Characterize conditions for well-conditioned Sylvester solutions

Establish conditions under which the solution matrix X of the Sylvester equation AX−XB=C is well conditioned, thereby supporting backward-stability analyses of numerical algorithms for Sylvester equations.

Background

For the Sylvester equation AX−XB=C, the conditioning of the solution matrix X is distinct from the conditioning of the associated Kronecker-form linear system. A small residual need not imply a small backward error, particularly when X itself is ill conditioned. Consequently, identifying assumptions on A, B, and C that guarantee a well-conditioned solution is relevant to the backward stability of numerical methods.

The paper states that Nick Higham identified this characterization as an open problem and reports only a partial answer. The results provide general a priori bounds and sufficient conditions in several special settings, but the general problem of characterizing when the solution is well conditioned remains unresolved.

References

This motivates our interest in deriving conditions under which the Sylvester equation has a well-conditioned solution. Higham identifies this as an open problem and states the problem explicitly in the analysis of the backward stability of a computed approximate solution $ X$ to~sylv:eq p.~319.

— Conditioning of solutions to the Sylvester equation  (2609.04050 - Fasi et al., 3 Sep 2026) in Section 1, paragraph beginning “This motivates our interest in deriving conditions…”