## 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, - 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**

*Elements of the Theory of Functions and Functional Analysis*, Graylock Press, Rochester, New York, 1957.

## 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