Abstract
We start by introducing a new class of structured matrix polynomials, namely, the class of M A -structured matrix polynomials, to providea common framework for many classes of structured matrix polynomials that are important in applications: the classes of (skew) symmetric, (anti-) palindromic, and alternating matrix polynomials. Then, we introduce the families of M A -structured strong block minimal bases pencils and of M A structured block Kronecker pencils, which are particular examples of block minimal bases pencils recently introduced by Dopico, Lawrence, Perez and Van Dooren, and show that any M A -structured odd-degree matrix polynomial can be strongly linearized via an M A -structured block Kronecker pencil. Finally, for the classes of (skew-)symmetric, (anti-)palindromic, and alternating odd-degree matrix polynomials, the M A -structured framework allows us to perform a global and structured backward stability analysis of complete structured polynomial eigenproblems, regular or singular, solved by applying to a M A -structured block Kronecker pencil a structurally backward stablealgorithm that computes its complete eigenstructure, like the palindromic-QR algorithm or the structured versions of the staircase algorithm. This analysis allows us to identify those M A -structured block Kronecker pencils that yield a computed complete eigenstructure which is the exact one of a slightly perturbed structured matrix polynomial. These pencils include (modulo permutations) the well-known block-tridiagonal and block-anti-tridiagonal structure-preserving linearizations. Our analysis incorporates structure to the recent (unstructured) backward error analysis performed for block Kronecker linearizations by Dopico, Lawrence, Perez and Van Dooren, and share with it its key features, namely, it is a rigorous analysis valid for finite perturbations, i.e., it is not a first order analysis, it provides precise bounds, and it is valid simultaneously for a large class of structure-preserving strong linearizations.
| Original language | English |
|---|---|
| Pages (from-to) | 1189-1228 |
| Number of pages | 40 |
| Journal | Mathematics of Computation |
| Volume | 88 |
| Issue number | 317 |
| DOIs | |
| State | Published - 2018 |
Funding
Received by the editor January 12, 2017, and, in revised form, July 25, 2017, January 10, 2018, and January 23, 2018. 2010 Mathematics Subject Classification. Primary 65F15, 15A18, 15A21, 15A22, 15A54, 93B18. Key words and phrases. Structured backward error analysis, complete polynomial eigenprob-lems, structured matrix polynomials, structure-preserving linearizations, Möbius transformations, matrix pertubation theory, dual minimal bases. The first author was supported by Ministerio de Economía, Industria y Competitividad of Spain and Fondo Europeo de Desarrollo Regional (FEDER) of EU through grants MTM-2015-68805-REDT, MTM-2015-65798-P (MINECO/FEDER, UE). The second author was supported by KU Leuven Research Council grant OT/14/074. The second and third authors were supported by the Belgian network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office.
| Funders | Funder number |
|---|---|
| European Commission | MTM-2015-68805-REDT, MTM-2015-65798-P |
| KU Leuven | OT/14/074 |
Keywords
- Complete polynomial eigenproblems
- Dual minimal bases
- Matrix pertubation theory
- Möbius transformations
- Structure-preserving linearizations
- Structured backward error analysis
- Structured matrix polynomials