Abstract
Computing an enclosure for the range of a rational function over an interval is one of the main goals of interval analysis. One way to obtain such an enclosure is to use interval arithmetic evaluation of a formula for the function. Often one would like to check how close the overestimation is to the correct range. Kreinovich, Nesterov, and Zheludeva (Reliable Computing 2(2) (1996)) suggested a new kind of twin arithmetic which produces a twin of intervals at the same time: the usual enclosure, i.e., an interval which is an overestimation for the range, and an interval which is contained in the range, i.e. an interval which is an underestimation for the range. We show in this paper that in certain cases the computed inner interval is much smaller than the correct range. For example, if the function has the same value in the two endpoints of the interval then the inner interval is either empty or contains only one point.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dimitrova, N. and Markov, S.: On the Interval-Arithmetic Presentation of the Range of a Class of Monotone Functions of Many Variables, in: Kaucher, E., Markov, S. M., and Mayer, G. (eds), Computer Arithmetic, Scientific Computation and Mathematical Modelling, IMACS, J.C. Baltzer AG, Scientific Publishing Co., 1991, pp. 213-228.
Gardeñes, E. and Trepat, A.: Fundamentals of SIGLA, an Interval Computing System over the Completed Set of Intervals, Computing 24 (1980), pp. 161-179.
Gardeñes, E., Trepat, A., and Janer. J. M.: SIGLA-PL1/(0). Development and Applications, in: Nickel, K. (ed.), Interval Mathematics 1980, Academic Press, New York, 1980, pp. 301-315.
Hansen, E. R.: The Centred Form, in: Hansen, E. R. (ed.), Topics in Interval Analysis, Clarendon Press, Oxford, 1969, pp. 102-106.
Hertling, P.: A Lower Bound for Range Enclosure in Interval Arithmetic, in: Chesnaux, J.-M., Jezequel, F., Lamotte, J.-L., and Vignes, J. (eds), Proceedings of the Third Real Numbers and Computers Conference, Paris, 1998, pp. 103-115.
Kaucher, E.: űber Eigenschaften und Anwendungsmöglichkeiten der erweiterten Intervallrechnung und des hyperbolischen Fastkörpers über R, in: Albrecht, R. and Kulisch, U. (eds), Grundlagen der Computer-Arithmetik, Computing Supplementum 1, Springer-Verlag, New York, 1977, pp. 81-94.
Kreinovich, V., Nesterov, V. M., and Zheludeva, N. A.: Interval Methods That Are Guaranteed to Underestimate (and the Resulting New Justification of Kaucher Arithmetic), Reliable Computing 2 (2) (1996), pp. 119-124.
Markov, S. M.: Extended Interval Arithmetic Involving Infinite Intervals, Mathematica Balkanica, New Series 6 (3) (1992), pp. 269-304.
Markov, S. M.: On Directed Interval Arithmetic and Its Applications, Journal of Universal Computer Science 1 (7) (1995), pp. 514-526.
Markov, S. M.: On the Presentation of Ranges of Monotone Functions Using Interval Arithmetic, in: Voschinin, A. (ed.), International Conference on Interval and Stochastic Methods in Science and Engineering (INTERVAL'92), volume 2, Moscow, 1992, pp. 66-74.
Nesterov, V. M.: How to Use Monotonicity-Type Information to Get Better Estimates of the Range of Real-Valued Functions, Interval Computations 4 (1993), pp. 3-12.
Nesterov, V. M.: Interval and Twin Arithmetics, Reliable Computing 3 (4) (1997), pp. 369-380.
Ratschek, H. and Rokne, J.: Computer Methods for the Range of Functions, Ellis Horwood, Chichester, 1984.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hertling, P. A Limitation for Underestimation Via Twin Arithmetic. Reliable Computing 7, 157–169 (2001). https://doi.org/10.1023/A:1011474231995
Issue Date:
DOI: https://doi.org/10.1023/A:1011474231995