Structural backward stability in rational eigenvalue problems solved via block Kronecker linearizations
(2023) Calcolo : a quarterly on numerical analysis and theory of computation — Vol. 60, n° 1 (2023)
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- Authors
Dopico, Froilán M. 
Quintana, María C. Van Dooren, Paul
- Abstract
- In this paper we study the backward stability of running a backward stable eigenstructure solver on a pencil that is a strong linearization of a rational matrix expressed in the form , where is a polynomial matrix and is a minimal state-space realization. We consider the family of block Kronecker linearizations of , which have the following structure where the blocks have some specific structures. Backward stable eigenstructure solvers, such as the QZ or the staircase algorithms, applied to will compute the exact eigenstructure of a perturbed pencil and the special structure of 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 that restores the original structure, and hence is a block Kronecker linearization of a perturbed rational matrix , where is a polynomial matrix with the same degree as . Moreover, we bound appropriate norms of , , and in terms of an appropriate norm of . 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 and 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 that can be expressed in exactly the same form as with the parameters defining the representation very near to those of . This shows that this approach is backward stable in a structured sense. Several numerical experiments confirm the obtained backward stability results.
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Citations
Dopico, F. M., Quintana, M. C., & Van Dooren, P. (2023). Structural backward stability in rational eigenvalue problems solved via block Kronecker linearizations. Calcolo : a quarterly on numerical analysis and theory of computation, 60(1). https://doi.org/10.1007/s10092-022-00502-4 (Original work published 2023)