scholar
  2. 2.
    Collatz, L.: Funktionalanalysis und numerische Mathematik. Berlin-Heidelberg-New York: Springer 1968Google Scholar
  3. 3.
    Feilmeier, M., Lory, P., Scheuring, H., Wacker, H.J.: “Einbettung”, DFG Berichte 1, 2, 3 (1973, 1974, 1975)Google Scholar
  4. 4.
    Gröbner, W.: Moderne algebraische Geometrie. Die Idealtheoretischen Grundlagen. Wien-Innsbruck: Springer 1949Google Scholar
  5. 5.
    Gröbner, W.: Algebraische Geometrie, Bd. II. Mannheim: Bibliographisches Institut 1970Google Scholar
  6. 6.
    Kantorowitsch, L.W., Akilow, G.P.: Funktionalanalysis in normierten Räumen. Berlin: Akademie Verlag 1964Google Scholar
  7. 7.
    Krasnosel'skii, M.A.: Approximate solution of operator equations. Groningen: Wolters-Noordoff 1972Google Scholar
  8. 8.
    Lugowski, H., Weinert, H.J.: Grundzüge der Algebra, Bd. III. Basel: Pfalz-Verlag 1967Google Scholar
  9. 9.
    Sather, D.: Branching of solutions of an equation in Hilbert space. Arch. Rational Mech. Anal.36, 47–64 (1970)CrossRefGoogle Scholar
  10. 10.
    Tutschke, W.: Grundlagen der Funktionentheorie, Berlin: 1967Google Scholar
  11. 11.
    Wacker, H.: Nichtlineare Homotopien zur Konstruktion von Startlösugen für Iterationsverfahren (Numerische Lösung nichtlinearer partieller Differential-und Integrodifferentialgleichungen). Lecture Notes in Mathematics, Vol. 267, pp. 51–67. Berlin-Heidelberg-New York: Springer 1972Google Scholar
  12. 12.
    Wainberg, M.M., Trenogin, W.A.: Theorie der Lösungsverzweigung bei nichtlinearen Gleichungen. Berlin: Akademie-Verlag 1973Google Scholar
  13. 13.
    Young, D.M.: Iterative solution of large linear systems. New York-London: Academic Press 1971Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • Franz-Josef Drexler
    • 1
  1. 1.München 60Germany (Fed. Rep.)

Personalised recommendations