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.

翻译：有理Krylov子空间投影方法在模型降阶领域已被证明是一种极为成功的途径，这主要归因于近似传递函数与原始传递函数的某些导数保持一致——此即著名的矩匹配理论。然而，对于远离插值节点的点所具有的性质，目前仍未得到充分研究。本文给出了模型降阶误差的显式表达式，该表达式同时包含位移参数与Ritz值。我们的发现相较于已知矩匹配理论的优越性，可类比于泰勒定理中拉格朗日型余项公式与佩亚诺型余项公式之间的差异。此外，本文针对预解函数中两个参数对应的误差公式提供了两种解释：一种解释表明模型降阶误差是插值余项，另一种解释则暗示该误差同时也是高斯-克里斯托弗尔求积余项。通过应用该误差公式，我们为插值型$H_{\infty}$范数模型降阶提出了一种贪婪算法。