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Reliable Computing

, Volume 3, Issue 1, pp 5–16 | Cite as

Inclusion of Zeros of Nowhere Differentiable n-Dimensional Functions

  • Siegfried M. Rump
Article

Abstract

A method is described for calculating verified error bounds for zeros of a system of nonlinear equations f(x) = 0 with continuous f : R" → R". We do not require existence of partial derivatives of f, and the function f may even be known only up to some finite precision ε > 0.

An inclusion may contain infinitely many zeros of f. An example of a continuous but nowhere differentiable function is given.

Keywords

Mathematical Modeling Computational Mathematic Partial Derivative Industrial Mathematic Nonlinear Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abramowitz, M. and Stegun, I.: Handbook of Mathematical Functions, Dover Publications, New York, 1968.Google Scholar
  2. 2.
    Griewank, A.: On Automatic Differentiation, in Mathematical Programming 88, Kluwer Academic Publishers, Boston, 1989.Google Scholar
  3. 3.
    Krawczyk, R. and Neumaier, A.: Interval Slopes for Rational Functions and Associated Centered Forms, SIAM J. Numer. Anal. 22 (3) (1985), pp. 604–616.Google Scholar
  4. 4.
    Neumaier, A.: Interval Methods for Systems of Equations, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1990.Google Scholar
  5. 5.
    Oliveira de, J. B. S.: A Backward Mode for Slope Evaluation and Sharper Inclusions for Interval Functions, Informatik III, TU Hamburg-Harburg, 1995.Google Scholar
  6. 6.
    PRO–MATLAB User's Guide, Vers. 32-SUN, The MathWorks Inc., 1992.Google Scholar
  7. 7.
    Rall, L. B.: Automatic Differentiation: Techniques and Applications, in Lecture Notes in Computer Science 120, Springer Verlag, Berlin–Heidelberg–New York, 1981.Google Scholar
  8. 8.
    Rump, S. M.: Expansion and Estimation of the Range of Nonlinear Functions, Math. Comp., to appear.Google Scholar
  9. 9.
    Rump, S.M.: Kleine Fehlerschranken bei Matrixproblemen, Dissertation, Universität Karlsruhe, 1980.Google Scholar
  10. 10.
    Rump, S.M.: VerificationMethods for Dense and Sparse Systems of Equations, in Herzberger, J. (ed.), Topics in Validated Computations—Studies in Computational Mathematics, Elsevier, Amsterdam, 1994, pp. 63–136.Google Scholar
  11. 11.
    Schmidt, J. W.: Die Regula-Falsi für Operatoren in Banachräumen, Angew. Math. Mech. 41 (1961), pp. 61–63.Google Scholar
  12. 12.
    Weierstraß, K.: Mathematische Werke, Band 2, Berlin, 1895.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Siegfried M. Rump
    • 1
  1. 1.Informatik IIITU Hamburg-HarburgHamburgGermany

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