Interpolatory model reduction of dynamical systems with root mean squared error

The paper’s Theorem 1 states that, for quadratic-output transfer functions restricted to the imaginary axis, if the projection spaces satisfy (ziE−A)−1 b ∈ span(V) and (ziE−A)−H Q (ziE−A)−1 b ∈ span(W), then both value and first-derivative interpolation hold at each selected point zi ∈ iR, providing a concise proof via projector identities and a derivative formula for Ĥ(z) (Theorem 1 and equations (19)–(20) in the paper) . The candidate solution proves the same result by differentiating the Petrov–Galerkin residual along the imaginary axis and using the quadratic-form identity H(z)=x(z)^H Q x(z) that follows from the specialization H(z)=b^H(−zE^H−A^H)^{-1}Q(zE−A)^{-1}b for z∈iR , consistent with the paper’s definition of H(s) . Both arguments are sound; the candidate’s proof is a different but correct derivation of the paper’s theorem.