Complete Interval Arithmetic and Its Implementation on the Computer

  • Ulrich W. Kulisch
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5492)

Abstract

Let \(I\textit{I \kern-.55em R}\) be the set of closed and bounded intervals of real numbers. Arithmetic in \(I\textit{I \kern-.55em R}\) can be defined via the power set \(\textit{I \kern-.54em P}\textit{I \kern-.55em R}\) of real numbers. If divisors containing zero are excluded, arithmetic in \(I\textit{I \kern-.55em R}\) is an algebraically closed subset of the arithmetic in \(\textit{I \kern-.54em P}\textit{I \kern-.55em R}\), i.e., an operation in \(I\textit{I \kern-.55em R}\) performed in \(\textit{I \kern-.54em P}\textit{I \kern-.55em R}\) gives a result that is in \(I\textit{I \kern-.55em R}\). Arithmetic in \(\textit{I \kern-.54em P}\textit{I \kern-.55em R}\) also allows division by an interval that contains zero. Such division results in closed intervals of real numbers which, however, are no longer bounded. The union of the set \(I\textit{I \kern-.55em R}\) with these new intervals is denoted by \((I\textit{I \kern-.55em R})\). This paper shows that arithmetic operations can be extended to all elements of the set \((I\textit{I \kern-.55em R})\).

Let \(F \subset \textit{I \kern-.55em R}\) denote the set of floating-point numbers. On the computer, arithmetic in \((I\textit{I \kern-.55em R})\) is approximated by arithmetic in the subset (IF) of closed intervals with floating-point bounds. The usual exceptions of floating-point arithmetic like underflow, overflow, division by zero, or invalid operation do not occur in (IF).

Keywords

computer arithmetic floating-point arithmetic interval arithmetic arithmetic standards 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, New York (1983)MATHGoogle Scholar
  2. 2.
    Kahan, W.: A More Complete Interval Arithmetic. Lecture Notes prepared for a summer course at the University of Michigan, June 17-21 (1968)Google Scholar
  3. 3.
    Kirchner, R., Kulisch, U.: Hardware support for interval arithmetic. Reliable Computing 12(3), 225–237 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Kulisch, U.W.: Computer Arithmetic and Validity – Theory, Implementation and Applications. De Gruyter, Berlin (2008)CrossRefMATHGoogle Scholar
  5. 5.
    IFIPWG-IEEE754R: Letter of the IFIP WG 2.5 to the IEEE Computer Arithmetic Revision Group (2007)Google Scholar
  6. 6.
    Moore, R.E.: Interval Analysis. Prentice Hall Inc., Englewood Cliffs (1966)MATHGoogle Scholar
  7. 7.
    Moore, R.E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)CrossRefMATHGoogle Scholar
  8. 8.
    Ratz, D.: On Extended Interval Arithmetic and Inclusion Isotony, Institut für Angewandte Mathematik, Universität Karlsruhe (preprint, 1999)Google Scholar
  9. 9.
    Rump, S.M.: Kleine Fehlerschranken bei Matrixproblemen. Dissertation, Universität Karlsruhe (1980)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Ulrich W. Kulisch
    • 1
  1. 1.Institut für Angewandte und Numerische MathematikUniversität KarlsruheGermany

Personalised recommendations