IRKA Is a Riemannian Gradient Descent Method

The iterative rational Krylov algorithm (IRKA) is a commonly used fixed point iteration developed to minimize the $\mathcal {H}_{2}$ model order reduction error. In this work, the IRKA is recast as a Riemannian gradient descent method with a fixed step size over the manifold of rational functions having fixed degree. This interpretation motivates the development of a Riemannian gradient descent method utilizing as a natural extension variable step size and line search. Comparisons made between the IRKA and this extension on a few examples demonstrate significant benefits.