Structural backward stability in rational eigenvalue problems solved via block Kronecker linearizations

In this paper 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(λIℓ-A)-1B, where D(λ) is a polynomial matrix and C(λIℓ-A)-1B is a minimal state-space realization. We consider the family of block Kronecker linearizations of R(λ) , which have the following structure S(λ):=[M(λ)K^2TCK2T(λ)BK^1A-λIℓ0K1(λ)00],where the blocks have some specific structures. Backward stable eigenstructure solvers, such as the QZ or the staircase algorithms, applied to S(λ) will compute the exact eigenstructure of a perturbed pencil S^ (λ) : = S(λ) + Δ_S(λ) and the special structure of S(λ) will be lost, including the zero blocks below the anti-diagonal. In order to link this perturbed pencil with a nearby rational matrix, we construct in this paper a strictly equivalent pencil S~ (λ) = (I- X) S^ (λ) (I- Y) that restores the original structure, and hence is a block Kronecker linearization of a perturbed rational matrix R~(λ)=D~(λ)+C~(λIℓ-A~)-1B~, where D~ (λ) is a polynomial matrix with the same degree as D(λ). Moreover, we bound appropriate norms of D~ (λ) - D(λ) , C~ - C, A~ - A and B~ - B in terms of an appropriate norm of Δ_S(λ). These bounds may be, in general, inadmissibly large, but we also introduce a scaling that allows us to make them satisfactorily tiny, by making the matrices appearing in both S(λ) and R(λ) have norms bounded by 1. Thus, for this scaled representation, we prove that the staircase and the QZ algorithms compute the exact eigenstructure of a rational matrix 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. Several numerical experiments confirm the obtained backward stability results.