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 ei...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 ω that contains an approximation to x, these two methods compute approximations (μ,x) and μ,x) to (λ,x), respectively. We establish three results. First, the refined eigenvector approximation or simply the refined Ritz vector x is unique as the deviation of x from ω approaches zero if A is simple. Second, in terms of residual norm of the refined approximate eigenpair (μ, x), we derive lower and upper bounds for the sine of the angle between the Ritz vector x and the refined eigenvector approximation x, and we prove that x≠x unless x = x. Third, we establish relationships between the residual norm ||AX -μx|| of the conventional methods and the residual norm ||Ax -μx|| of the refined methods, and we show that the latter is always smaller than the former if (μ, x) is not an exact eigenpair of A, indicating that the refined projection method is superior to the corresponding conventional counterpart.展开更多
§1.引言
设A∈RM×N,定义增广矩阵
(A~)=(O A AT O),(1)
其中上标T表示转置.不失一般性,假设M≥N,设σi,i=1,2,…,N是A的奇异值,ui和ui分别是对应的左右奇异向量,奇异值按从小到大或从大到小的顺序排列,则A的特征值恰好为±...§1.引言
设A∈RM×N,定义增广矩阵
(A~)=(O A AT O),(1)
其中上标T表示转置.不失一般性,假设M≥N,设σi,i=1,2,…,N是A的奇异值,ui和ui分别是对应的左右奇异向量,奇异值按从小到大或从大到小的顺序排列,则A的特征值恰好为±σi,i=1,2,…,N和M-N个零,±σi对应的特征向量分别为1/√2(uT i,vT i)T和1/√2(uT i,-vT i)T.展开更多
文摘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 ω that contains an approximation to x, these two methods compute approximations (μ,x) and μ,x) to (λ,x), respectively. We establish three results. First, the refined eigenvector approximation or simply the refined Ritz vector x is unique as the deviation of x from ω approaches zero if A is simple. Second, in terms of residual norm of the refined approximate eigenpair (μ, x), we derive lower and upper bounds for the sine of the angle between the Ritz vector x and the refined eigenvector approximation x, and we prove that x≠x unless x = x. Third, we establish relationships between the residual norm ||AX -μx|| of the conventional methods and the residual norm ||Ax -μx|| of the refined methods, and we show that the latter is always smaller than the former if (μ, x) is not an exact eigenpair of A, indicating that the refined projection method is superior to the corresponding conventional counterpart.
