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

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(a 1 — α < x/x < 1/(1 - a)), x = 0, x/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(xk, xk) < k \ d(x, x) is not true for 0 < k < 1. (ii) d(x, x) is not defined for complex x, x. In this paper the definition of d(¦, ¦) is replaced by d(x, x) = x — x /max(\x,\x). This definition is equivalent to the first in case x ¥= 0, x/x > 0, and is free of (ii). The inequality d(xk, xk) <\k\d(x, x) is replaced by the more universally valid inequality d(xk, xk) < k \ d(x, x)/(\ — 8), S = max(d(x, x),\k\ d(x, x)). The favorable properties of d(x, x) are preserved in the complex case. Moreover, its definition may be generalized to linear normed spaces by d(x, x) = Ilx — x II/max( 3c, x). 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. © 1982 American Mathematical Society.