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Mathematics of Computation

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Relative distance--an error measure in round-off error analysis


Author: Abraham Ziv
Journal: Math. Comp. 39 (1982), 563-569
MSC: Primary 65G05
MathSciNet review: 669649
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Abstract: Olver (SIAM J. Numer. Anal., v. 15, 1978, pp. 368-393) suggested relative precision as an attractive substitute for relative error in round-off error analysis. He remarked that in certain respects the error measure $ d(\bar x,x) = \min \{ \alpha \vert 1 - \alpha \leqslant x/\bar x \leqslant 1/(1 - \alpha )\} $, $ \bar x \ne 0$, $ x/\bar x > 0$ is even more favorable, through it seems to be inferior because of two drawbacks which are not shared by relative precision: (i) the inequality $ d({\bar x^k},{x^k}) \leqslant \vert k\vert d(\bar x,x)$ is not true for $ 0 < \vert k\vert < 1$. (ii) $ d(\bar x,x)$ is not defined for complex $ \bar x,x$. In this paper the definition of $ d( \cdot , \cdot )$ is replaced by $ d(\bar x,x) = \vert\bar x - x\vert/\max \{ \vert\bar x\vert,\vert x\vert\} $. This definition is equivalent to the first in case $ \bar x \ne 0$, $ x/\bar x > 0$, and is free of (ii). The inequality $ d({\bar x^k},{x^k}) \leqslant \vert k\vert d(\bar x,x)$ is replaced by the more universally valid inequality $ d({\bar x^k},{x^k}) \leqslant \vert k\vert d(\bar x,x)/(1 - \delta ),\delta = \max \{ d(\bar x,x),\vert k\vert d(\bar x,x)\} $. The favorable properties of $ d( \cdot , \cdot )$ are preserved in the complex case. Moreover, its definition may be generalized to linear normed spaces by $ d(\bar x,x) = \left\Vert {\bar x - x} \right\Vert/\max \{ \left\Vert {\bar x} \right\Vert,\left\Vert x \right\Vert\} $. Its properties in such spaces raise the possibility that with further investigation it might become the basis for error analysis in some vector, matrix, and function spaces.


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  • [1] Vijay B. Aggarwal and James W. Burgmeier, A round-off error model with applications to arithmetic expressions, SIAM J. Comput. 8 (1979), no. 1, 60–72. MR 522970, 10.1137/0208005
  • [2] A. N. Kolmogorov & S. C. Fomin, Elements of the Theory of Functions and Functional Analysis, Graylock Press, Rochester, New York, 1957.
  • [3] F. W. J. Olver, A new approach to error arithmetic, SIAM J. Numer. Anal. 15 (1978), no. 2, 368–393. MR 0483379
  • [4] Pat H. Sterbenz, Floating-point computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. Prentice-Hall Series in Automatic Computation. MR 0349062
  • [5] G. W. Stewart, Introduction to matrix computations, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Computer Science and Applied Mathematics. MR 0458818
  • [6] J. H. Wilkinson, Rounding errors in algebraic processes, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0161456

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1982-0669649-2
Keywords: Relative error, round-off error analysis, metric
Article copyright: © Copyright 1982 American Mathematical Society