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BIT Numerical Mathematics

Erschienen in:

01.12.2013

Some issues related to double rounding

verfasst von: Érik Martin-Dorel, Guillaume Melquiond, Jean-Michel Muller

Erschienen in: BIT Numerical Mathematics | Ausgabe 4/2013

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Abstract

Double rounding is a phenomenon that may occur when different floating-point precisions are available on the same system. Although double rounding is, in general, innocuous, it may change the behavior of some useful small floating-point algorithms. We analyze the potential influence of double rounding on the Fast2Sum and 2Sum algorithms, on some summation algorithms, and Veltkamp’s splitting.

Vorheriger Artikel Numerical integration based on trivariate C 2 quartic spline quasi-interpolants

Nächster Artikel Sparse tensor edge elements

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The FMA instruction evaluates expressions of the form xy+z with one final rounding only.

Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development. Coq’Art: The Calculus of Inductive Constructions. Texts in Theoretical Computer Science. Springer, Berlin (2004) CrossRef

Boldo, S.: Pitfalls of a full floating-point proof: example on the formal proof of the Veltkamp/Dekker algorithms. In: Furbach, U., Shankar, N. (eds.) Proceedings of the 3rd International Joint Conference on Automated Reasoning. Lecture Notes in Computer Science, vol. 4130, pp. 52–66 (2006) CrossRef

Boldo, S., Daumas, M.: Representable correcting terms for possibly underflowing floating point operations. In: Bajard, J.C., Schulte, M. (eds.) Proceedings of the 16th Symposium on Computer Arithmetic, pp. 79–86. IEEE Comput. Soc. Press, Los Alamitos (2003)

Boldo, S., Daumas, M., Moreau-Finot, C., Théry, L.: Computer validated proofs of a toolset for adaptable arithmetic. Tech. rep, École Normale Supérieure de Lyon (2001). Available at http://arxiv.org/pdf/cs.MS/0107025

Boldo, S., Melquiond, G.: Emulation of FMA and correctly rounded sums: proved algorithms using rounding to odd. IEEE Trans. Comput. 57(4), 462–471 (2008) MathSciNetCrossRef

Cornea, M., Harrison, J., Anderson, C., Tang, P.T.P., Schneider, E., Gvozdev, E.: A software implementation of the IEEE 754R decimal floating-point arithmetic using the binary encoding format. IEEE Trans. Comput. 58(2), 148–162 (2009) MathSciNetCrossRef

Dekker, T.J.: A floating-point technique for extending the available precision. Numer. Math. 18(3), 224–242 (1971) MathSciNetCrossRefMATH

Figueroa, S.A.: When is double rounding innocuous? ACM SIGNUM Newsl. 30(3) (1995)

Figueroa, S.A.: A rigorous framework for fully supporting the IEEE standard for floating-point arithmetic in high-level programming languages. Ph.D. thesis, Department of Computer Science, New York University (2000)

10.

Goldberg, D.: What every computer scientist should know about floating-point arithmetic. ACM Comput. Surv. 23(1), 5–47 (1991). An edited reprint is available at http://www.physics.ohio-state.edu/~dws/grouplinks/floating_point_math.pdf from Sun’s Numerical Computation Guide; it contains an addendum ”Differences among IEEE 754 implementations”, also available at http://www.validlab.com/goldberg/addendum.html CrossRef

11.

Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002) CrossRefMATH

12.

IEEE Computer Society: IEEE standard for floating-point arithmetic. IEEE Standard 754-2008 (2008). Available at http://ieeexplore.ieee.org/servlet/opac?punumber=4610933

13.

International Organization for Standardization: Programming languages—C. ISO/IEC Standard 9899:1999, Geneva, Switzerland (1999)

14.

Kahan, W.: Pracniques: further remarks on reducing truncation errors. Commun. ACM 8(1), 40 (1965) CrossRef

15.

Kahan, W.: Lecture notes on the status of IEEE-754 (1996). PDF file accessible at http://www.cs.berkeley.edu/~wkahan/ieee754status/IEEE754.PDF

16.

Knuth, D.: The Art of Computer Programming vol. 2, 3rd edn. Addison-Wesley, Reading (1998)

17.

Møller, O.: Quasi double-precision in floating-point addition. BIT Numer. Math. 5, 37–50 (1965) CrossRefMATH

18.

Monniaux, D.: The pitfalls of verifying floating-point computations. ACM TOPLAS 30(3), 12 (2008) CrossRef

19.

Muller, J.M., Brisebarre, N., de Dinechin, F., Jeannerod, C.P., Lefèvre, V., Melquiond, G., Revol, N., Stehlé, D., Torres, S.: Handbook of Floating-Point Arithmetic. Birkhäuser, Boston (2010) CrossRefMATH

20.

Neumaier, A.: Rundungsfehleranalyse einiger Verfahren zur Summation endlicher Summen. Z. Angew. Math. Mech. 54, 39–51 (1974) (in German) MathSciNetCrossRefMATH

21.

Nievergelt, Y.: Scalar fused multiply-add instructions produce floating-point matrix arithmetic provably accurate to the penultimate digit. ACM Trans. Math. Softw. 29(1), 27–48 (2003) MathSciNetCrossRefMATH

22.

Ogita, T., Rump, S.M., Oishi, S.: Accurate sum and dot product. SIAM J. Sci. Comput. 26(6), 1955–1988 (2005) MathSciNetCrossRefMATH

23.

Pichat, M.: Correction d’une somme en arithmétique à virgule flottante. Numer. Math. 19, 400–406 (1972) (in French) MathSciNetCrossRefMATH

24.

Priest, D.M.: Algorithms for arbitrary precision floating point arithmetic. In: Kornerup, P., Matula, D.W. (eds.) Proceedings of the 10th IEEE Symposium on Computer Arithmetic (Arith-10), pp. 132–144. IEEE Comput. Soc. Press, Los Alamitos (1991)

25.

Priest, D.M.: On properties of floating-point arithmetics: numerical stability and the cost of accurate computations. Ph.D. thesis, University of California at Berkeley (1992)

26.

Rump, S.M., Ogita, T., Oishi, S.: Accurate floating-point summation, part I: faithful rounding. SIAM J. Sci. Comput. 31(1), 189–224 (2008) MathSciNetCrossRefMATH

27.

Rump, S.M., Ogita, T., Oishi, S.: Accurate floating-point summation, part II: sign, K-fold faithful and rounding to nearest. SIAM J. Sci. Comput. 31(2), 1269–1302 (2008) MathSciNetCrossRefMATH

28.

Shewchuk, J.R.: Adaptive precision floating-point arithmetic and fast robust geometric predicates. Discrete Comput. Geom. 18, 305–363 (1997) MathSciNetCrossRefMATH

29.

Sterbenz, P.H.: Floating-Point Computation. Prentice-Hall, Englewood Cliffs (1974)

Titel: Some issues related to double rounding
verfasst von: Érik Martin-Dorel
Guillaume Melquiond
Jean-Michel Muller
Publikationsdatum: 01.12.2013
Verlag: Springer Netherlands
Erschienen in: BIT Numerical Mathematics / Ausgabe 4/2013
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-013-0436-2

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