Skip to main content
SpringerLink
Log in

Safe bounds for the solutions of nonlinear problems using a parallel multisplitting method

Sichere Schranken für die Lösung nichtlinearer Probleme mit einem parallelen Multisplitting-Verfahren

  • Published:
Computing Aims and scope Submit manuscript

Abstract

For some systems of nonlinear equationsF(x)=0 we derive an algorithm which iteratively constructs tight lower and upper bounds for the zeros ofF. The algorithm is based on a multisplitting of certain matrices thus showing a natural parallelism. We prove criteria for the convergence of the bounds towards the zeros and we investigate the speed of convergence.

Zusammenfassung

Für gewisse Systeme nichtlinearer GleichungenF(x)=0 entwicklen wir ein Verfahren, welches iterativ enge untere und obere Schranken für die Nullstellen vonF berechnet. Das Verfahren beruht auf einem Multisplitting für bestimmte Matrizen und weist so in natürlcher Weise Parallelität auf. Wir geben Kriterien für die Konvergenz der Schranken gegen die Nullstellen an und untersuchen die Konvergenzgeschwindigkeit.

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

References

  1. Alefeld, G.: Über Zerlegungen bei nichtlinearen Abbildungen. Z. Angew. Math. Mech.52, 233–238 (1972).

    Google Scholar 

  2. Alefeld, G.: Intervallanalytische Methoden bei nichtlinearen Gleichungen. In: Überblicke Mathematik 1979, pp.63–78. Mannheim: Bibliographisches Institut 1979.

    Google Scholar 

  3. Alefeld, G.: On the convergence of some interval-arithmetic modifications of Newton's method. SIAM J. Numer. Anal.21, 363–372 (1984).

    Article  Google Scholar 

  4. Alefeld, G., Herzberger, J.: Introduction to Interval Computations. New York: Academic Press 1983.

    Google Scholar 

  5. Barth, W., Nuding, E.: Optimale Lösung von Intervallgleichungssystemen. Computing12, 117–125 (1974).

    Google Scholar 

  6. Bru, R., Elsner, L., Neumann, M.: Models of parallel chaotic iteration methods. Linear Algebra Appl.103, 175–192 (1988).

    Article  Google Scholar 

  7. Frommer, A.: Parallel nonlinear multisplitting methods. To appear in Numer. Math.

  8. Frommer, A., Mayer, G.: Parallel interval multisplittings. To appear in Numer. Math.

  9. Frommer, A., Mayer, G.: Convergence of relaxed parallel multisplitting methods. Linear Algebra Appl., in press.

  10. Frommer, A., Mayer, G.: On theR-order of Newton-like methods for enclosing solutions of nonlinear equations. To appear in SIAM J. Numer. Anal.

  11. Hockney, R. W., Jesshope, C. R.: Parallel Computers. Bristol: Adam Hilger 1981.

    Google Scholar 

  12. Mayer, G.: Reguläre Zerlegungen und der Satz von Stein und Rosenberg für Intervallmatrizen. Habilitationsschrift, Karlsruhe (1986).

  13. Mayer, G.: On Newton-like methods to enclose solutions of nonlinear equations. Apl. Mat.34, 67–84 (1989).

    Google Scholar 

  14. Neumaier, A.: Interval iteration for zeros of systems of equations. BIT25, 256–273 (1985).

    Article  Google Scholar 

  15. Neumann, M., Plemmons, R. J.: Convergence of parallel multisplitting methods forM-matrices. Linear Algebra Appl.88/89, 559–573 (1987).

    Article  Google Scholar 

  16. O'Leary, D. P., White, R. E.: Multi-splittings of matrices and parallel solution of linear systems. SIAM J. Alg. Disc. Meth.6, 630–640 (1985).

    Google Scholar 

  17. Ortega, J. M., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press 1970.

    Google Scholar 

  18. Rall, L. B.: Computational Solution of Nonlinear Operator Equations. New York: Krieger Publishing Company 1979.

    Google Scholar 

  19. Schneider, H.: Theorems onM-splittings of a singularM-matrix which depend on graph structure. Linear Algebra Appl.58, 407–424 (1984).

    Article  Google Scholar 

  20. Schwandt, H.: Schnelle, fast global konvergente Verfahren für die Fünf-Punkt-Diskretisierung der Poissongleichung mit Dirichletschen Randbedingungen auf Rechteckgebieten. Dissertation, Techn. Univ. Berlin (1981).

  21. Sherman, A. H.: On Newton-iterative methods for the solution of systems of nonlinear equations. SIAM J. Numer. Anal.15, 755–771 (1978).

    Article  Google Scholar 

  22. White, R. E.: A nonlinear parallel algorithm with applications to the Stefan problem. SIAM J. Numer. Anal.23, 639–652 (1986).

    Article  Google Scholar 

  23. White, R. E.: Parallel algorithms for nonlinear problems. SIAM J. Alg. Disc. Meth.7, 137–149 (1986).

    Google Scholar 

  24. Varga, R. S.: Matrix Iterative Analysis. Englewood Cliffs, N. J.: Prentice-Hall 1962.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Karlsruhe, Kaiserstrasse 12, D-7500, Karlsruhe, Federal Republic of Germany

    A. Frommer & G. Mayer

Authors
  1. A. Frommer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Mayer
    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

Frommer, A., Mayer, G. Safe bounds for the solutions of nonlinear problems using a parallel multisplitting method. Computing 42, 171–186 (1989). https://doi.org/10.1007/BF02239746

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02239746

AMS (MOS) Subject Classifications

Key words

Search

Navigation