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Computation and presentation of graphs displaying closure hierarchies of Jordan and Kronecker structures
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 Computing Center North (HPC2N).
2001 (English)In: Numerical Linear Algebra with Applications, ISSN 1070-5325, E-ISSN 1099-1506, Vol. 8, no 6-7, 381-399 p.Article in journal (Refereed) Published
Abstract [en]

StratiGraph, a Java-based tool for computation and presentation of closure hierarchies of Jordan and Kronecker structures is presented. The tool is based on recent theoretical results on stratifications of orbits and bundles of matrices and matrix pencils. A stratification reveals the complete hierarchy of nearby structures. information critical for explaining the qualitative behaviour of linear systems under perturbations. StratiGraph facilitates the application of these theories and visualizes the resulting hierarchy as a graph. Nodes in the graph represent orbits or bundles of matrices or matrix pencils. Edges represent covering relations in the closure hierarchy. Given a Jordan or Kronecker structure, a user can obtain the complete information of nearby structures simply by mouse clicks on nodes of interest. This contribution gives an overview of the StratiGraph tool, presents its main functionalities and other features, and illustrates its use by sample applications.

Copyright (C) 2001 John Wiley & Sons, Ltd.

Place, publisher, year, edition, pages
John Wiley & Sons, 2001. Vol. 8, no 6-7, 381-399 p.
Keyword [en]
stratification, Jordan and Kronecker canonical form, structure hierarchy, orbit, bundle, StratiGraph, controllability, observability, matrix pencils, perturbation theory
National Category
Computer Science
Identifiers
URN: urn:nbn:se:umu:diva-5399DOI: 10.1002/nla.253OAI: oai:DiVA.org:umu-5399DiVA: diva2:144904
Note

Special Issue: Numerical Linear Algebra Techniques for Control and Signal Processing

Available from: 2006-10-11 Created: 2006-10-11 Last updated: 2012-05-15Bibliographically approved
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

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Publisher's full texthttp://dx.doi.org/10.1002/nla.253

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Elmroth, ErikJohansson, PedherKågström, Bo

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