解大规模非对称矩阵特征问题的精化Arnoldi方法的一种变形  

A VARIANT OF THE REFINED ARNOLDI METHOD FOR LARGE SCALE MATRIX EIGENPROBLEMS

作　　者：贾仲孝[1] 陈桂芝[2] 

机构地区：[1]清华大学数学科学系,100084 [2]南京大学数学系,210008

出　　处：《数值计算与计算机应用》2003年第2期101-110,共10页Journal on Numerical Methods and Computer Applications

基　　金：国家重点基础研究发展规划项目(973)(G1999032805);高等学校骨干教师基金资助

摘　　要：The refined Arnoldi method proposed by Jia is used for computing some eigen-pairs of large matrices. In contrast to the Arnoldi method, the fundamental dif-ference is that the refined method seeks certain refined Ritz vectors, which aredifferent from the Ritz vectors obtained by the Arnoldi method, from a projection space with minimal residuals to approximate the desired eigenvectors. In com-parison with the Ritz vectors, the refined Ritz vectors are guaranteed to converge theoretically and can converge much faster numerically. In this paper we propose to replace the Ritz values, obtained by the Arnoldi method with respect to a Krylovsubspace, by the ones obtained with respect to the subspace spanned by the refined Ritz vectors. We discuss how to compute these new approximations cheaply and reliably. Theoretical error bounds between the original Ritz values and the new Ritz values are established. Finally, we present a variant of the refined Arnoldi al-gorithm for an augmented Krylov subspace and discuss restarting issue. Numerical results confirm efficiency of the new algorithm.The refined Arnoldi method proposed by Jia is used for computing some eigen-pairs of large matrices. In contrast to the Arnoldi method, the fundamental difference is that the refined method seeks certain refined Ritz vectors, which are different from the Ritz vectors obtained by the Arnoldi method, from a projection space with minimal residuals to approximate the desired eigenvectors. In comparison with the Ritz vectors, the refined Ritz vectors are guaranteed to converge theoretically and can converge much faster numerically. In this paper we propose to replace the Ritz values, obtained by the Arnoldi method with respect to a Krylov subspace, by the ones obtained with respect to the subspace spanned by the refined Ritz vectors. We discuss how to compute these new approximations cheaply and reliably. Theoretical error bounds between the original Ritz values and the new Ritz values are established. Finally, we present a variant of the refined Arnoldi algorithm for an augmented Krylov subspace and discuss restarting issue. Numerical results confirm efficiency of the new algorithm.

关 键 词：大规模非对称矩阵 特征问题 精化Arnoldi方法 Ritz向量 RITZ值 精化投影方法 近似特征值 

分 类 号：O241.6[理学—计算数学]