Complete stagnation of GMRES

We study problems for which the iterative method gmres for solving linear systems of equations makes no progress in its initial iterations. Our tool for analysis is a nonlinear system of equations, the stagnation system, that characterizes this behavior. We focus on complete stagnation, for which there is no progress until the last iteration. We give necessary and sufficient conditions for complete stagnation of systems involving unitary matrices, and show that if a normal matrix completely stagnates then so does an entire family of nonnormal matrices with the same eigenvalues. Finally, we show that there are real matrices for which complete stagnation occurs for certain complex right-hand sides but not for real ones.