Componentwise perturbation analyses for the QR factorization

This paper gives componentwise perturbation analyses for Q and R in the QR factorization A = QR, QTQ = I , R upper triangular, for a given realm×nmatrixA of rank n. Such specific analyses are important for examplewhen the columns ofA 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∆A are explained and extended. Weaker rigorous bounds are also given.