Block Kronecker linearizations of matrix polynomials and their backward errors
Publisher:
Springer Nature
Issued date:
2018-10
Citation:
Dopico, F. M., Lawrence, P. W., Pérez, J., Dooren, P. V. (2018). Block Kronecker linearizations of matrix polynomials and their backward errors. Numerische Mathematik, 140(2), pp. 373–426.
ISSN:
0029-599X
xmlui.dri2xhtml.METS-1.0.item-contributor-funder:
Ministerio de Economía y Competitividad (España)
Sponsor:
This work was partially supported by the “Ministerio de Economía, Industria y Competitividad of Spain” and “Fondo Europeo de Desarrollo Regional (FEDER) of EU” through Grants MTM-2012-32542, MTM-2015-68805-REDT, MTM-2015-65798-P, MTM2017-90682-REDT, by the Belgian network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office, and by the Engineering and Physical Sciences Research Council of UK through Grant EP/I005293.
Project:
Gobierno de España. MTM2012-32542
Gobierno de España. MTM2015-65798-P
Gobierno de España. MTM2015-68805-REDT
Gobierno de España. MTM2017-90682-REDT
Keywords:
Backward error analysis
,
Polynomial eigenvalue problems
,
Complete eigenstructure
,
Dual minimal bases
,
Linearization
,
Matrix polynomials
,
Matrix perturbation theory
,
Minimal indices
Rights:
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract:
We introduce a new family of strong linearizations of matrix polynomialswhich we call block Kronecker pencilsand perform a backward stability analysis of complete polynomial eigenproblems. These problems are solved by applying any backward stable algorithm to
We introduce a new family of strong linearizations of matrix polynomialswhich we call block Kronecker pencilsand perform a backward stability analysis of complete polynomial eigenproblems. These problems are solved by applying any backward stable algorithm to a block Kronecker pencil, such as the staircase algorithm for singular pencils or the QZ algorithm for regular pencils. This stability analysis allows us to identify those block Kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial. The global backward error analysis in this work presents for the first time the following key properties: it is a rigorous analysis valid for finite perturbations (i.e., it is not a first order analysis), it provides precise bounds, it is valid simultaneously for a large class of linearizations, and it establishes a framework that may be generalized to other classes of linearizations. These features are related to the fact that block Kronecker pencils are a particular case of the new family of strong block minimal bases pencils, which are robust under certain perturbations and, so, include certain perturbations of block Kronecker pencils.
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