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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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Keywords: Relative error, round-off error analysis, metric
Article copyright: © Copyright 1982 American Mathematical Society