Advertisement

Proving Conjectures by Use of Interval Arithmetic

  • Andreas Frommer

Abstract

Machine interval arithmetic has become an important tool in computer assisted proofs in analysis. Usually, an interval arithmetic computation is just one of many ingredients in such a proof. The purpose of this contribution is to highlight and to summarize the role of interval arithmetic in some outstanding results obtained in computer assisted analysis. ‘Outstanding’ is defined through the observation that the importance of a mathematical result is at least to some extent indicated by the fact that it has been formulated as a ‘conjecture’ prior to its proof.

Keywords

Arithmetic Operation Interval Arithmetic Floating Point Floating Point Number Interval Vector 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Aigner and E. Behrends. Alles Mathematik. Vieweg, Braunschweig, 2000.Google Scholar
  2. 2.
    G. Alefeld, A. Gienger, and F. A. Potra. Efficient numerical validation of solutions of nonlinear systems. SIAM J. Numer. Anal., 31:252–260, 1994.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    G. Alefeld and J. Herzberger. Introduction to Interval Computations. Academic Press, New York, 1983.MATHGoogle Scholar
  4. 4.
    American National Standard Institute. IEEE standard for binary floating point arithmetic IEEE/ANSI 754-1985. Technical report, New York, 1985.Google Scholar
  5. 5.
    K. Appel and W. Haken. Every planar graph is four colorable, part I: Discharging. Illinois J. of Mathematics, 21:429–490, 1977.MathSciNetMATHGoogle Scholar
  6. 6.
    K. Appel and W. Haken. Every planar graph is four colorable, part 11: Reducibility. Illinois J. of Mathematics, 21:491–567, 1977.MathSciNetMATHGoogle Scholar
  7. 7.
    C. V. Boys. Soap Bubbles. Dover Publ. Inc., New York, 1959. (first edition 1911).Google Scholar
  8. 8.
    C. Camacho and L. de Figueiredo. The dynamics of the Jouanolou foliation on the complex projective 2-space. Ergodic Theory Dyn. Sys., to appear.Google Scholar
  9. 9.
    C. L. Fefferman and L. A. Seco. Interval arithmetic in quantum mechanics. In Kearfott, B. R. et al., editor, Applications olInterval Computations, volume 3 of Appl. Optim., pages 145-167. Kluwer, Dordrecht, 1995. Proceedings of an international workshop.Google Scholar
  10. 10.
    S. P. Ferguson. Sphere Packings, V. PhD thesis, Department of Mathematics, University of Michigan, 1997.Google Scholar
  11. 11.
    J. HaIes. The Kepler conjecture. Technical report, 1998. http://www.math.1sa.umich.edu/-hales/countdovn/.Google Scholar
  12. 12.
    J. Hass, M. Hutchings, and R. Schlafly. The double bubble conjecture. Electron. Res. Announc. Am. Math. Soc., 1:98–102, 1995.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    J. Hass and R. Schlafly. Double bubbles minimize. Ann. Math. (2), 151:459–515, 2000.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    N. J. Higham. Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia, 1996.MATHGoogle Scholar
  15. 15.
    R. B. Kearfott. Rigorous Global Search. Kluwer Academic Publishers, 1996.Google Scholar
  16. 16.
    U. Kulisch and W. Miranker. Computer Arithmetic in Theory and Practice. Academic Press, New York, 1981.MATHGoogle Scholar
  17. 17.
    O. E. Lanford III. A computer-assisted proof of the feigenbaum conjectures. Bull. Am. Math. Soc. New Ser., 6:427–434, 1982.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    R. Lohner. AWA: Software for the computation of guaranteed bounds for solutions of ordinary initial value problems. Technical report, Institut für Angewandte Mathematik, Universität Karlsruhe, 1994. Software available at ftp://ftp.iam.uni-karlsruhe.de/pub/awa/.Google Scholar
  19. 19.
    J. J. Moré. A collection of nonlinear model problems. In E. L. Allgower and K. Georg, editors, Computational Solution of Nonlinear Systems of Equations, volume 26 of Lectures in Applied Mathematics. American Mathematical Society, Providence, 1990.Google Scholar
  20. 20.
    A. Neumaier and T. Rage. Rigorous verification in discrete dynamical systems. Physica D, 67:327–346, 1993.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    J. Schwinger. Thomas-Fermi model: The second correction. Physical Review, A24:2253–2361, 1981.MathSciNetGoogle Scholar
  22. 22.
    S. Singh. Fermat’s Enigma: The Quest to Solve the World’s Greatest Mathematical Problem. Walker & Company, 1997.Google Scholar
  23. 23.
    J. H. Wilkinson. The Algebraic Eigenvalue Problem. Oxford University Press, Oxford, 1965.MATHGoogle Scholar
  24. 24.
    J. H. Wilkinson. Rounding Errors in Algebraic Processes. Dover, New York, 1994. Originally published as Notes on Applied Science No. 32, Her Majesty’s Stationery Office, London, 1963.Google Scholar

Copyright information

© Springer-Verlag Wien 2001

Authors and Affiliations

  • Andreas Frommer
    • 1
  1. 1.Universität WuppertalFachbereich MathematikWuppertalGermany

Personalised recommendations