COMPUTING EIGENVECTORS OF NORMAL MATRICES WITH SIMPLE INVERSE ITERATION

• 来源期刊: 计算数学（英文版）
• 关键词: Eigenvector Inverse iteration Accuracy Error estimation.
• 年，卷（期）: 2003,（5）
• 页码范围: 657-670
• 页数: 14页
• 语种: 英文

摘要

It is well-known that if we have an approximate eigenvalue λ of a normal matrix Aof order n, a good approximation to the corresponding eigenvector u can be computedby one inverse iteration provided the position, say kmax, of the largest component of u isknown. In this paper we give a detailed theoretical analysis to show relations between thesolution to the system (A - λI)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 propertiesrelated to the smallest residual and smallest approximation error to u in spectral norm andFrobenius norm. We also prove that the normalized absolute vector v = |u|/‖u‖∞ of uupper bounds of |u(k)| for those "optimal" indexes such as Fernando's heuristic for kmaxwithout any assumptions. A stable double orthogonal factorization method and a simplerbut may less stable approach are proposed for locating the largest component of u.