Block Kronecker Linearizations of Matrix Polynomials and their Backward Errors

Dopico, Froilán M. and Lawrence, Piers W. and Pérez, Javier and Van Dooren, Paul (2016) Block Kronecker Linearizations of Matrix Polynomials and their Backward Errors. [MIMS Preprint]

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We introduce a new family of strong linearizations of matrix polynomials---which we call "block Kronecker pencils"---and 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 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. These favorable pencils include the famous Fiedler linearizations, which are just a very particular case of block Kronecker pencils. Thus, our analysis offers the first proof available in the literature of global backward stability for Fiedler pencils. In addition, the theory developed for block Kronecker pencils is much simpler than the theory available for Fiedler pencils, especially in the case of rectangular matrix polynomials. 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 include certain perturbations of block Kronecker pencils; this allows us to extend the results in this paper to other contexts.

Item Type: MIMS Preprint
Uncontrolled Keywords: backward error analysis, polynomial eigenvalue problems, complete eigenstructure, dual minimal bases, Fiedler pencils, linearization, matrix polynomials, matrix perturbation theory, minimal indices, recovery of minimal indices and bases, recovery of eigenvectors
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
MSC 2010, the AMS's Mathematics Subject Classification > 93 Systems theory; control
Depositing User: Dr. Piers W. Lawrence
Date Deposited: 31 May 2016
Last Modified: 08 Nov 2017 18:18

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