Simple backward error bounds for linear least-squares problems

We consider two upper bounds on the normwise backward error (BE) for linear least-squares problems. The advantage of these bounds is their simplicity. Their behaviour in commonly-used iterative methods can be analyzed more easily than that of the BE itself, and the bounds can also be estimated very cheaply in such methods. It is known that each of these upper bounds can be orders of magnitude larger than the BE. Then one may ask: under which conditions is each of the bounds a good estimate of the BE? We partially answer this question by giving sufficient conditions for each bound to be a good estimate of the BE. We illustrate these results with some numerical examples.