Structural backward stability in rational eigenvalue problems solved via block Kronecker linearizations
We study the backward stability of running a backward stable eigenstructure solver on a pencil $S(λ)$ that is a strong linearization of a rational matrix $R(λ)$ expressed in the form $R(λ)=D(λ)+ C(λI_\ell-A)^{-1}B$, where $D(λ)$ is a polynomial matrix and $C(λI_\ell-A)^{-1}B$ is a minimal state-space realization. We consider the family of block Kronecker linearizations of $R(λ)$, which are highly structured pencils. Backward stable eigenstructure solvers applied to $S(λ)$ will compute the exact eigenstructure of a perturbed pencil $\widehat S(λ):=S(λ)+Δ_S(λ)$ and the special structure of $S(λ)$ will be lost. In order to link this perturbed pencil with a nearby rational matrix, we construct a strictly equivalent pencil $\widetilde S(λ)$ to $\widehat S(λ)$ that restores the original structure, and hence is a block Kronecker linearization of a perturbed rational matrix $\widetilde R(λ) = \widetilde D(λ)+ \widetilde C(λI_\ell- \widetilde A)^{-1} \widetilde B$, where $\widetilde D(λ)$ is a polynomial matrix with the same degree as $D(λ)$. Moreover, we bound appropriate norms of $\widetilde D(λ)- D(λ)$, $\widetilde C - C$, $\widetilde A - A$ and $\widetilde B - B$ in terms of an appropriate norm of $Δ_S(λ)$. These bounds may be inadmissibly large, but we also introduce a scaling that allows us to make them satisfactorily tiny. Thus, for this scaled representation, we prove that the staircase and the $QZ$ algorithms compute the exact eigenstructure of a rational matrix $\widetilde R(λ)$ that can be expressed in exactly the same form as $R(λ)$ with the parameters defining the representation very near to those of $R(λ)$. This shows that this approach is backward stable in a structured sense.