关键词: 无误差变换, 浮点数, 高精度, 求和, 点乘

Abstract: In large-scale and long-term numerical calculations, the cumulative effect of rounding errors in floating-point operations may lead to unreliable numerical results. Sum and dot multiplication are the most basic operations in floating-point numerical calculations. They are frequently called during large-scale scientific calculations, and the accuracy of their numerical results is very important. Oriented to the domestic Phytium processor, based on OpenBLAS, this paper uses error-free transformation technology to design efficient assembly kernel functions, and implements and optimizes the high-precision sum and dot product algorithms. Numerical experiments show that the accuracy of the numerical results of our high-precision algorithms is the same as that of the original algorithm under double working accuracy, which verifies the effectiveness of the algorithm. The running time of our algorithms is 1.57 and 1.76 times the running time of the original algorithms in the single-threaded case, and the efficiency is not significantly reduced while the accuracy is improved. In the case of multi-threading, it has almost the same running time as the original algorithms, which reflects the efficiency of our algorithms. Theoretical error analysis further ensures the reliability of our algorithms.

Key words: error-free transformation, float-point number, high-precision, summation, dot product ,  ,