Reliable Computing

, Volume 1, Issue 1, pp 9–14 | Cite as

Formulas for the width of interval products

  • Helmut Ratschek
  • Jon G. Rokne
Mathematical Research


Sharp formulas for the width of the product of intervals are derived which are simpler and more effective than the ones previously known. These formulas are useful in applications and they are appropriate tools for estimating the quality of interval evaluations. Proofs of such formulas will, in general, result in a number of different cases involving longwinded calculations. By utilizing certain functionals which are invariants of appropriate interval transformations the calculations are reduced to the ones required for a minimum number of cases.


Mathematical Modeling Computational Mathematic Industrial Mathematic Interval Evaluation Interval Product 
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.

Формулы для вычисления ширины интервальных произведений


Прелставлены точные формулы для вычнсления шнрнны произвеления нитервалов, являющиеся как более простыми, так н более зффективными, чем нспользовавшиеся до снх пор. Зти формулы могут быть полезны для приложений; кроме того, они прелставляют собой хорошнй ииструмент для оценки качества нитервального оиенивания. Доказтезательства таких формпл, вообше говоря, требппт разбора большого числа числа случаеВ и, соотсвенио, обБемных вычислений. Применение функкпионалов, инвариантных по отношений к определеннум интервальным преобразоаниям, позволило сократить сократить до минимума число слудаев иобБем вычислений.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Alefeld, G. and Herzberger, J.Einführung in die Intervallrechnung Bibliographisches Institut, Zürich, 1974.Google Scholar
  2. [2]
    Alefeld, G. and Herzberger, J.Introduction to interval computations. Academic Press, New York. 1983.Google Scholar
  3. [3]
    Alefeld, G., Gienger, A., and Potra, F.Efficient numerical validation of solutions of nonlinear systems. SIAM J. Numer. Anal.31 (1994), pp. 252–260.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Cornelius, H. andLohner, R. Computing the range of values of real functions with accuracy higher than second order. Computing33 (1984), pp. 331–347.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Kearfou, R. B. Abstract generalized bisection and a cost bound. Math. of Comput.49 (1987), pp. 187–202.Google Scholar
  6. [6]
    Kearfott, R. B. Preconditioners for the interval Gauss-Seidel method. SIAM J. Numer. Anal.27 (1990), pp. 804–822.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    Krawczyk, R. andNeumaier, A. An improved interval Newton operator J. Math. Anal. Appl.118 (1986), pp. 194–207.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Moore, R. E. The automatic analysis and control of error in digital computation based on the use of interval numbers.In: Rall, L. B. (ed.) “Error in Digital Computation”, Vol. 1, Proceedings of an advanced seminar. University of Madison, 1965, pp. 61–130.Google Scholar
  9. [9]
    Moore, R. E. Interval analysis. Prentice-Hall, Englewood Cliffs, N. J., 1966.Google Scholar
  10. [10]
    Olver, F. W. A new approach to error arithmetic. SIAM J. Number. Analysis2 (1982), pp. 368–392.Google Scholar
  11. [11]
    Olver, F. W. Further developments of rp and ap analysis. IMA J. Number. Analysis2 (1982), pp. 249–274.zbMATHMathSciNetGoogle Scholar
  12. [12]
    Rall, L. B. Representations of intervals and optimal error bounds. Math. of Computation41 (1983), pp. 219–227.zbMATHMathSciNetGoogle Scholar
  13. [13]
    Ratschek, H. Die binären Systeme der Intervallmathematik. Computing6 (1970), pp. 295–308.zbMATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    Ratschek, H. Die Subdistributivität in der Intervallmathematik. Z. Angew. Math. Mech.51 (1971), pp. 189–192.zbMATHMathSciNetGoogle Scholar
  15. [15]
    Ratschek, H. andRokne, J. The transistor modeling problem again. Microelectronics and Reliability32 (1992), pp. 1725–1740.CrossRefGoogle Scholar
  16. [16]
    Ratschek, H. andSauer, W. Linear interval equations. Computing28 (1982), pp. 105–115.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Ris, F. N. Interval analysis and applications to linear algebra. Ph. D. Thesis, Oxford University, 1972.Google Scholar
  18. [18]
    Rump, S. Personal communication. See [2] p. 18.Google Scholar

Copyright information

© H. Ratschek, J. Rokne 1995

Authors and Affiliations

  • Helmut Ratschek
    • 1
  • Jon G. Rokne
    • 2
  1. 1.Mathematisches Institut der Universität DüsseldorfGermany
  2. 2.Dept. of Computer ScienceThe University of CalgaryCanada

Personalised recommendations