Preconditioning and convergence in the right norm

The convergence of numerical approximations to the solutions of differential equations is a key aspect of Numerical Analysis and Scientific Computing. Iterative solution methods for the systems of linear(ised) equations which often result are also underpinned by analyses of convergence. In the function space setting, it is widely appreciated that there are appropriate ways in which to assess convergence and it is well-known that different norms are not equivalent. In the finite dimensional linear algebra setting, however, all norms are equivalent and little attention is often payed to the norms used. In this paper, we highlight this consideration in the context of preconditioning for minimum residual methods (minres and gmres/gcr/ orthomin) and argue that even in the linear algebra setting there is a ‘right’ norm in which to consider convergence: stopping an iteration which is rapidly converging in an irrelevant or highly scaled norm at some tolerance level may still give a poor answer.