IRKA Is a Riemannian Gradient Descent Method

The iterative rational Krylov algorithm (IRKA) is a commonly used fixed point iteration developed to minimize the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula> 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.