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Computing

, Volume 53, Issue 3–4, pp 323–335 | Cite as

Safe starting regions by fixed points and tightening

  • H. Hong
  • V. Stahl
Article

Abstract

In this paper, we present a method for finding safe starting regions for a given system of non-linear equations. The method is an improvement of the usual method which is based on the fixed point theorem. The improvement is obtained by enclosing the components of the equation system by univariante interval polynomials whose zero sets are found. This operation is called “tightening”. Preliminary experiments show that the tightening operation usually reduces the number of bisections, and thus the computing time. The reduction seems to become more dramatic when the number of variables increases.

Key words

Nonlinear equation systems interval arithmetic safe starting regions tightening 

Startintervalle mit garantierter Konvergenz durch Fixpunktiteration und Einengung

Zusammenfassung

In dieser Arbeit wird eine Methode zur Bestimmung von Startintervallen mit garantierter Konvergenz für ein gegebenes nichtlineares Gleichungssystem vorgestellt. Die Methode ist eine verbesserung der gebräuchlichen, auf dem Fixpunkt Theorem basierenden Methode. Die Verbesserung wird durch Einschließen der Komponenten des Gleichungssystems durch univariate Intervallpolynome, deren Lösungsmengen berechnet werden, erzielt. Diese Operation wird “Einengung” genannt. Erste experimentelle Untersuchungen zeigen, daß Einengung im allgemeinen die Anzahl der Intervallhalbierungen und somit die Rechenzeit reduziert. Die Reduktion scheint umso signifikanter, je höher die Anzahl der Variablen ist.

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References

  1. [1]
    Alefeld, G., Herzberger, J.: Introduction to interval computations. New York: Academic Press 1983.Google Scholar
  2. [2]
    Cleary, J. G.: Logical arithmetic. Future Comput. Syst. 125–149 (1987).Google Scholar
  3. [3]
    Hansen, E., Sengupta, S.: Bounding solutions of systems of equations using interval analysis. BIT21, 203–211 (1981).CrossRefMathSciNetGoogle Scholar
  4. [4]
    Krawczyk, R.: Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing4, 187–201 (1969).zbMATHMathSciNetGoogle Scholar
  5. [5]
    Mackworth, A. K.: Consistency in networks of relations. Art. Intell.8, 99–118 (1977).zbMATHGoogle Scholar
  6. [6]
    Moore, R. E.: Interval analysis. Englewood Cliffs: Prentice-Hall 1966.Google Scholar
  7. [7]
    Moore, R. E.: A test for existence of solution to nonlinear systems. ISAM J. Numer. Anal.14, 611–615 (1977).zbMATHGoogle Scholar
  8. [8]
    Moore, R. E.: A computational test for convergence of iterative methods for nonlinear systems. SIAM J. Numer. Anal.15, 1194–1196 (1978).zbMATHMathSciNetGoogle Scholar
  9. [9]
    Moore, R. E., Jones, S. T.: Safe starting regions for iterative methods. SIAM J. Numer. Anal.14, 1051–1065 (1977).MathSciNetGoogle Scholar
  10. [10]
    Moore, R. E., Qi, L.: A successive interval test for nonlinear systems. SIAM J. Numer. Anal.19, 845–850 (1982).CrossRefMathSciNetGoogle Scholar
  11. [11]
    Morgan, A.: Solving polynomial systems using continuation for engineering and scientific problems. Englewood Cliffs: Prentice-Hall 1987.Google Scholar
  12. [12]
    Neumaier, A.: Interval methods for systems of equations. Cambridge: Cambridge University Press 1990.Google Scholar
  13. [13]
    Older, W., Vellino, A.: Extending Prolog with constraint arithmetic on real intervals. In: Proceedings of the Eight Biennial Conference of the Canadian Society for Computational Studies of Intelligence, 1990.Google Scholar
  14. [14]
    Qi, L.: A note on the Moore test for nonlinear system. SIAM J. Numer. Anal.19, 851–857 (1982).CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    Rump, S. M.: Solving nonlinear systems with least significant bit accuracy. Computing29, 183–200 (1982).CrossRefzbMATHMathSciNetGoogle Scholar
  16. [16]
    Rump, S. M.: Solution of linear and nonlinear algebraic problems with sharp, guaranteed bounds. Computing [Suppl.]5, 147–168 (1984).zbMATHMathSciNetGoogle Scholar
  17. [17]
    Shearer, J. M., Wolfe, M. A.: Some computable existence, uniquness, and convergence tests for nonlinear systems. SIAM J. Numer. Anal.22, 1200–1207 (1985).CrossRefMathSciNetGoogle Scholar
  18. [18]
    Wolfe, M. A.: Interval methods for algebraic equations. In: Reliability in computing, pp. 229–248. London: Academic Press 1988.Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • H. Hong
    • 1
  • V. Stahl
    • 1
  1. 1.Research Institute for Symbolic ComputationJohannes Kepler UniversityLinzAustria

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