Some theoretical comparisons of refined Ritz vectors and Ritz vectors

作     者：JIA Zhongxiao 

作者机构：Department of Mathematical Sciences Tsinghua University Beijing 100084 China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2004年第47卷第Z1期

页      面：222-233页

核心收录：

学科分类：07[理学] 08[工学] 0701[理学-数学] 

基　　金：国家重点基础研究发展计划(973计划)(G19990328) 

主　　题：large matrix, conventional projection, refined projection, eigenvalue, eigenvector, Ritz value, Ritz vector,refined Ritz vector. 

摘      要：Refined projection methods proposed by the author have received attention internationally. We are concerned with a conventional projection method and its refined counterpart for computing approximations to a simple eigenpair (λ,x)of a large matrix A. Given a subspace w that contains anapproximation to x, these two methods compute approximations(μ(x~)) and (μ(x^)) to (λ,x),respectively. We establish three results. First, the refinedeigenvector approximation or simply the refined Ritz vector (x^) is unique as the deviation of x from w approaches zero if λ is simple. Second, interms of residual norm of the refined approximate eigenpair (μ,(x^)), we derive lower and upper bounds for the sine of the angle betweenthe Ritz vector (x~) and the refined eigenvector approximation (x^), and we prove that (x~)≠(x^) unless (x^)=x. Third, we establish relationships between theresidual norm ‖A(x~)-μ(x^)‖ of the conventionalmethods and the residual norm ‖A(x^)-μ(x^)‖ of therefined methods, and we show that the latter is always smallerthan the former if (μ,(x^)) is not an exact eigenpair ofA, indicating that the refined projection method is superiorto the corresponding conventional counterpart.