An Efficient Summation Algorithm for the Accuracy, Convergence and Reproducibility of Parallel Numerical Methods

Farah Benmouhoub (1) , Pierre-Loïc Garoche (2) , Matthieu Martel (1)

1 UPVD - Université de Perpignan Via Domitia
2 ENAC - Ecole Nationale de l'Aviation Civile

Nowadays, parallel computing is ubiquitous in several application fields, both in engineering and science. The computations rely on the floating-point arithmetic specified by the IEEE754 Standard. In this context, an elementary brick of computation, used everywhere, is the sum of a sequence of numbers. This sum is subject to many numerical errors in floating-point arithmetic. To alleviate this issue, we have introduced a new parallel algorithm for summing a sequence of floating-point numbers. This algorithm which scales up easily with the number of processors, adds numbers of the same exponents first. In this article, our main contribution is an extensive analysis of its efficiency with respect to several properties: accuracy, convergence and reproducibility. In order to show the usefulness of our algorithm, we have chosen a set of representative numerical methods which are Simpson, Jacobi, LU factorization and the Iterated power method.

Mots clés: floating-point arithmetic; accurate summation; numerical accuracy; numerical methods; convergence; reproducibility

Domaines: Informatique et langage [cs.CL]; Génie logiciel [cs.SE]; Théorie et langage formel [cs.FL]

Farah Benmouhoub, Pierre-Loïc Garoche, Matthieu Martel. An Efficient Summation Algorithm for the Accuracy, Convergence and Reproducibility of Parallel Numerical Methods. Numerical Software Verification, Jul 2021, Los Angeles, United States. pp.165-181, ⟨10.1007/978-3-030-95561-8_10⟩. ⟨hal-03664170⟩