
## 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.