Computing

, Volume 15, Issue 3, pp 263–273 | Cite as

Round-off error in products

  • R. Goodman
  • A. Feldstein
Article

Abstract

LetA1 andA2 be floating point numbers represented in arbitrary base β and randomly chosen from a logarithmic distribution. Letr denote the round-off error
$$r = fl(A_1 * A_2 ) - (A_1 * A_2 )$$
where * is floating point multiplication and wherefl(A1*A2) denotes the normalizedN digit computer result of forming (A1*A2). This paper analyzes the mean and variance of both the actual round-off error and the fraction round-off error. This analysis relies upon sharp order estimates for the digit by digit deviation of logarithmically distributed numbers from uniformly distributed numbers. This completely resolves open questions of Kaneko and Liu and of Tsao. Also included is a generalization to arbitrary base (from binary) of an important round-off theorem of Henrici.

Rundungsfehler in Produkten

Zusammenfassung

SeienA1 undA2 zufällige Gleitkommazahlen zu einer beliebigen Basis β mit einer logarithmischen Verteilung. Seir der Rundungsfehler
$$r = fl(A_1 * A_2 ) - (A_1 * A_2 )$$
, wo * die Gleitkommamultiplikation bedeutet undfl(A1*A2) das normalisierteN-stellige Computerresultat für (A1*A2). Die Arbeit analysiert Mittelwert und Varianz des Rundungsfehlers sowohl des Ergebnisses wie auch dessen Mantisse. Die Analyse beruht auf scharfen Ordnungsabschätzungen der Abweichung pro Mantissenstelle zwischen logarithmisch verteilten Zahlen und gleichverteilten Zahlen. Offene Probleme von Kaneko und Liu und von Tsao werden vollständig gelöst. Ferner wird ein wichtiger Rundungsfehler-Satz von Henrici auf beliebige Basis (von der Basis 2) verallgemeinert.

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References

  1. [1]
    Dickson, L. E.: Introduction to the Theory of Numbers. University of Chicago Press, 1929.Google Scholar
  2. [2]
    Feldstein, A., Goodman, R.: Convergence Estimates for the Distribution of Trailing Digits. JACM, To appear.Google Scholar
  3. [3]
    Feldstein, A., Goodman, R.: Round-Off Error in Floating Point Addition. (To be submitted for publication.)Google Scholar
  4. [4]
    Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. New York: J. Wiley, 1962.Google Scholar
  5. [5]
    Kaneko, T., Liu, B.: On Local Round-Off Errors in Floating Point Arithmetic. JACM20, 391–398 (1973).CrossRefGoogle Scholar
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    Knuth, D.: The Art of Computer Programming, Vol. 2. Seminumerical Algorithms. Reading: Addison-Wesley, 1969.Google Scholar
  7. [7]
    Tsao, N.: On the distributions of Significant Digits and Round-Off Errors. CACM17, 269–271 (1974).Google Scholar

Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • R. Goodman
    • 1
  • A. Feldstein
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of MiamiCoral GablesU.S.A.
  2. 2.Department of MathematicsArizona State UniversityTempeU.S.A.
  3. 3.Mathematics Research CenterU.S. Naval Research LaboratoryWashington, D.C.U.S.A.

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