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

In this paper we study the backward stability of running a stable eigenstructure solver on pencil $$S(\lambda )$$ that is strong linearization rational matrix $$R(\lambda expressed in form )=D(\lambda )+ C(\lambda I_\ell -A)^{-1}B$$ , where $$D(\lambda polynomial and $$C(\lambda minimal state-space realization. We consider family block Kronecker linearizations which have following structure $$\begin{aligned} S(\lambda ):=\left[ \begin{array}{ccc} M(\lambda ) &{} {\widehat{K}}_2^T C K_2^T(\lambda \\ B {\widehat{K}}_1 A- \lambda 0\\ K_1(\lambda 0 \end{array}\right] \end{aligned}$$ blocks some specific structures. Backward solvers, such as QZ or staircase algorithms, applied to will compute exact perturbed $$\widehat{S}(\lambda ):=S(\lambda )+\varDelta _S(\lambda special be lost, including zero below anti-diagonal. order link with nearby matrix, construct strictly equivalent $$\widetilde{S}(\lambda )=(I-X)\widehat{S}(\lambda )(I-Y)$$ restores original structure, hence $${{\widetilde{R}}}(\lambda = {{\widetilde{D}}}(\lambda {{\widetilde{C}}}(\lambda - {{\widetilde{A}}})^{-1} {{\widetilde{B}}}$$ $${{\widetilde{D}}}(\lambda same degree . Moreover, bound appropriate norms )- D(\lambda $${{\widetilde{C}}} C$$ $${{\widetilde{A}}} A$$ $${{\widetilde{B}}} B$$ terms an norm $$\varDelta These bounds may be, general, inadmissibly large, but also introduce scaling allows us make them satisfactorily tiny, by making matrices appearing both bounded 1. Thus, for scaled representation, prove algorithms can exactly parameters defining representation very near those This shows approach structured sense. Several numerical experiments confirm obtained results.