This talk will be accessible to anyone interested in solving large sparse systems of linear equations. We give a gentle introduction to such strange sounding (but in fact logical and quite simple) concepts as the ``normwise relative backward error'' (NRBE) of an approximate solution $y$ to the linear system of equations $Ax=b$. We describe the use of the NRBE in determining when to stop an iterative process for solving $Ax=b$ where $A$ is a large sparse matrix. An efficient implementation of the generalized minimum residual (GMRES) method for solving $Ax=b$ uses modified Gram-Schmidt orthogonalization (MGS-GMRES). We indicate why this behaves so well despite loss of orthogonality, and use it to illustrate the effectiveness of the NRBE in designing stopping criteria.