| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

9. Inverse numerical range problems

    1. A possible counterexample via FOM

          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).

      Problem 9.1.

      [Jurjen Duintjer Tebbens] To extend the above inequality to a counterexample of Crouzeix’s conjecture, one may try among other ideas to (1) find matrices $A$ where (most of) the prescribed Ritz values lie on the boundary of the field of values ; (2) numerically follow the influence of the chosen prescribed residual norm $\| r_k\| $ on the max of $ p_k $ for the field of values, with prescribed Ritz values.

          Cite this as: AimPL: Crouzeix's conjecture, available at http://aimpl.org/crouzeix.