Componentwise perturbation analyses for the QR factorization

TLDR: In this paper, componentwise perturbation analyses for Q and R in the QR factorization A=QR, ��$Q^\mathrm{T}Q=I$¯¯¯¯, R upper triangular, for a given real $m\times n$ matrix A of rank n.

Abstract: This paper gives componentwise perturbation analyses for Q and R in the QR factorization A=QR, $Q^\mathrm{T}Q=I$ , R upper triangular, for a given real $m\times n$ matrix A of rank n. Such specific analyses are important for example when the columns of A are badly scaled. First order perturbation bounds are given for both Q and R. The analyses more accurately reflect the sensitivity of the problem than previous such results. The condition number for R is bounded for a fixed n when the standard column pivoting strategy is used. This strategy also tends to improve the condition of Q, so usually the computed Q and R will both have higher accuracy when we use the standard column pivoting strategy. Practical condition estimators are derived. The assumptions on the form of the perturbation $\Delta A$ are explained and extended. Weaker rigorous bounds are also given.