计算大型实对称特征问题的 Lanczos-QR 算法  

Lanczos QR Methods for Computation of a Large Real Symmetric Eigenvalue Problems

作　　者：汪晓虹[1] 周传荣[1] 

机构地区：[1]南京航空航天大学振动工程研究所

出　　处：《南京航空航天大学学报》1998年第5期501-506,共6页Journal of Nanjing University of Aeronautics & Astronautics

摘　　要：为了计算大型实对称特征值问题Ｋｘ＝λＭｘ的少数低阶特征值对，本文给出Ｌａｎｃｚｏｓ－ＱＲ迭代方法。首先，给定初始迭代向量ｖ１，作ｍ步Ｌａｎｃｚｏｓ分解：ＫＶｍ＝ＭＶｍＴｍ＋ｈｍｅｍＴ。取Ｔｍ的ｄ个最大特征值为移位量，对Ｔｍ进行ｄ步带原点位移的ＱＲ分解。然后，修改初始迭代向量ｖ１。迭代地重新开始这一过程，迫使初始迭代向量ｖ１进入需求的特征子空间，从而使残量‖Ｋｘ－θＭｘ‖→０。数值例子表明，该方法收敛性强，且稳定、有效。The recursion formulas of the Lanczos QR mixed method are used to determine a few smallest eigenvalues and associated eigenvectors of a large sparse real symmetric eigenvalue problems: Kx=λMx . After m steps of the Lanczos process with initial vector v 1 , we have KV m=MV mT m+h me T m . Choosing d shifts——the largest eigenvalues of T m , we carry out d steps shifted QR factorization for T m , and initial vector v 1 is then updated. The truncate Lanczos process is iteratively restarted to force the initial vector closer and closer to the desired invariant subspace, and the residual ‖Kx-θMx‖→0 . Numerical examples illustrate that the method can give rapid convergence, and the result is stable, effective.

关 键 词：QR因子分解 LANCZOS方法 结构动力学 实对称 

分 类 号：TB123[理学—工程力学] O241.6[理学—力学]