Year: 2025
Type: article
Abstract: . Floating-point summation is a fundamental task at the heart of many scientific computing applications. When the sum is very ill conditioned, computing it accurately can become challenging. One can employ distillation methods, which consist in transforming an ill-conditioned sum into an equivalent but well-conditioned one. However, distillation is a very expensive process. In this article, we propose Condense & Distill, a new distillation method that relies on a preprocessing step that we call condensation, because it transforms the original sum into a far smaller sum, which can then be distilled inexpensively. This condensation step exploits a new, key observation that floating-point addition is exact when the addends have both the same exponent and the same least significant bit. Condense & Distill thus requires accessing the exponent field of the summands. Compared with state-of-the-art summation methods with the same requirement such as the Demmel–Hida method [J. Demmel and Y. Hida , SIAM J. Sci. Comput., 25 (2003) , pp. 1214 –1248], Condense & Distill presents the significant benefit of running entirely in the working precision, with no need for extra precision. At the same time, it preserves the main advantages of the Demmel–Hida method compared with other methods, in particular those based on error-free transformations such as AccSum [S. M. Rump, T. Ogita, and S. Oishi , SIAM J. Sci. Comput., 31 (2008), pp. 189 –224]: namely, its cost is independent of the conditioning, and it exhibits near perfect parallel scaling. We present numerical experiments that confirm that Condense & Distill can reliably and efficiently distill large ill-conditioned sums and performs favorably compared with other state-of-the-art summation methods.
DOI: 10.1137/24m1647217
Authors: Stef Graillat, Theo Mary
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4407068355