A Storage Efficient WY Representation for Products of Householder Transformations

Abstract

A product $Q = P_{1} \cdots P_{r}$ of m-by-m Householder matrices can be written in the form $Q = I + WY^{T}$ where W and Y are each m-by-r. This is called the WY representation of Q. It is of interest when implementing Householder techniques in high-performance computing environments that "like" matrix-matrix multiplication. In this note we describe a storage efficient way to implement the WY representation. In particular, we show how the matrix Q can be expressed in the form $Q = I + YTY^{T}$ where $Y \epsilon R^{mxr}$ and $T \epsilon R^{rxr}$ with T upper triangular. Usually r less than less than m and so this "compact" WY representation requires less storage. When compared with the recent block-reflector strategy proposed by Schreiber and Parlett the new technique still has a storage advantage and involves a comparable amount of work.