On Graded QR Decompositions of Products of Matrices

Abstract

This paper is concerned with the singular values and vectors of a product $M_{m}=A_{1}A_{2}\cdots A_{m}$ of matrices of order $n$. The chief difficulty with computing them from directly from $M_{m}$ is that with increasing $m$ the ratio of the small to the large singular values of $M_{m}$ may fall below the rounding unit, so that the former are computed inaccurately. The solution proposed here is to compute recursively the factorization $M_{m} = QRP\trp$, where $Q$ is orthogonal, $R$ is a graded upper triangular, and $P\trp$ is a permutation. (Also cross-referenced as UMIACS-TR-94-53)