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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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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


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.
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Math. Comp. 39 (1982), 563-569
  • MSC: Primary 65G05
  • DOI:
  • MathSciNet review: 669649