一类特殊类型子空间上Ritz对的性质及其应用

ON THE RITZ PAIRS WITH RESPECT TO A KIND OF SPECIAL SUBSPACES WITH AN APPLICATION

§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.

We study some properties of the Ritz pairs of an argumented matrix with respect to a kind of special subspaces. It is proved that the projected eigenproblem can be reduced to a half dimensional singular value problem. As an important application, this equivalence allows one to compute a partial SVD of a large scale matrix and refined shifts for use within an implicitly restarted refined bidiagonal-ization Lanczos algorithm (IRRBL), so that the computational cost and the storage requirement can be saved significantly.