物理化学学报 >> 2008, Vol. 24 >> Issue (05): 823-826.doi: 10.3866/PKU.WHXB20080515

研究论文 上一篇    下一篇

大型广义特征值问题的部分特征值和特征向量的块迭代求解

赵小红; 陈飞武; 吴健; 周巧龙   

  1. 北京科技大学化学系, 北京 100083
  • 收稿日期:2007-11-22 修回日期:2008-02-28 发布日期:2008-05-05
  • 通讯作者: 陈飞武 E-mail:chenfeiwu@sas.ustb.edu.cn

Iterative Calculations of a Few Lowest Eigenvalues and Corresponding Eigenvectors of Large Generalized Eigenvalue Problem

ZHAO Xiao-Hong; CHEN Fei-Wu; WU Jian; ZHOU Qiao-Long   

  1. Department of Chemistry, University of Science and Technology Beijing, Beijing 100083, P. R. China
  • Received:2007-11-22 Revised:2008-02-28 Published:2008-05-05
  • Contact: CHEN Fei-Wu E-mail:chenfeiwu@sas.ustb.edu.cn

摘要: 将求解标准特征值问题的Davidson方法推广到求解大型广义特征值问题, 并给出了相应的块迭代算法. 经过理论分析和数值计算发现, 如果迭代过程不发散, 则块迭代算法经过有限次迭代一定收敛. 设矩阵的维数为n, 要求的特征值和相应特征向量的个数为k, 初始的子空间大小为r(r≥k),迭代次数为m,则它们之间满足关系n=r+km. 通过调节子空间大小, 就得到迭代次数m的正整数解.

关键词: 广义特征值问题, 特征值, 特征向量

Abstract: We extended the Davidson method, which was used to solve the standard eigenvalue problem, to solve the generalized eigenvalue problem and proposed the corresponding block iterative algorithm. Through theoretical analysis and numerical calculation, we found that the block iterative algorithm was doomed to converge after finite iterations if the process of iteration was not divergent. If the dimension of the matrix is n, the number of the eigenvalues and corresponding eigenvectors to be calculated is k, the size of the initial subspace is r(r≥k), the number of iteration is m, then they will fit in with the equation n=r+km. The positive integer root m could be obtained by regulating the size of the subspace.

Key words: Generalized eigenvalue problem, Eigenvalue, Eigenvector

MSC2000: 

  • O641