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 ( x ¯ , x ) = min { α | 1 − α ⩽ x / x ¯ ⩽ 1 / ( 1 − α ) } d(\bar x,x) = \min \{ \alpha |1 - \alpha \leqslant x/\bar x \leqslant 1/(1 - \alpha )\} , x ¯ ≠ 0 \bar x e 0 , x / x ¯ > 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 ( x ¯ k , x k ) ⩽ | k | d ( x ¯ , x ) d({\bar x^k},{x^k}) \leqslant |k|d(\bar x,x) is not true for 0 > | k | > 1 0 > |k| > 1 . (ii) d ( x ¯ , x ) d(\bar x,x) is not defined for complex x ¯ , x \bar x,x . In this paper the definition of d ( ⋅ , ⋅ ) d( \cdot , \cdot ) is replaced by d ( x ¯ , x ) = | x ¯ − x | / max { | x ¯ | , | x | } d(\bar x,x) = |\bar x - x|/\max \{ |\bar x|,|x|\} . This definition is equivalent to the first in case x ¯ ≠ 0 \bar x e 0 , x / x ¯ > 0 x/\bar x > 0 , and is free of (ii). The inequality d ( x ¯ k , x k ) ⩽ | k | d ( x ¯ , x ) d({\bar x^k},{x^k}) \leqslant |k|d(\bar x,x) is replaced by the more universally valid inequality d ( x ¯ k , x k ) ⩽ | k | d ( x ¯ , x ) / ( 1 − δ ) , δ = max { d ( x ¯ , x ) , | k | d ( x ¯ , x ) } 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 ( ⋅ , ⋅ ) d( \cdot , \cdot ) are preserved in the complex case. Moreover, its definition may be generalized to linear normed spaces by d ( x ¯ , x ) = ‖ x ¯ − x ‖ / max { ‖ x ¯ ‖ , ‖ x ‖ } 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.