Crouzeix's conjecture

    The following inequality holds for FOM polynomials $ p^F $ when applying the method to a linear system $ Ax=b $ with $ b $ a unit vector such that FOM does not break down at or before the $ k $th iteration: \[ 0=2\max_{\rho\in{\cal H}_k}|p_k^F(\rho)| < \| r_k^F\|\leq \|p_k^F(A)\|, \] where $ {\cal H}_k $ are the Ritz values for the $ k $th FOM iteration, $ p_k^F $ is the corresponding FOM polynomial of degree at most $ k $ with the value one in the origin and $ r_k^F $ the residual vector. It is possible to construct matrices yielding any complex Ritz values $ {\cal H}_k $ while the corresponding residual norm takes any positive real value (in fact, infinity might be included).