计算大规模矩阵最大最小奇异值和奇异向量的两个精化Lanczos算法  

TWO REFINED LANCZOS ALGORITHMS FOR COMPUTING THE LARGEST/SMALLEST SINGULAR VALUES AND ASSOCITED SINGULAR VECTORS OF A LARGE MATRIX

作　　者：贾仲孝[1] 张萍[2] 

机构地区：[1]清华大学数学科学系,北京100084 [2]大连理工大学应用数学系,大连116024

出　　处：《计算数学》2003年第3期293-304,共12页Mathematica Numerica Sinica

基　　金：国家重点基础研究专项基金(G19990328);高等学校骨干教师基金资助项目

摘　　要：1.引言 在科学工程计算中经常需要计算大规模矩阵的少数最大或最小的奇异值及其所对应的奇异子空间.This paper concerns the computation of a few large (or small) singular values and the associated singular vectors of an l×n matrix A. They are the square roots of the large (or the small) eigenvalues and the eigenvectors of the cross-product matrix AT A. So instead of solving the full SVD problem we solve the eigenproblem of the cross-product matrix using projection methods, and then revert it to the original one. For the cross-product matrix ATA, an explicitly restarted refined Lanczos algorithm and an implicitly restarted refined Lanczos algorithm are proposed. A convergence analysis is presented for the Ritz value, Ritz vector and refined Ritz vector. Numerical experiments show that two refined algorithms are far superior to their conventional counterparts.

关 键 词：大规模矩阵 奇异值 奇异向量 精化Lanczos算法 收敛性 显式重新启动 正交投影 RITZ值 奇异值分解 Ritz向量 

分 类 号：O241.6[理学—计算数学]