Abstract
The equality relation (more generally, the ordering relations) in floating point arithmetic is the exact translation of the mathematical equality relation. Because of the propagation of round-off errors, the floating point arithmetic is not the exact representation of the theoretical arithmetic which is continuous on the real numbers.
This leads to some incoherence when the equality concept is used in floating point arithmetic. A well known example is the detection of a zero element in the pivoting column and equation when applying Gaussian elimination, which is almost impossible in floating point arithmetic.
We shall begin by showing the inadequacy of the equality relation used in floating point arithmetic (we will call it floating point equality), and then introduce two new concepts: stochastic numbers and the equality relation between such numbers which will be called the stochastic equality. We will show how these concepts allow to recover the coherence between the arithmetic operators and the ordering relations that was missing in floating point computations.
Similar content being viewed by others
References
R. Alt, Floating point error propagation in iterative methods, Math. Comp. Simul. 30 (1988) 505–517.
J.-M. Chesneaux, Modélisation et conditions de validité de la méthode CESTAC, C.R. Acad. Sci., Paris, sér. 1, 307 (1988) 417–422.
J.-M. Chesneaux, Etude théorique et implémentation en ADA de la méthode CESTAC, Thesis, Université P. et M. Curie, Paris (1988).
J.-M. Chesneaux and J. Vignes, Sur la robustesse de la méthode CESTAC, C.R. Acad. Sci., Paris, sér. 1, 307 (1988) 855–860.
J.-M. Chesneaux, Study of the computing accuracy by using probabilistic approach, in:Contribution to Computer Arithmetic and Self Validating Numerical Methods, ed. C. Ulrich (J.C. Baltzer, 1990) pp. 19–30.
J.-M. Chesneaux, CADNA: An ADA tool for round-off error analysis and numerical debugging,Proc. Conf. ADA in Aerospace, Barcelona (December 1990).
J.-M. Chesneaux, Stochastic arithmetic properties, in:Computational and Applied Mathematics, I — Algorithms and Theory, ed. C. Brezinski (North-Holland, 1992) pp. 81–91.
J.-M. Chesneaux, C.A.D.N.A.: Contrôle des arrondis et débogage numérique en ADA, MASI Report, no. 92-31 (1992).
J.-M. Chesneaux, Descriptif d'utilisation du logiciel CADNA_F, MASI Report, no. 92.32 (1992).
J.-M. Chesneaux and J. Vignes, Les fondements de l'arithmétique stochastique, C.R. Acad. Sci., Paris, sér. 1, 315 (1992) 1435–1440.
J.-M. Chesneaux and J. Vignes, L'algorithme de Gauss en arithmétique stochastique. C.R. Acad. Sci., Paris, sér. II, 316 (1993) 171–176.
a. Feldstein and R. Goodman, Convergence estimates for the distribution of trailing digits, J. ACM 23 (1976) 287–297.
R.W. Hamming, On the distribution of numbers, Bell Syst. Tech. J., 9 (1970) 1609–1625.
T.E. Hull and J.R. Swenson, Test of probabilistic models for propagation of round-off errors, Commun. ACM 9 (1966).
D.E. Knuth,The Art of Computer Programming, vol. 2 (Addison-Wesley, Reading, 1969).
J. Vignes and M. La Porte,Algorithmes Numériques, Analyse et Mise en Oeuvre, vol. 1 and 2 (Editions Technip, Paris, 1980).
J. Vignes and M. La Porte, Error analysis in computing, in:Information Processing 74 (North-Holland, 1974).
J. Vignes, New methods for evaluating the validity of the results of mathematical computation, Math. Comp. Simul. 20 (1978) 221–249.
J. Vignes, Zéro mathématique et zéro informatique, La Vie des Sciences, Comptes Rendus, série générale, 4 (1987) 1–13.
J. Vignes, Optimal implementation of optimization methods and estimation of the accuracy of the results, I.S.N.M. 87 (1989) 219–227.
J. Vignes, Estimation de la précision des résultats de logiciels numériques, La Vie des Sciences, Comptes Rendus, série générale, 7 (1990) 93–115.
J. Vignes, A stochastic arithmetic for reliable scientific computation, Math. Comp. Simul. 35 (1993) 233–261.
Author information
Authors and Affiliations
Additional information
Communicated by J.Vignes
Rights and permissions
About this article
Cite this article
Chesneaux, JM. The equality relations in scientific computing. Numer Algor 7, 129–143 (1994). https://doi.org/10.1007/BF02140678
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02140678