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OP-10459Status unassessedAnalysis

A possible counterexample via FOM

Crouzeix's conjecture · AIMPL workshop record

MSC 47 · Operator theory

Editorial mapping of AIMPL list topic: Crouzeix's conjecture. http://aimpl.org/crouzeix/ Not an author-supplied MSC code.

Workshop record · status unassessedAs of 2026-09-10 source snapshot; not a current-status certificationSource & evidence ↓
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Problem statement

To extend the above inequality to a counterexample of Crouzeix's conjecture, one may try among other ideas to (1) find matrices AA 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 ∥rk∥\| r_k\| on the max of pkp_k for the field of values, with prescribed Ritz values.

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Background

Question context

The following inequality holds for FOM polynomials pFp^F when applying the method to a linear system Ax=bAx=b with bb a unit vector such that FOM does not break down at or before the kkth iteration:

0=2max⁡ρ∈Hk∣pkF(ρ)∣<∥rkF∥≤∥pkF(A)∥, 0=2\max_{\rho\in{\cal H}_k}|p_k^F(\rho)| < \| r_k^F\|\leq \|p_k^F(A)\|,

where Hk{\cal H}_k are the Ritz values for the kkth FOM iteration, pkFp_k^F is the corresponding FOM polynomial of degree at most kk with the value one in the origin and rkFr_k^F the residual vector. It is possible to construct matrices yielding {\em any} complex Ritz values
Hk{\cal H}_k while the corresponding residual norm takes {\em any} positive real value (in fact, infinity might be included).

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Dated source record, inspected 2026-09-10. Current status has not been independently reassessed. Source-reported results have not been independently verified.

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References
  1. [1]Crouzeix's conjecture · Original question link
  2. [2]AIMPL source bibliography link
  3. [3]AIMPL text license · CC BY-SA 3.0 linkAIM Problem Lists (AIMPL), original contributors; CC BY-SA 3.0. Adaptation: structural TeX converted to Markdown, documented source macros expanded, and source bibliography links added. Mathematical wording retained. These adapted source texts remain under CC BY-SA 3.0.

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