Abstract
Floating-point arithmetic is prone to accuracy problems due to the round-off errors. The combination of the round-off errors and of the out of order execution of arithmetic operations due to the scheduling of parallel tasks, introduces additional numerical accuracy issues. In this article, we address the problem of improving the numerical accuracy and reproducibility of summation operators. We propose two efficient parallel algorithms for summing n floating-point numbers. The first objective of our algorithms is to obtain an accurate result without increasing the linear complexity of the naive algorithm. The second objective is to improve the reproducibility of the summations compared to those computed by the naive algorithm and this regardless of the number of processors used for the computations.
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References
ANSI/IEEE. IEEE Standard for Binary Floating-Point Arithmetic. SIAM (2008)
Bohlender, G.: Floating-point computation of functions with maximum accuracy. IEEE Trans. Comput. 26(7), 621–632 (1977)
De Luca, P., Galletti, A., Giunta, G., Marcellino, L., Raei, M.: Performance analysis of a multicore implementation for solving a two-dimensional inverse anomalous diffusion problem. In: Sergeyev, Y.D., Kvasov, D.E. (eds.) NUMTA 2019. LNCS, vol. 11973, pp. 109–121. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-39081-5_11
Demmel, J., Hida, Y.: Accurate floating point summation (2002)
Demmel, J., Hida, Y.: Accurate and efficient floating point summation. SIAM J. Sci. Comput. 25(4), 1214–1248 (2003)
Demmel, J., Nguyen, H.D.: Parallel reproducible summation. IEEE Trans. Comput. 64(7), 2060–2070 (2015)
Goldberg, D.: What every computer scientist should know about floating-point arithmetic. ACM Comput. Surv. 23(1), 5–48 (1991)
Graillat, S., Langlois, P., Louvet, N.: Algorithms for accurate, validated and fast polynomial evaluation. Jpn. J. Ind. Appl. Math. 26(2–3), 191–214 (2009)
Graillat, S., Ménissier-Morain, V.: Compensated Horner scheme in complex floating point arithmetic. In: Proceedings. 8th Conference on Real Numbers and Computers, pp. 133–146. Santiago de Compostela, Spain (2008)
Higham, N.: The accuracy of floating point summation. SIAM J. Sci. Comput. 14(4), 783–799 (1993)
Higham, N.: Accuracy and Stability of Numerical Algorithms. Society for Industrial and Applied Mathematics (1996)
Jeannerod, C.-P., Rump, S.: On relative errors of floating-point operations: optimal bounds and applications. Math. Comput. 87, 01 (2014)
Kahan, W.: A survey of error analysis. In: IFIP Congress (1971)
Langlois, P., Martel, M., Thévenoux, L.: Accuracy versus time: a case study with summation algorithms. In: Proceedings of the 4th International Workshop on Parallel and Symbolic Computation, PASCO 2010, pp. 121–130. Association for Computing Machinery, New York (2010)
Leuprecht, H., Oberaigner, W.: Parallel algorithms for the rounding exact summation of floating point numbers. Computing 28(2), 89–104 (1982)
Malcolm, M.: On accurate floating-point summation. Commun. ACM 14(11), 731–736 (1971)
Muller, J.M., et al.: Handbook of Floating-Point Arithmetic, Birkhäuser (2010)
Ogita, T., Rump, S., Oishi, S.: Accurate sum and dot product. SIAM J. Sci. Comput. 26(6), 1955–1988 (2005)
Pichat, M.: Correction d’une somme en arithmetique a virgule flottante. Numer. Math. 19(5), 400–406 (1972)
Rump, S.: Ultimately fast accurate summation. SIAM J. Sci. Comput. 31(5), 3466–3502 (2009)
Rump, S., Ogita, T.: Fast high precision summation. Nonlinear Theor. Its Appl. IEICE 1, 01 (2010)
Rump, S., Ogita, T., Oishi, S.: Accurate floating-point summation part I: faithful rounding. SIAM J. Sci. Compu. 31(1), 189–224 (2008)
Thévenoux, L., Langlois, P., Martel, M.: Automatic source-to-source error compensation of floating-point programs: code synthesis to optimize accuracy and time. Concurrency Comput. Pract. Experience 29(7), e3953 (2017)
Acknowledgments
This work was supported by a regional funding (Region Occitanie) and partially by project ANR-17-CE25-0018 FEANICSES.
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Benmouhoub, F., Garoche, PL., Martel, M. (2022). Parallel Accurate and Reproducible Summation. In: Arai, K. (eds) Intelligent Computing. Lecture Notes in Networks and Systems, vol 283. Springer, Cham. https://doi.org/10.1007/978-3-030-80119-9_21
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DOI: https://doi.org/10.1007/978-3-030-80119-9_21
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