Using Compensated Summation Algorithms to Improve the Accuracy of a Parallel BVP Solver

This paper investigates the effectiveness of compensated summation techniques in improving the numerical accuracy of algorithms involving the summation of long sequences of floating‐point numbers, such as boundary value problems for ordinary differential equations. Three parallel programming frameworks—OpenACC, CUDA, and SYCL—are evaluated in terms of their performance, numerical accuracy, and cross‐platform applicability. Experiments are conducted on both GPU and CPU architectures. Two data layouts, namely column‐wise and row‐wise formats, are introduced to improve memory access efficiency through better coalescing. Three summation methods are considered: ordinary summation, Kahan compensated summation, and Neumaier compensated summation. Parallel implementations are developed using both single and double precision arithmetic. The results are compared with their sequential counterparts and selected library functions. The experimental results demonstrate that parallel implementations with compensated summation techniques significantly improve numerical accuracy while maintaining a negligible impact on the execution time of BVP parallel solvers. Furthermore, the proposed approach provides results comparable to library functions, while outperforming the evaluated reference implementations for the largest considered problem sizes.