Estimating the Backward Error in LSQR

We propose practical stopping criteria for the iterative solution of sparse linear least squares (LS) problems. Although we focus our discussion on the algorithm LSQR of Paige and Saunders, the ideas discussed here may also be applicable to other algorithms. We review why the 2-norm of the projection of the residual vector onto the range of A is a useful measure of convergence, and we show how this projection can be estimated efficiently at every iteration of LSQR. We also give practical and cheaply computable estimates of the backward error for the LS problem.