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Interval Arithmetic with Containment Sets

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Abstract

The idea of containment sets (csets) is due to Walster and Hansen, and the theory is mainly due to the first author. Now that floating point computation with infinities is widely accepted, it is necessary to achieve the same for interval computation. The cset of a function over a set in its domain space is the set of all limits of normal function values over that set. Csets form a sound basis for defining a number of practical models for interval arithmetic that handle division by zero and related operations in an intuitive and consistent manner. Cset-based systems offer new opportunities for compiler optimization by rearranging interval expressions, achieving tighter bounds by reducing dependencies within the expression. This paper presents basic theory. It discusses division by zero, the case for a global flag to support ``loose'' evaluation, performance, and semantics. It presents numerical examples using a trial Matlab implementation.

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References

  • American National Standards Institute: IEEE standard for binary floating-point arithmetic, ANSI/IEEE Std. 754–1985. New York, 1985.

  • American National Standards Institute: Proposed revision of IEEE 754. http://754r.ucbtest.org.

  • Beeson, M., Wiedijk, F.: The meaning of infinity in calculus and computer algebra systems. In: Proc. Joint Int. Conf. on Artificial Intelligence, Automated Reasoning, and Symbolic Computation, pp. 246–258. Springer 2002.

  • Kahan, W. M.: A more complete interval arithmetic. Lecture Notes for a Summer Course at the University of Michigan, 1968.

  • Kearfott, R. B., Dawande, M., Du, K.-S., Hu, C.-Y.: Algorithm 737: INTLIB, a portable FORTRAN 77 interval standard function library. ACM Trans. Math. Software 20(4), 447–459, December 1994.

  • Knüppel, O.: BIAS – basic interval arithmetic subroutines. Technical Report 99.3, Berichte des Forschungsschwerpunktes Informations- und Kommunikationstechnik, Technische Universität Hamburg-Harburg, 1993.

  • Lambov, B.: Talk at Dagstuhl seminar 06021, January 2006. BRICS, Department of Computer Science, University of Aarhus, Denmark (barnie@brics.dk).

  • Lerch, M., Tischler, G., Wolff von Gudenberg, J.: FILIB++, a fast interval library supporting containment computations. ACM Trans. Math. Software 32(2), 299–324, June 2006. http://doi.acm.org/10.1145/1141885.1141893.

  • R. E. Moore (1966) Interval a analysis Prentice-Hall Englewood, N.J.

    Google Scholar 

  • Pryce, J. D.: An introduction to containment sets. Technical report DoIS/TR01/05, Department of Information Systems, Shrivenham Campus, Cranfield University, Swindon, UK, 2005.

  • Ratz, D.: Inclusion-isotone extended interval arithmetic – a toolbox update. Institut für Angewandte Mathematik, Univ. Karlsruhe (TH), Germany, 1996. Available at ftp://iamk4515.mathematik.uni-karlsruhe.de/pub/documents/reports.

  • Rump, S. M.: INTLAB Interval Toolbox, version 5.2. Tech. Univ. Hamburg-Harburg, Germany, 1999–2006. www.ti3.tu-harburg.de/intlab.ps.gz.

  • Sun Microsystems: Interval arithmetic in the Forte[tm] C++ compiler. www.sun.com/forte/cplusplus/interval/, 2000.

  • Sun Microsystems: Interval arithmetic in the Forte[tm] Fortran 95 compiler. www.sun.com/forte/ fortran/interval/, 2000.

  • Winter, L.: Postings to e-discussion group ``For discussions concerning the C++ standardization of intervals'', std-interval@compgeom.poly.edu, March–July 2006. Sharp Software, 5 Bellingrath Place, Nashua, New Hampshire, USA.

  • Wolff von Gudenberg, J.: Private communication at Dagstuhl seminar 06021, January 2006. Institut für Informatik III, Universität Würzburg, Würzburg, Germany. wolff@informatik.uni-wuerzburg.de.

  • Zemke, J.: b4m: A free interval arithmetic toolbox for MATLAB. Technical report, Technische Universitaet Hamburg-Harburg, Technische Informatik III, Eissendorfer Str. 38, 21071 Hamburg, Germany, 1998. Available at http://www.ti3.tu-harburg.de/ zemke/b4m/.

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Authors and Affiliations

  1. Department of Information Systems Shrivenham Campus, Cranfield University, Swindon, SN6 8LA, UK

    J. D. Pryce

  2. Electrical and Computer Engineering, Marquette University, P.O. Box 1881, Milwaukee, WI, 53201-1881, USA

    G. F. Corliss

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  1. J. D. Pryce
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  2. G. F. Corliss
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Correspondence to J. D. Pryce.

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Pryce, J.D., Corliss, G.F. Interval Arithmetic with Containment Sets. Computing 78, 251–276 (2006). https://doi.org/10.1007/s00607-006-0180-4

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  • DOI: https://doi.org/10.1007/s00607-006-0180-4

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