umu.sePublications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Bounds for the distance between nearby Jordan and Kronecker structures in a closure hierarchy
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Compting Center North (HPC2N).
2003 (English)In: Journal of Mathematical Science, ISSN 1072-3374, Vol. 114, no 6, 1765-1779 p.Article in journal (Refereed) Published
Abstract [en]

Computing the fine-canonical-structure elements of matrices and matrix pencils are ill-posed problems. Therefore, besides knowing the canonical structure of a matrix or a matrix pencil, it is equally important to know what are the nearby canonical structures that explain the behavior under small perturbations. Qualitative strata information is provided by our StratiGraph tool. Here, we present lower and upper bounds for the distance between Jordan and Kronecker structures in a closure hierarchy of an orbit or bundle stratification. This quantitative information is of importance in applications, e.g., distance to more degenerate systems (uncontrollability). Our upper bounds are based on staircase regularizing perturbations. The lower bounds are of EckartYoung type and are derived from a matrix representation of the tangent space of the orbit of a matrix or a matrix pencil. Computational results illustrate the use of the bounds.

Place, publisher, year, edition, pages
2003. Vol. 114, no 6, 1765-1779 p.
Identifiers
URN: urn:nbn:se:umu:diva-5400OAI: oai:DiVA.org:umu-5400DiVA: diva2:144905
Available from: 2006-10-11 Created: 2006-10-11 Last updated: 2011-02-22
In thesis
1. Software tools for matrix canonical computations and web-based software library environments
Open this publication in new window or tab >>Software tools for matrix canonical computations and web-based software library environments
2006 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This dissertation addresses the development and use of novel software tools and environments for the computation and visualization of canonical information as well as stratification hierarchies for matrices and matrix pencils.

The simplest standard shape to which a matrix pencil with a given set of eigenvalues can be reduced is called the Kronecker canonical form (KCF). The KCF of a matrix pencil is unique, and all pencils in the manifold of strictly equivalent pencils - collectively termed the orbit - can be reduced to the same canonical form and so have the same canonical structure. For a problem with fixed input size, all orbits are related under small perturbations. These relationships can be represented in a closure hierarchy with a corresponding graph depicting the stratification of these orbits. Since degenerate canonical structures are common in many applications, software tools to determine canonical information, especially under small perturbations, are central to understanding the behavior of these problems.

The focus in this dissertation is the development of a software tool called StratiGraph. Its purpose is the computation and visualization of stratification graphs of orbits and bundles (i.e., union of orbits in which the eigenvalues may change) for matrices and matrix pencils. It also supports matrix pairs, which are common in control systems. StratiGraph is extensible by design, and a well documented plug-in feature enables it, for example, to communicate with Matlab(TM). The use and associated benefits of StratiGraph are illustrated via numerous examples. Implementation considerations such as flexible software design, suitable data representations, and good and efficient graph layout algorithms are also discussed.

A way to estimate upper and lower bounds on the distance between an input S and other orbits is presented. The lower bounds are of Eckhart-Young type, based on the matrix representation of the associated tangent spaces. The upper bounds are computed as the Frobenius norm F of a perturbation such that S + F is in the manifold defining a specified orbit. Using associated plug-ins to StratiGraph this information can be computed in Matlab, while visualization alongside other canonical information remains within StratiGraph itself.

Also, a proposal of functionality and structure of a framework for computation of matrix canonical structure is presented. Robust, well-known algorithms, as well algorithms improved and developed in this work, are used. The framework is implemented as a prototype Matlab toolbox. The intention is to collect software for computing canonical structures as well as for computing bounds and to integrate it with the theory of stratification into a powerful new environment called the MCS toolbox.

Finally, a set of utilities for generating web computing environments related to mathematical and engineering library software is presented. The web interface can be accessed from a standard web browser with no need for additional software installation on the local machine. Integration with the control and systems library SLICOT further demonstrates the efficacy of this approach.

Place, publisher, year, edition, pages
Umeå: Datavetenskap, 2006. 30 p.
Series
Report / UMINF, ISSN 0348-0542 ; 06.30
Keyword
Canonical structure, Jordan canonical form, controllability, StratiGraph, Matlab toolbox, Kronecker canonical form, matrix, matrix pencil, perturbation theory, closure hirerarchy, matrix stratification, control system, observability
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-890 (URN)91-7264-144-X (ISBN)
Public defence
2006-11-03, MA121, MIT, Umeå Universitet, Umeå, 13:15 (English)
Opponent
Supervisors
Available from: 2006-10-11 Created: 2006-10-11 Last updated: 2010-04-19Bibliographically approved

Open Access in DiVA

No full text

Other links

http://dx.doi.org/10.1023/A:1022498301583

Search in DiVA

By author/editor
Elmroth, ErikJohansson, PedherKågström, Bo
By organisation
Department of Computing ScienceHigh Performance Compting Center North (HPC2N)

Search outside of DiVA

GoogleGoogle Scholar

urn-nbn

Altmetric score

urn-nbn
Total: 114 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf