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Mathematics of Computation

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Relative distance—an error measure in round-off error analysis
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by Abraham Ziv PDF
Math. Comp. 39 (1982), 563-569 Request permission

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 $d(\bar x,x) = \min \{ \alpha |1 - \alpha \leqslant x/\bar x \leqslant 1/(1 - \alpha )\}$, $\bar x \ne 0$, $x/\bar x > 0$ is even more favorable, through it seems to be inferior because of two drawbacks which are not shared by relative precision: (i) the inequality $d({\bar x^k},{x^k}) \leqslant |k|d(\bar x,x)$ is not true for $0 < |k| < 1$. (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 $d(\bar x,x) = |\bar x - x|/\max \{ |\bar x|,|x|\}$. This definition is equivalent to the first in case $\bar x \ne 0$, $x/\bar x > 0$, and is free of (ii). The inequality $d({\bar x^k},{x^k}) \leqslant |k|d(\bar x,x)$ is replaced by the more universally valid inequality $d({\bar x^k},{x^k}) \leqslant |k|d(\bar x,x)/(1 - \delta ),\delta = \max \{ d(\bar x,x),|k|d(\bar x,x)\}$. The favorable properties of $d( \cdot , \cdot )$ are preserved in the complex case. Moreover, its definition may be generalized to linear normed spaces by $d(\bar x,x) = \left \| {\bar x - x} \right \|/\max \{ \left \| {\bar x} \right \|,\left \| x \right \|\}$. 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.
References
  • 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, DOI 10.1137/0208005
  • A. N. Kolmogorov & S. C. Fomin, Elements of the Theory of Functions and Functional Analysis, Graylock Press, Rochester, New York, 1957.
  • F. W. J. Olver, A new approach to error arithmetic, SIAM J. Numer. Anal. 15 (1978), no. 2, 368–393. MR 483379, DOI 10.1137/0715024
  • Pat H. Sterbenz, Floating-point computation, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR 0349062
  • G. W. Stewart, Introduction to matrix computations, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR 0458818
  • J. H. Wilkinson, Rounding errors in algebraic processes, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0161456
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Math. Comp. 39 (1982), 563-569
  • MSC: Primary 65G05
  • DOI: https://doi.org/10.1090/S0025-5718-1982-0669649-2
  • MathSciNet review: 669649