Sharp estimates for perturbation errors in summations

Error Bounds for Computer Arithmetics

“…As exploited in [24], this Theorem 4.4 has a number of consequences. Denote by s " floatpexpressionq the result of an expression with each operation replaced by the corresponding floating-point operation in some nearest rounding.…”

“…Theorem 4.4: [24] Let an α-ary tree T with root r and height h be given. For an inner node j of T , denote the set of leaves of the corresponding subtree by L j and the set of all its inner nodes including j by N j .…”

“…Corollary 4.5: [24] For an IEEE-754 p-digit base-β floatingpoint arithmetic, let s be the result of a floating-point summation of p 1 , . .…”

“…By means of explicit examples (cf. [24]) it is easy to see that some restriction on n is mandatory for (36). But although it was common belief that this is the worst case, it could not be proved.…”

“…The upper bound on n is almost sharp. For rounding ties away from zero it cannot be replaced by the next larger integer; for rounding ties to even, p ě 3 mantissa digits and even β the upper bound cannot be increased by 2`β 2 , see [24]. OPEN PROBLEM 3.…”