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.

翻译：特征值问题是科学计算中的基本问题。本文首先分析了浮点运算下单步或单轮雅可比法的误差。随后，我们提出了一种求解对称特征值问题的混合精度预处理雅可比法：首先通过低精度特征求解器计算实对称矩阵的特征值分解，获得低精度特征向量矩阵；接着采用高精度修正格拉姆-施密特正交化过程，得到高精度正交矩阵作为雅可比法的初始猜测。在特定条件下，我们建立了该方法的舍入误差分析理论。针对奇异值问题，我们还提出了混合精度预处理单边雅可比法，并讨论了相应的舍入误差分析。在CPU和GPU上的数值实验表明，该方法相较于原始雅可比法具有更高的计算效率。