Jed, I just remembered something I should have said when we talked about the rewrite of the KSPSPECEST stuff. If one has the bound on the smallest eigenvalue of the (preconditioned) operator then ones convergence test can take that into account and know that the 2-norm of the error of the linear solver (as opposed to the 2 norm of the residual) is less than some tolerance. Of course the KSPSPECEST can give us this information, even though Mark doesn't believe it is accurate enough :-), with enough iterations it can be. So ideally we'd have options that allowed convergence tests to use the estimate of the error norm instead of just the residual norm. Barry