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

TL;DR

This paper introduces a new parallel summation algorithm for floating-point numbers that improves accuracy, convergence, and reproducibility in numerical computations across multiple processors.

Contribution

The paper presents a novel parallel summation algorithm that scales efficiently and enhances numerical accuracy and reproducibility in scientific computing.

Findings

Algorithm improves summation accuracy in parallel computations

Enhanced reproducibility of numerical results across different runs

Demonstrated effectiveness with methods like Simpson, Jacobi, LU, and power iteration

Abstract

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.