On graded QR decompositions of products of matrices

This paper is concerned with the singular values and vectors of a product Mm =A1A2 ... Am of matrices of order n. The chief difficulty with computing them directly from Mm
is that with increasing m the ratio of the small to the large singular values of Mm may fall below
the rounding unit, so that the former are computed inaccurately. The solution proposed here is to
compute recursively the factorization Mm = QRPT, where Q is orthogonal, R is a graded upper
triangular, and PT is a permutation.