[Paper Review] Block Kronecker Linearizations of Matrix Polynomials and their Backward Errors

Froilán M. Dopico, Piers W. Lawrence+2 more|MIMS EPrints (University of Southampton)|Jul 16, 2017

Matrix Theory and Algorithms45 references5 citations

TL;DR

This paper introduces block Kronecker linearizations—a new family of strong linearizations for matrix polynomials—that enable backward stable solution of complete polynomial eigenproblems. It provides a rigorous, finite-perturbation backward error analysis showing that computed eigenstructures are exact for slightly perturbed polynomials, with precise bounds valid across a broad class 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.

Motivation & Objective

• To develop a new class of strong linearizations for matrix polynomials that ensure backward stability in solving complete polynomial eigenproblems.
• To perform a rigorous backward error analysis valid for finite perturbations, avoiding first-order approximations.
• To establish precise backward error bounds applicable simultaneously to a large family of linearizations, including block Kronecker pencils.
• To identify conditions under which the computed eigenstructure corresponds exactly to that of a slightly perturbed matrix polynomial.
• To lay a foundation for generalizing the analysis to other classes of linearizations via the framework of strong block minimal bases pencils.

Proposed method

• Proposes block Kronecker pencils as a new family of strong linearizations derived from the structure of dual minimal bases and block minimal bases.
• Applies orthogonal transformations and row/column permutations to transform coefficient matrices into block diagonal forms with identifiable submatrices.
• Uses singular value analysis of structured matrices $\hat{C}_{\varepsilon-1}(L_{\varepsilon}(\lambda))$ and $\hat{C}_{\varepsilon}(L_{\varepsilon}(\lambda))$ to bound backward errors.
• Establishes that the smallest singular values of these coefficient matrices correspond to $\sigma_{\min}(M_\varepsilon) = 2\sin\left(\frac{\pi}{4\varepsilon+2}\right)$, which determines the backward error bound.
• Leverages the orthogonality and equivalence of transformed matrices to preserve singular values and enable precise error quantification.
• Uses the framework of strong block minimal bases pencils to ensure robustness under perturbations, enabling generalization to other linearization families.

Experimental results

Research questions

• RQ1 How can a rigorous backward error analysis be developed for matrix polynomial eigenproblems that is valid for finite perturbations rather than first-order approximations?
• RQ2 What properties must a linearization possess to ensure that the computed eigenstructure is exact for a slightly perturbed matrix polynomial?
• RQ3 How do block Kronecker pencils compare to other linearizations in terms of backward error bounds and stability?
• RQ4 Can the backward error analysis be generalized to other classes of linearizations beyond block Kronecker pencils?
• RQ5 What is the role of minimal indices and dual minimal bases in constructing robust and backward stable linearizations?

Key findings

• The backward error analysis is rigorous and valid for finite perturbations, not limited to first-order approximations.
• The smallest singular value of the key matrix $M_\varepsilon$ is $\sigma_{\min}(M_\varepsilon) = 2\sin\left(\frac{\pi}{4\varepsilon+2}\right)$, which determines the backward error bound.
• This smallest singular value satisfies $\sigma_{\min}(M_\varepsilon) \geq \frac{3}{2(\varepsilon+1)}$, providing a useful lower bound for error analysis.
• The analysis shows that the computed eigenstructure of a block Kronecker pencil is exact for a slightly perturbed matrix polynomial, with the perturbation size bounded by the backward error.
• The framework of strong block minimal bases pencils ensures robustness under perturbations, enabling extension of the results to other linearization families.
• The singular values of the coefficient matrices $C_{\varepsilon-1}(L_{\varepsilon}(\lambda))$ and $C_{\varepsilon}(L_{\varepsilon}(\lambda))$ are preserved under orthogonal transformations, allowing precise error quantification.