On the Accuracy of the Karlson-Waldén Estimate of the Backward Error for Linear Least Squares Problems

We consider the backward error associated with a given approximate solution of a linear least squares problem. The backward error can be very expensive to compute, as it involves the minimal singular value of certain matrix that depends on the problem data and the approximate solution. An estimate based on a regularized projection of the residual vector has been proposed in the literature and analyzed by several authors. Although numerical experiments in the literature suggest that it is a reliable estimate of the backward error for any given approximate LS solution, to date no satisfactory explanation for this behavior had been found. We derive new bounds which confirm this experimental observation.