基于多精度计算的新型、高效的高精度算法研究

Floating-point arithmetic is by far the most widely used way of representing real numbers in modern computers which is required in the fields of science, engineering and financial computing. Specifying floating-point formats require us to find compromises between speed, precision, dynamic representation range and storage volume.How to make full use of the advantages of half, single and double precision floating-point formats in numerical algorithms is the key problem that needs to be solved in high performance computing. The main focus of this project is on developing the new and efficient high precision algorithms based on multiple precision computations. There are two innovations in our research program. First, the project adopts a strategy of accelerating the algorithm with low precision. On the basis of the algorithm being stable and executable, the main part of the algorithm will be calculated using half precision or single precision as much as possible. Second, the traditional error-free transformation is only applicable to the floating-point operations of addition, subtraction and multiplication. This project extends the error-free transformation to more operations, and then designs more new low-complexity and high-precision compensation algorithms. In order to solve the typical numerical algorithms in the linear system, this project analyzes the numerical sensitivity of each part of the algorithms with round-off error and then exploits different precisions to achieve new faster algorithms with higher accuracy. This project takes the opportunity of hardware support for half precision floating-point operation to provide more efficient computing performance, and the research results will advance the theoretical and technical development of numerical calculation.

浮点运算广泛应用于科学、工程与金融计算领域，它是考虑了运算速度、精度、动态表示范围与存储体积来近似表示实数的折中方案，数值算法中如何充分利用半、单和双精度浮点格式的优势是高性能计算亟需解决的关键性问题。本项目集中研究基于多精度浮点格式的新型、高效的高精度算法，与传统方法的区别在于：i）强调低精度加速，在保持算法稳定、可执行的前提下，将算法的主体部分尽可能使用半精度或单精度来计算；ii）设计新型的低复杂度高精度补偿算法，突破了仅基于浮点数加、减、乘法无误差变换的传统补偿模式。本项目针对求解线性系统中的典型数值算法，分析算法各部分对舍入误差的数值敏感性，在保证精确性的前提下以尽可能使用低精度为原则，分别对其进行高精度和低精度计算，最终形成新型、高效的高精度数值算法。本项目以硬件支持半精度可提供更高效的计算性能为契机，研究成果将推进数值计算的理论与技术发展。

在大规模科学与工程计算中，由于浮点运算需要兼顾运算速度、精度、动态表示范围和存储体积，传统数值算法为了提高精度往往会牺牲一定的速度。基于多精度浮点格式设计新型、高效的高精度算法将有效解决这一问题。本项目主要运用无误差变换技术，结合高性能计算及应用需求，针对若干数值线性代数问题开展高精度算法研究。本项目研究累计发表学术论文11篇，其中SCI检索5篇，EI检索3篇，达到了申请书中预期的研究目标，按照研究计划实现了申请书中承诺的各项研究成果指标。项目取得的主要成果有：1）针对复系数多项式计算问题，设计了高精度Goertzel算法；2）针对多项式函数求根问题，设计了多精度Newton算法；3）针对矩阵正交分解问题，设计了高精度Gram-Schmidt正交化算法；4）针对求解特征值/奇异值问题，设计了高精度dqds算法与多精度dqds算法，并设计了精度可调的奇异值计算框架。这些高精度算法都较原算法有明显的精度提升，多精度算法都较原算法有明显的性能提升。