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

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.