Multithreaded Reproducible Banded Matrix-Vector Multiplication

Reproducibility refers to getting bitwise identical floating point results from multiple runs of the same program, is an important basis for debugging or correctness checking in many codes. However the round-off error and non-associativity of floating point makes attaining reproducibility a challenge in large-scale, long-term parallel computing or solving ill conditioned problems. The $\mathrm{dgbmv}$ performs general banded matrix-vector multiplication for double precision, is the most basic Level-2 operation in BLAS. First, we designed a reproducible algorithm for banded matrix-vector multiplication $\mathrm{repro}\_\mathrm{dgbmv}$ based on the technique of error-free transformation. Then the error of the algorithm is analyzed. Second, the algorithm is parallelized into $\mathrm{repro}\_\mathrm{dgbmv}\_\mathrm{thread}$ on ARM and x86 platforms. The numerical test results verify that $\mathrm{repro}\_\mathrm{dgbmv}\_\mathrm{thread}$ is reproducible and has higher accuracy than ordinary $\mathrm{dgbmv}$. In numerical experiments on ARM platform, as the number of threads increases from 1 to 8, the run time of this algorithm is reduced by 5.2–7 times, while the run time of multithreaded $\mathrm{dgbmv}$ is only reduced by 2.2–3.8 times. In numerical experiments on x86 platform, as the number of threads increases from 1 to 15, the run time of this algorithm is reduced by 7.7–10.6 times, while the run time of multithreaded $\mathrm{dgbmv}$ is only reduced by 4.2–6.8 times.