基于多精度SIMD的线性方程组迭代细化求解

Iterative Refinement of Linear Equations Based on Multi-Precision SIMD

随着低精度和多精度计算的发展,多精度计算已经成为一种有效的加速计算方法。在求解线性方程组时,为了保证方程解的精度,大多采用双精度浮点数进行乘法运算,但是在现代处理器中,单精度浮点数的乘法吞吐率都是双精度浮点数的两倍。因此,提出一种使用多精度SIMD加速求解线性方程组的方法,该方法在对精度要求不高的LU分解步骤使用速度更快的单精度浮点乘法,在对精度要求严格的计算残差步骤使用双精度浮点乘法。通过实验表明,在取得具有相同精度的方程解时,使用多精度SIMD求解线性方程组可以带来大约60%的性能提升。

With the development of low-precision and multi-precision calculations,multi-precision calculations have become an effective method to speed up calculations.When solving linear equations,double-precision floating-point numbers are mostly used for multiplication to ensure the accuracy of the solution.However,in modern processors,the multiplication throughput of single-precision floating-point is twice that of double-precision floating-point.Therefore,this paper proposes a method to use multi-precision SIMD to speed up the solution of linear equations.This method uses faster single-precision floating-point multiplication in the LU decomposition step that does not require high precision and uses double-precision multiplication to calculate residuals that require high precision.Experiments show that when obtaining equation solutions with the same accuracy,using multi-precision SIMD to solve linear equations can achieve about 60%performance improvement.