RT Journal Article
T1 Structural backward stability in rational eigenvalue problems solved via block Kronecker linearizations
A1 Martínez Dopico, Froilán César
A1 Quintana Ponce, María del Carmen
A1 Van Dooren, Paul
AB 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 [...] 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 Sˆ(λ):=S(λ)+ΔS(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)Sˆ(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 ΔS(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.
PB Springer
SN 0008-0624
YR 2023
FD 2023-03
LK https://hdl.handle.net/10016/37022
UL https://hdl.handle.net/10016/37022
LA eng
NO The first and second authors were partially supported by "Ministerio de Economía, Industria y Competitividad (MINECO)" of Spain and "Fondo Europeo de Desarrollo Regional (FEDER)" of EU through grant MTM2015-65798-P, by the "Proyecto financiado por la Agencia Estatal de Investigación de España" (PID2019-106362GB-I00 / AEI / 10.13039/501100011033) and by the Madrid Government (Comunidad de Madrid-Spain) under the "Multiannual Agreement with Universidad Carlos III de Madrid in the line of Excellence of University Professors (EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation)". The second author was funded by the "contrato predoctoral" BES-2016-076744 of MINECO and by an Academy of Finland grant (Suomen Akatemian päätös 331240). This work was developed while the third author held a "Chair of Excellence UC3M - Banco de Santander" at Universidad Carlos III de Madrid in the academic year 2019-2020.
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RD 1 sept. 2024