COMPUTING EIGENVECTORS OF NORMAL MATRICES WITH SIMPLE INVERSE ITERATION  

作　　者：Zhen-yue Zhang Tiang-wei Ouyang (Department of Mathematics, Zhejiang University, Yuquan Campus, Hangzhou 310027, China) 

出　　处：《Journal of Computational Mathematics》2003年第5期657-670,共14页计算数学（英文）

基　　金：The work of this author was supported in part by NSFC (project 19771073);Special Funds for Major State Basic Research Projects of China (project G19990328); Zhejiang Provincial Natural Science Foundation of China;Foundation for University Key Te

摘　　要：It is well-known that if we have an approximate eigenvalue A of a normal matrix A of order n, a good approximation to the corresponding eigenvector u can be computed by one inverse iteration provided the position, say kmax, of the largest component of u is known. In this paper we give a detailed theoretical analysis to show relations between the eigenvector u and vector Xk, k = 1, … ,n, obtained by simple inverse iteration, i.e., the solution to the system (A - I)x = ek with ek the kth column of the identity matrix I. We prove that under some weak conditions, the index kmax is of some optimal properties related to the smallest residual and smallest approximation error to u in spectral norm and Probenius norm. We also prove that the normalized absolute vector v = |u|/||u|| of u can be approximated by the normalized vector of (||x1||2, … ||xn||2)T- We also give some upper bounds of |u(k)| for those 'optimal' indexes such as Fernando's heuristic for kmax without any assumptions. A stable double orthogonal factorization method and a simpler but may less stable approach are proposed for locating the largest component of u.It is well-known that if we have an approximate eigenvalue A of a normal matrix A of order n, a good approximation to the corresponding eigenvector u can be computed by one inverse iteration provided the position, say kmax, of the largest component of u is known. In this paper we give a detailed theoretical analysis to show relations between the eigenvector u and vector Xk, k = 1, … ,n, obtained by simple inverse iteration, i.e., the solution to the system (A - I)x = ek with ek the kth column of the identity matrix I. We prove that under some weak conditions, the index kmax is of some optimal properties related to the smallest residual and smallest approximation error to u in spectral norm and Probenius norm. We also prove that the normalized absolute vector v = |u|/||u|| of u can be approximated by the normalized vector of (||x1||2, … ||xn||2)T- We also give some upper bounds of |u(k)| for those 'optimal' indexes such as Fernando's heuristic for kmax without any assumptions. A stable double orthogonal factorization method and a simpler but may less stable approach are proposed for locating the largest component of u.

关 键 词：EIGENVECTOR Inverse iteration ACCURACY Error estimation. 

分 类 号：O241.6[理学—计算数学]