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The exact dot product as basic tool for long interval arithmetic

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Abstract

Computing with guarantees is based on two arithmetical features. One is fixed (double) precision interval arithmetic. The other one is dynamic precision interval arithmetic, here also called long interval arithmetic. The basic tool to achieve high speed dynamic precision arithmetic for real and interval data is an exact multiply and accumulate operation and with it an exact dot product. Pipelining allows to compute it at the same high speed as vector operations on conventional vector processors. Long interval arithmetic fully benefits from such high speed. Exactitude brings very high accuracy, and thereby stability into computation. This document, which has been incorporated into the draft standard for interval arithmetic being developed by IEEE P1788, specifies the implementation of an exact multiply and accumulate operation.

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References

  1. Lohner R (1993) Interval arithmetic in staggered correction Format. In: Adams E, Kulisch U (eds) Scientific computing with automatic result verification. Academic Press, London, pp 301–321

    Chapter  Google Scholar 

  2. Blomquist F, Hofschuster W, Krämer W et al (2009) A modified staggered correction arithmetic with enhanced accuracy and very wide exponent range. In: Cuyt A (eds) Numerical validation in current hardware architectures. Lecture notes in computer science LNCS, vol 5492. Springer, Berlin, pp 41–67

    Chapter  Google Scholar 

  3. Klatte R, Kulisch U, Lawo C, Rauch M, Wiethoff A (1993) C–XSC, A C++ class library for extended scientific computing. Springer, Berlin. See also: http://www.math.uni-wuppertal.de/~xsc/resp; http://www.xsc.de/

  4. Kulisch U (2008) Computer arithmetic and validity—theory, implementation, and applications. de Gruyter, Berlin

  5. The IFIP WG-IEEE 754R letter, dated 4 September 2007

  6. The IFIP WG-IEEE P1788 letter, dated 9 September 2009

  7. IEEE Floating-Point Arithmetic Standard (2008) 754

  8. Neumaier A. Vienna Proposal for Interval Standardization

  9. IBM System/370 RPQ (1984) High accuracy arithmetic. SA 22-7093-0, IBM Deutschland GmbH (Department 3282, Schönaicher Strasse 220, D-71032 Böblingen)

  10. ACRITH-XSC: IBM High Accuracy Arithmetic, Extended Scientific Computation. Version 1, Release 1, General Information (GC33-6461-01), Reference (SC33-6462-00), Sample Programs (SC33-6463-00), How To Use (SC33-6464-00), Syntax Diagrams (SC33-6466-00). IBM Deutschland GmbH (Schönaicher Strasse 220, D-71032 Böblingen), 1990

  11. Rump SM (1980) Kleine Fehlerschranken bei Matrixproblemen. Dissertation, Universität Karlsruhe

  12. Oishi S, Tanabe K, Ogita T, Rump SM (2007) Convergence of Rump’s method for inverting arbitrarily ill-conditioned matrices. J Comput Appl Math 205: 533–544

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Angewandte und Numerische Mathematik, Universität Karlsruhe, Englerstraße 2, 76128, Karlsruhe, Germany

    Ulrich Kulisch

  2. California Institute of Technology, Jet Propulsion Laboratory, 4800 Oak Grove Drive, Mail Stop 183-701, Pasadena, CA, 91109-8099, USA

    Van Snyder

Authors
  1. Ulrich Kulisch
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  2. Van Snyder
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Correspondence to Ulrich Kulisch.

Additional information

Communicated by G. Alefeld

This document is an identical copy of a motion accepted by the international standards committee IEEE P1788 on interval arithmetic. Its contents will be published in a few years when the development of the standard is completed. The new floating-point arithmetic standard IEEE 754 (available since 2008) provides a function for accumulation of the dot product of two vectors with no accuracy requirement. Manufacturers who support the dot product by hardware should be aware that IEEE P1788 requires the exact result. Once a weak solution has been put into hardware it may be difficult to change it later. It is therefore important to bring this decision to public attention as soon as possible. Actually the simplest and fastest way for computing a dot product is to compute it exactly [4].

Part of this work supported by Institut für Angewandte und Numerische Mathematik, Universität Karlsruhe.

Part of this work supported by NASA at Jet Propulsion Laboratory, California Institute of Technology.

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Kulisch, U., Snyder, V. The exact dot product as basic tool for long interval arithmetic. Computing 91, 307–313 (2011). https://doi.org/10.1007/s00607-010-0127-7

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  • DOI: https://doi.org/10.1007/s00607-010-0127-7

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