Stability of Householder QR Factorization for Weighted Least Squares Problems
Anthony J. Cox, Nicholas John Higham · OpenGrey (Institut de l'Information Scientifique et Technique) · 2010
For least squares problems in which the rows of the coefficient matrix vary widely in norm, Householder QR factorization (without pivoting) has unsatisfactory backward stability properties. Powell and Reid showed in 1969 that the use of both row and column pivoting leads to a desirable row-wise backward error result. We give a reworked backward error analysis in modern notation and prove two new results. First, sorting the rows by decreasing ∞-norm at the start of the factorization obviates the need for row pivoting. Second, row-wise backward stability is obtained for only one of the two possible choices of sign in the Householder vector.