A Storage-Efficient $WY$ Representation for Products of Householder Transformations 论文

摘要

A product $Q = P_1 \cdots P_r $, of $m \times m$ Householder matrices can be written in the form $Q = I + WY^T $, where W and Y are each $m \times r$. This is called the $WY$ representation of Q. It is of interest when implementing Householder techniques in high-performance computing environments that are especially good at matrix-matrix multiplication. In this note a storage-efficient way to implement the $WY$ representation is described. In particular, it is shown how the matrix Q can be expressed in the form $Q = I + YTY^T $, where $Y \in R^{m \times r} $ and $T \in R^{r \times r} $ with T upper triangular. Usually $r \ll m$ and so this “compact” $WY$ representation requires less storage. When compared with the recent block-reflector strategy proposed by Schreiber and Parlett [SIAM J. Numer. Anal, 25 (1988), pp. 189–205], the new technique still has a storage advantage and involves a comparable amount of work.