一种混合精精度雅各比方法,用于应对对称电子元值问题 (A mixed precision Jacobi method for the symmetric eigenvalue problem)

The eigenvalue problem is a fundamental problem in scientific computing. In this paper, we propose a mixed precision 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 modified Gram-Schmidt orthogonalization process to the low-precision eigenvector matrix in high precision, a high-precision orthogonal matrix is obtained, which is used as an initial guess for the Jacobi method. We give the rounding error analysis for the proposed method and the quadratic convergence of the proposed method is established under some sufficient conditions. We also present a mixed precision one-side Jacobi method for the singular value problem and the corresponding rounding error analysis and quadratic convergence are discussed. Numerical experiments on CPUs and GPUs are conducted to illustrate the efficiency of the proposed mixed precision Jacobi method over the original Jacobi method.

翻译：egenvaly 问题是科学计算中的一个根本问题。在本文中, 我们提出一种对称性egenvaly问题的混合精密 Jacobi 方法。我们首先用一个低精度的egensolt 来计算一个真实的对称矩阵的精度分解, 我们得到一个低精度的对称矩阵的低精度矩阵。然后, 我们用经过修改的Gram- Schmidt 或thotocal化过程来对低精度egenvict 矩阵进行精密的混合精度分析, 获得一种高精度或色度矩阵矩阵, 用作对 Jacobi 方法的初步猜测。我们给出了拟议方法的圆形错误分析, 并在一些足够的条件下确定了拟议方法的四面趋同。我们还提出了一种混合精度单面的叶科比方法, 并讨论了相应的圆性误差分析和四面融合。正在对CPUs和GPUs进行纽实验, 以说明提议的混合精度方法相对于原Jacobi 方法的效率。