Error amplification in block-CMV Schur recursion

Determine whether the sequential Schur recursion used in the block-CMV realization of Cayley-transformed self-energy moments amplifies numerical errors when the recursively generated matrices approach singularity.

Background

The paper constructs a Hermitian upfolded Hamiltonian for the GW self-energy by representing Cayley-transformed particle and hole moments through a unitary auxiliary matrix. One implementation uses block-CMV matrix-valued orthogonal polynomials, whose Verblunsky coefficients are generated sequentially by a Schur recursion.

The authors report that block-CMV is empirically preferable at low moment orders but can exhibit increasing moment-reconstruction errors at higher orders, including poles on the wrong semicircle or near the singular point of the inverse Cayley map. They conjecture that this behavior results from error amplification in the sequential Schur recursion as the matrices become nearly singular; the issue is unresolved, and the paper instead uses a non-recursive Toeplitz construction as a fallback for high moment orders.