Analyze the boundary of the Schur modal-stability domain

Determine the convergence or divergence behavior of the non-autonomous quadratic IGAHD recurrence at the boundary of the Schur-stability domain for its limiting modal matrices.

Background

For finite-dimensional convex quadratic objectives, the IGAHD iteration decomposes into scalar spectral modes governed by non-autonomous recurrences whose coefficient matrices converge to limiting modal matrices. The paper derives necessary and sufficient conditions for Schur stability of those limiting matrices and obtains geometric decay when the condition is strict.

At the boundary of the uniform spectral condition, at least one limiting modal matrix can have an eigenvalue of modulus one. The contraction argument used in the paper then fails, and the behavior of the original recurrence with variable coefficients is not determined by the limiting-matrix analysis.