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Kaj Madsen - DTU Informatics•Head of Department


DTU Informatics Interval Homepage

Interval Arithmetic and Analysis

When we use a computer to make calculations involving real numbers, we have to use the finite set of floating-point numbers that the hardware makes available. In such a situation there are two main choices for the approximation of a real number by the floating-point number system: to use one floating-point number, or to use two. Interval Arithmetic is based on the latter choice. Whenever an operation on reals is specified, the corresponding operation on their intervals is executed. By ensuring that any rounding is outward, it is guaranteed that the computed interval contains the mathematically correct result.

Thus interval arithmetic was introduced as an automatic method to keep track of the rounding errors made in computer calculations, but it has also been widely used to find upper and lower bounds for the effect caused by truncation errors and by errors (or uncertainties) in the input data to a numerical calculation. Another major application is rigorous global optimization.

The science of making these ideas practical is a research programme called Interval Analysis. It was introduced in the book "Interval Analysis" by Ramon Moore, Prentice-Hall, Englewood Cliffs, NJ, 1966.

An introduction to Interval Analysis and interval global optimization is given in the DTU course



General information

  • Interval Computations, is a site at the University of Texas, El Paso. This site contains a lot of information on interval analysis.
  • Interval Bibliography. A bibliography of interval analysis papers collected by Nelson Beebe, Center for Scientific Computing, University of Utah, USA.


  • INTLAB - an interval toolbox for Matlab 5. INTLAB is entirely written in Matlab except for 3 routines for switching the rounding mode downwards, upwards and to nearest.
    Free for non-commercial use. Technical University of Hamburg-Harburg.
  • INTLAB at the G-Databar of DTU.
  • PC version of PROFIL/BIAS.
  • Sun's Fortran 95 compiler. The early access (beta test) version of Sun's Fortran 95 compiler with support for intrinsic interval data types is now available for test purposes. The early access is scheduled to run for 16 weeks from mid November 1999.
  • FADBAD/TADIFF. An automatic differentiation package.
    Authors: Ole Stauning, DTU Informatics, and Claus Bendsen, UNI-C.
Email to Hans Bruun Nielsen