PERTURBATION ANALYSES FOR THE QR FACTORIZATION

This paper gives perturbation analyses for Q(1) and R in the QR factor ization A = Q(1)R, Q(1)(T)Q(1) = I for a given real m x n matrix A of rank n and general perturbations in A which are sufficiently small in norm. The analyses more accurately reflect the sensitivity of the prob lem than previous such results. The condition numbers here are altered by any column pivoting used in AP = Q(1)R, and the condition number f or R is bounded for a fixed n when the standard column pivoting strate gy is used. This strategy also tends to improve the condition of Q(1), so the computed Q(1) and R will probably both have greatest accuracy when we use the standard column pivoting strategy. First-order perturb ation analyses are given for both Q(1) and R. It is seen that the anal ysis for R may be approached in two ways-a detailed ''matrix-vector eq uation'' analysis which provides a tight bound and corresponding condi tion number, which unfortunately is costly to compute and not very int uitive, and a simpler ''matrix equation'' analysis which provides resu lts that are usually weaker but easier to interpret and which allows t he efficient computation of satisfactory estimates for the actual cond ition number. These approaches are powerful general tools and appear t o be applicable to the perturbation analysis of any matrix factorizati on.