低计算精度下实对称矩阵的特征值分解  

EIGENVALUE DECOMPOSITION OF REAL SYMMETRIC MATRICES WITH LOW COMPUTATIONAL PRECISION

作　　者：胡乐宇 蔡邢菊[1] Hu Leyu;Cai Xingju(School of Mathematical Sciences,Nanjing Normal University,Nanjing 210023)

机构地区：[1]南京师范大学数学科学学院,南京210023

出　　处：《高等学校计算数学学报》2021年第2期117-133,共17页Numerical Mathematics A Journal of Chinese Universities

摘　　要：1引言1.1背景简介设A∈R~(n×n)为n阶实对称矩阵,矩阵A的特征值分解是找正交矩阵U∈R^(n×n),使得A=UAU^(T),(1.1)其中U^(T)指U的转置,Λ为对角矩阵,且Λ=diag(λ_(1),λ_(2),…,λ_(n)),其中λ_(i),i=1,…,n是矩阵A的特征值.矩阵A的奇异值分解为A=UEU^(H),(1.2)其中,U∈C^(n×n)是酉矩阵.Dominant Eigenvalue Decomposition(EVD)plays an important role in scientific computing.When it is restricted in low computational precision,dominant EVD becomes a difficult task,even for the low dimensional case.In this paper,we consider the dominant eigenvalue decomposition of real symmetric matrix under low computational precision.We show that by making appropriate improvements based on the original EVD method for large-scale matrices,we can successfully complete the task.We also provide an iterative refinement strategy that yields high-precision results with low computational precision.We provide computational results to illustrate the assertion.

关 键 词：singular value decomposition low precision computational precision iterative refinement Lanczos method inverse iteration 

分 类 号：O241.6[理学—计算数学]