GMBACK: A generalised minimum backward error algorithm for nonsymmetric linear systems A drawback of employing the residual error norm as a stopping condition in an iterative process is that the error must be small if the approximation is accurate. However, the converse need not be true. The main aims of this paper are to consider an alternative criterion for judging whether a given approximation is acceptable and to present an algorithm which computes an approximate solution to the linear system $Ax = b$ such that the normwise backward error meets some optimality condition.