Stagnation of block GMRES and its relationship to block FOM

We examine the the convergence behavior of block GMRES and block FOM and characterize the phenomenon of stagnation in block GMRES. Stagnation is then related to the behavior of the block FOM method. Following from [Brown, SIAM J. Sci. Statist. Comput., '91], we generalize the block FOM method to generate well-defined approximations in the case that block FOM would normally break down, and these generalized solutions are used in our analysis. Our analysis covers both the cases of normal iterations as well as block Arnoldi breakdown. Two toy numerical examples are given to illustrate what we have proven, and we also apply both block methods to a small application problem to demonstrate the validity of the analysis in a less non-pathological case.