Transfer function interpolation remainder formula of rational Krylov subspace methods

Rational Krylov subspace projection methods have proven to be a highly successful approach in the field of model order reduction (MOR), primarily due to the fact that some derivatives of the approximate and original transfer functions are identical.This is the well-known theory of moments matching. Nevertheless, the properties of points situated at considerable distances from the interplation nodes remain underexplored. In this paper, we present the explicit expression of the MOR error, which involves both the shifts and the Ritz values.The superiority of our discoveries over the known moments matching theory can be likened to the disparity between the Lagrange and Peano type remainder formulas in Taylor's theorem. Furthermore, two explanations are provided for the error formula with respect to the two parameters in the resolvent function. One explanation reveals that the MOR error is an interplation remainder, while the other explanation implies that the error is also a Gauss-Christoffel quadrature remainder.By applying the error formula, we suggest a greedy algorithm for the interpolatory $H_{\infty}$ norm MOR.