An Efficient Summation Algorithm for the Accuracy, Convergence and Reproducibility of Parallel Numerical Methods
This work addresses numerical stability issues in parallel computing for engineering and science applications, but it is incremental as it builds on existing summation techniques.
The authors tackled the problem of numerical errors in parallel floating-point summation by introducing a new algorithm that groups numbers by exponent first, and demonstrated its efficiency through extensive analysis of accuracy, convergence, and reproducibility across methods like Simpson and Jacobi.
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.