AI 中文总结
提出一种计算矩阵平方根的混合精度算法，结合低精度舒尔分解与近似牛顿法迭代精化，对近似牛顿法做收敛性分析，针对对称正定矩阵在温和条件下可恢复全精度，x86-64架构实验显示常能减少执行时间。
AI 中文摘要
混合精度算法可通过利用日益强大的低精度硬件显著提高线性代数求解器的性能，同时通过迭代精化恢复工作精度。本文提出一种计算矩阵平方根的新型混合精度算法。该算法将低精度的舒尔分解方法与通过近似牛顿法进行的迭代精化相结合，并对近似牛顿法进行了详细的收敛性分析。对于对称正定矩阵的特殊情况，分析表明在温和条件下可恢复全工作精度。x86-64架构上的数值实验表明，该算法与固定工作精度的舒尔算法相比，经常能减少执行时间。
英文摘要
Mixed precision algorithms can significantly enhance the performance of linear algebra solvers by leveraging increasingly powerful low precision hardware while recovering working precision accuracy through, for example, iterative refinement. In this paper, we propose a novel mixed precision algorithm for computing matrix square roots. Our algorithm combines a Schur decomposition approach in low precision with iterative refinement performed through an approximate Newton method. We perform a detailed convergence analysis of the approximate Newton method. For the special case of symmetric positive definite matrices, this analysis implies that one can recover full working precision accuracy under mild conditions. Numerical experiments on x86-64 architectures indicate that our algorithm frequently reduces execution time compared with a fixed working-precision Schur algorithm.
Comments21 pages, 3 figures
