On accurate floating-point summation

Abstract

cumulation of floating-point sums is considered on a computer which performs t -digit base β floating-point addition with exponents in the range — m to M . An algorithm is given for accurately summing n t -digit floating-point numbers. Each of these n numbers is split into q parts, forming q · n t -digit floating-point numbers. Each of these is then added to the appropriate one of η auxiliary t -digit accumulators. Finally, the accumulators are added together to yield the computed sum. In all, q · n + η - 1 t -digit floating-point additions are performed. Let ν = ⌈( M + m + 1)/( η + 1)⌉. If n ≤ (1/ q ) β ⌈(( q -1)/ q ) t ⌈- ν +1 (*), then the relative error in the computed sum is at most ⌈( t + 1)/ ν ⌉ β 1- t . Further, with an additional q + η - 1 t -digit additions, the computed sum can be corrected to full t -digit accuracy. For example, for the IBM/360 ( β = 16, t = 14, M = 63, m = 64), typical values for q and η are q = 2 and η = 32. In this case, (*) becomes n ≤ 1/2 × 16 4 = 32,768, and we have ⌈( t + 1)/ ν ⌉ β 1- t = 4 × 16 -13 .