Two-step Gram-Schmidt downdating methods

The problem of deleting a row from the QR factorization X = UR by Gram-Schmidt techniques is intimately connected to solving the least squares problem (formula available in paper) by classical iterative methods. Past approaches to this problem have focused upon accurate computation of the residual (formula available in paper), thereby finding a vector (formula available in paper) that is orthogonal to U. This work shows that it is also important to accurately compute the vector f and that it must be used in the downdating process to maintain good backward error in the new factorization. New algorithms are proposed based upon this observation.