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Software tools for matrix canonical computations and web-based software library environments
Umeå University, Faculty of Science and Technology, Department of Computing Science.
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 [en]
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: urn:nbn:se:umu:diva-890ISBN: 91-7264-144-X OAI: oai:DiVA.org:umu-890DiVA: diva2:144913
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
List of papers
1. Computation and presentation of graphs displaying closure hierarchies of Jordan and Kronecker structures
Open this publication in new window or tab >>Computation and presentation of graphs displaying closure hierarchies of Jordan and Kronecker structures
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
Keyword
stratification, Jordan and Kronecker canonical form, structure hierarchy, orbit, bundle, StratiGraph, controllability, observability, matrix pencils, perturbation theory
National Category
Computer Science
Identifiers
urn:nbn:se:umu:diva-5399 (URN)10.1002/nla.253 (DOI)
Note

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

Available from: 2006-10-11 Created: 2006-10-11 Last updated: 2017-12-14Bibliographically approved
2. Bounds for the distance between nearby Jordan and Kronecker structures in a closure hierarchy
Open this publication in new window or tab >>Bounds for the distance between nearby Jordan and Kronecker structures in a closure hierarchy
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.

Identifiers
urn:nbn:se:umu:diva-5400 (URN)
Available from: 2006-10-11 Created: 2006-10-11 Last updated: 2011-02-22
3. Orbit and bundle stratification of controllability and observability matrix pairs in StratiGraph
Open this publication in new window or tab >>Orbit and bundle stratification of controllability and observability matrix pairs in StratiGraph
2004 (English)In: Proceedings MTNS 2004Article in journal (Refereed) Published
Identifiers
urn:nbn:se:umu:diva-5401 (URN)
Available from: 2006-10-11 Created: 2006-10-11 Last updated: 2011-02-22
4. StratiGraph software design and algorithms
Open this publication in new window or tab >>StratiGraph software design and algorithms
2006 (English)Report (Other (popular science, discussion, etc.))
Series
Report / UMINF, ISSN 0348-0542 ; 06:16
Keyword
StratiGraph, Software Development
National Category
Software Engineering
Research subject
Computer Science
Identifiers
urn:nbn:se:umu:diva-5402 (URN)
Available from: 2006-10-11 Created: 2006-10-11 Last updated: 2010-04-20Bibliographically approved
5. StratiGraph user's guide
Open this publication in new window or tab >>StratiGraph user's guide
2003 (English)Report (Other academic)
Place, publisher, year, edition, pages
Umeå: University, 2003
Series
Phonum : reports in phonetics, ISSN 1101-2714 ; 03:23
Keyword
StratiGraph, Software Developement
National Category
Software Engineering
Research subject
Computer Science
Identifiers
urn:nbn:se:umu:diva-5403 (URN)
Available from: 2006-10-11 Created: 2006-10-11 Last updated: 2010-04-20Bibliographically approved
6. StratiGraph developer's guide
Open this publication in new window or tab >>StratiGraph developer's guide
2006 (English)Report (Other academic)
Place, publisher, year, edition, pages
Umeå: Umeå Universitet, 2006
Series
Report / UMINF, ISSN 0348-0542 ; 06:14
Keyword
Canonical structures, StratiGraph, Sowftware development
National Category
Software Engineering
Research subject
Computer Science
Identifiers
urn:nbn:se:umu:diva-5404 (URN)
Available from: 2006-10-11 Created: 2006-10-11 Last updated: 2010-04-20Bibliographically approved
7. Matrix canonical structure toolbox
Open this publication in new window or tab >>Matrix canonical structure toolbox
2006 (English)Report (Other (popular science, discussion, etc.))
Place, publisher, year, edition, pages
Umeå: Universitet, 2006
Series
Report / UMINF, ISSN 0348-0542 ; 06:15
Keyword
Matlab, StratiGraph, Canonical Structures
National Category
Software Engineering
Identifiers
urn:nbn:se:umu:diva-5405 (URN)
Available from: 2006-10-11 Created: 2006-10-11 Last updated: 2010-04-20Bibliographically approved
8. A Web Computing Environment for the SLICOT Library
Open this publication in new window or tab >>A Web Computing Environment for the SLICOT Library
2001 (English)In: The Third NICONET Workshop on Numerical Control Software, 53-61 p.Article in journal (Refereed) Published
Abstract [en]

A prototype web computing environment for computations related to the design and analysis of control systems using the SLICOT software library is presented. The web interface can be accessed from a standard world wide web browser with no need for additional software installations on the local machine. The environment provides user-friendly access to SLICOT routines where run-time options are specified by mouse clicks on appropriate buttons. Input data can be entered directly into the web interface by the user or uploaded from a local computer in a standard text format or in Matlab binary format. Output data is presented in the web browser window and possible to download in a number of different formats, including Matlab binary. The environment is ideal for testing the SLICOT software before performing a software installation or for performing a limited number of computations. It is also highly recommended for education as it is easy to use, and basically self-explanatory, with the users' guide integrated in the user interface.

National Category
Computer Science
Research subject
Computing Science
Identifiers
urn:nbn:se:umu:diva-5406 (URN)
Available from: 2006-10-11 Created: 2006-10-11 Last updated: 2011-02-22
9. Semi-automatic generation of web-based computing environments for software libraries
Open this publication in new window or tab >>Semi-automatic generation of web-based computing environments for software libraries
2002 (English)In: Lecture Notes in Computer Science, ISSN 0302-9743, E-ISSN 1611-3349, Vol. 2329, 872-880 p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:umu:diva-5407 (URN)10.1007/3-540-46043-8 (DOI)
Available from: 2006-10-11 Created: 2006-10-11 Last updated: 2017-12-14Bibliographically approved

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