论文标题
对称特征值问题的混合精度预处理方法
A mixed precision preconditioned Jacobi method for the symmetric eigenvalue problem
论文作者
论文摘要
特征值问题是科学计算中的一个基本问题。在本文中，我们首先给出误差分析的误差分析，或在浮点算术中对雅各比的方法进行扫描。然后，我们为对称特征值问题提出了一种混合精度预处理方法：我们首先通过低精度来计算真实对称矩阵对真实对称矩阵的特征值分解，并获得了特征向量的低级矩阵；然后，通过使用高精度修改的革兰氏式正交过程，获得了高精度正交矩阵，该基质被用作Jacobi方法的初始猜测。该方法的四舍五入误差分析是在某些条件下建立的。我们还为单一值问题提供了混合精度的预处理单侧雅各比方法，并讨论了相应的舍入误差分析。据报道，对CPU和GPU的数值实验说明了所提出的方法对原始Jacobi方法的效率。
The eigenvalue problem is a fundamental problem in scientific computing. In this paper, we first give the error analysis for a single step or sweep of Jacobi's method in floating point arithmetic. Then we propose a mixed precision preconditioned Jacobi method for the symmetric eigenvalue problem: We first compute the eigenvalue decomposition of a real symmetric matrix by an eigensolver at low precision and we obtain a low-precision matrix of eigenvectors; Then by using the high-precision modified Gram-Schmidt orthogonalization process, a high-precision orthogonal matrix is obtained, which is used as an initial guess for Jacobi's method. The rounding error analysis of the proposed method is established under some conditions. We also present a mixed precision preconditioned one-sided Jacobi method for the singular value problem and the corresponding rounding error analysis is discussed. Numerical experiments on CPUs and GPUs are reported to illustrate the efficiency of the proposed method over the original Jacobi method.
