There are some emails I sent to Prateek Sharma around Aug. 28-29, 2010, with thoughts on possible improvements to the dr_gmres.f routine. In addition: Another possible improvement is the choice of the initial guess and/or the initial two search directions. As Phil Colella mentioned in an teleconference yesterday, for PDE applications, one can use extrapolation from previous time steps or previous Runge-Kutta stages to estimate the solution better. Another possible improvement is to add backtracking, to make it a "Globally-Convergent" method, though I need to think a little bit about how to do that for this method where there are essentially 2 search directions and not just the one search direction of a Newton method. Could perhaps combine this 2cd order Richardson with Newton with Levenberg-Marquadt in some way. I've forgotten exactly how I extended the 2cd order Richardson / Conjugate Gradient method from the linear problem to nonlinear. Now that I understand globally-convergent Newton-Krylov methods better, might want to think about this again.