Block Kronecker Linearizations of Matrix Polynomials and their Backward Errors

TL;DR

This paper introduces block Kronecker pencils as a new family of strong linearizations for matrix polynomials and provides a rigorous backward stability analysis, demonstrating their robustness and potential for generalization.

Contribution

The paper presents a new class of strong linearizations called block Kronecker pencils and offers the first rigorous, finite-perturbation backward error analysis for these linearizations.

Findings

Block Kronecker pencils are robust under certain perturbations.

The backward error analysis provides precise bounds for eigenstructure recovery.

The framework can be extended to other classes of linearizations.

Abstract

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 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 rigurous 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. We hope that this robustness property will allow us to extend the results in this paper to other contexts.