From ADI Tuning to Topological Redefinition: A Banach–Space View of Indefinite Lyapunov Equations

Classical low-rank Alternating Direction Implicit (ADI) algorithms for large-scale Lyapunov equations rely on Frobenius-norm residual minimization inside a Hilbert geometry.
When the right-hand side becomes indefinite, these methods lose stability: the quadratic energy functional is no longer positive definite, and iterative updates oscillate or blow up in rank.
Recent “tangential ADI” variants patch this behavior through direction selection and shift optimization, yet all remain confined to the same metric structure.

Here I propose a Banach-space reformulation that replaces signed quadratic energy with absolute and nuclear norms, converting the problem from spectral optimization to convex minimization.
This geometric update removes the need for complex shifts, ensures monotone residual decay even for indefinite systems, and provides intrinsic rank adaptation through nuclear-norm regularization.
Conceptually, the change corresponds to moving from algorithmic tuning to topological redefinition—a kernel-level update of the Lyapunov solver.

Solving indefinite Lyapunov equations by changing the geometry, not tuning the algorithm.
A Banach-space Lyapunov solver — the kernel update to ADI.

#BanachSpaceLyapunov #NumericalLinearAlgebra #ConvexOptimization #LowRankADI #BanachNotHilbert