Toward Accurate and Fast Summation

Abstract

We introduce a new accurate summation algorithm based on the error-free summation into floating-point buckets. Our algorithm exploits ideas from Zhu and Hayes’ OnlineExactSum , but it uses a significantly smaller number of accumulators and has a better instruction-level parallelism. In the default setting, our implementation aaaSum returns a faithfully rounded floating-point approximation of the true sum. We also discuss possible modifications for the computation of reproducible, correctly rounded, and multiple precision floating-point approximations. The computational overhead for any of these modifications is kept comparably small. Numerical tests demonstrate that aaaSum performs well for very small till large problem sizes, independent of the condition number of the problem. We compare our algorithm with other accurate and high-precision summation approaches.

Floating-point arithmetic

“…More recently, Lange andRump (2017, 2019) introduced a general arithmetic framework for FP summation, which in particular makes it possible to go beyond (4.11) in at least three ways, by considering roundings other than to nearest, by incorporating the height of the summation tree into the error bounds, and by addressing the issue of the attainability of these bounds. Specifically, they gave the following analogue of (4.11) for more general roundings, such as faithful rounding (and thus also directed roundings), for which property (4.10) no longer holds.…”

“…Let us illustrate these algorithms in the case of the computation of the sum of floating-point numbers. We do not claim exhaustiveness here, as there is a large body of literature on summation algorithms; see for instance the works by Rump et al (2008), Demmel and Nguyen (2015), Higham (1993Higham ( , 2002, Blanchard et al (2020) and Lange (2022). We give a few examples below.…”