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Software tools for matrix canonical computations and web-based software library environments
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.ORCID-id: 0000-0002-9957-7728
2006 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Umeå: Datavetenskap , 2006. , s. 30
Serie
Report / UMINF, ISSN 0348-0542 ; 06.30
Nyckelord [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
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
URN: urn:nbn:se:umu:diva-890ISBN: 91-7264-144-X (tryckt)OAI: oai:DiVA.org:umu-890DiVA, id: diva2:144913
Disputation
2006-11-03, MA121, MIT, Umeå Universitet, Umeå, 13:15 (Engelska)
Opponent
Handledare
Tillgänglig från: 2006-10-11 Skapad: 2006-10-11 Senast uppdaterad: 2019-06-26Bibliografiskt granskad
Delarbeten
1. Computation and presentation of graphs displaying closure hierarchies of Jordan and Kronecker structures
Öppna denna publikation i ny flik eller fönster >>Computation and presentation of graphs displaying closure hierarchies of Jordan and Kronecker structures
2001 (Engelska)Ingår i: Numerical Linear Algebra with Applications, ISSN 1070-5325, E-ISSN 1099-1506, Vol. 8, nr 6-7, s. 381-399Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
John Wiley & Sons, 2001
Nyckelord
stratification, Jordan and Kronecker canonical form, structure hierarchy, orbit, bundle, StratiGraph, controllability, observability, matrix pencils, perturbation theory
Nationell ämneskategori
Datavetenskap (datalogi)
Identifikatorer
urn:nbn:se:umu:diva-5399 (URN)10.1002/nla.253 (DOI)
Anmärkning

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

Tillgänglig från: 2006-10-11 Skapad: 2006-10-11 Senast uppdaterad: 2019-06-26Bibliografiskt granskad
2. Bounds for the distance between nearby Jordan and Kronecker structures in a closure hierarchy
Öppna denna publikation i ny flik eller fönster >>Bounds for the distance between nearby Jordan and Kronecker structures in a closure hierarchy
2003 (Engelska)Ingår i: Journal of Mathematical Science, ISSN 1072-3374, Vol. 114, nr 6, s. 1765-1779Artikel i tidskrift (Refereegranskat) 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.

Identifikatorer
urn:nbn:se:umu:diva-5400 (URN)
Tillgänglig från: 2006-10-11 Skapad: 2006-10-11 Senast uppdaterad: 2019-06-26
3. Orbit and bundle stratification of controllability and observability matrix pairs in StratiGraph
Öppna denna publikation i ny flik eller fönster >>Orbit and bundle stratification of controllability and observability matrix pairs in StratiGraph
2004 (Engelska)Ingår i: Proceedings MTNS 2004Artikel i tidskrift (Refereegranskat) Published
Identifikatorer
urn:nbn:se:umu:diva-5401 (URN)
Tillgänglig från: 2006-10-11 Skapad: 2006-10-11 Senast uppdaterad: 2019-06-26
4. StratiGraph software design and algorithms
Öppna denna publikation i ny flik eller fönster >>StratiGraph software design and algorithms
2006 (Engelska)Rapport (Övrig (populärvetenskap, debatt, mm))
Serie
Report / UMINF, ISSN 0348-0542 ; 06:16
Nyckelord
StratiGraph, Software Development
Nationell ämneskategori
Programvaruteknik
Forskningsämne
data- och systemvetenskap
Identifikatorer
urn:nbn:se:umu:diva-5402 (URN)
Tillgänglig från: 2006-10-11 Skapad: 2006-10-11 Senast uppdaterad: 2019-06-26Bibliografiskt granskad
5. StratiGraph user's guide
Öppna denna publikation i ny flik eller fönster >>StratiGraph user's guide
2003 (Engelska)Rapport (Övrigt vetenskapligt)
Ort, förlag, år, upplaga, sidor
Umeå: University, 2003
Serie
Phonum : reports in phonetics, ISSN 1101-2714 ; 03:23
Nyckelord
StratiGraph, Software Developement
Nationell ämneskategori
Programvaruteknik
Forskningsämne
data- och systemvetenskap
Identifikatorer
urn:nbn:se:umu:diva-5403 (URN)
Tillgänglig från: 2006-10-11 Skapad: 2006-10-11 Senast uppdaterad: 2019-06-26Bibliografiskt granskad
6. StratiGraph developer's guide
Öppna denna publikation i ny flik eller fönster >>StratiGraph developer's guide
2006 (Engelska)Rapport (Övrigt vetenskapligt)
Ort, förlag, år, upplaga, sidor
Umeå: Umeå Universitet, 2006
Serie
Report / UMINF, ISSN 0348-0542 ; 06:14
Nyckelord
Canonical structures, StratiGraph, Sowftware development
Nationell ämneskategori
Programvaruteknik
Forskningsämne
data- och systemvetenskap
Identifikatorer
urn:nbn:se:umu:diva-5404 (URN)
Tillgänglig från: 2006-10-11 Skapad: 2006-10-11 Senast uppdaterad: 2019-06-26Bibliografiskt granskad
7. Matrix canonical structure toolbox
Öppna denna publikation i ny flik eller fönster >>Matrix canonical structure toolbox
2006 (Engelska)Rapport (Övrig (populärvetenskap, debatt, mm))
Ort, förlag, år, upplaga, sidor
Umeå: Universitet, 2006
Serie
Report / UMINF, ISSN 0348-0542 ; 06:15
Nyckelord
Matlab, StratiGraph, Canonical Structures
Nationell ämneskategori
Programvaruteknik
Identifikatorer
urn:nbn:se:umu:diva-5405 (URN)
Tillgänglig från: 2006-10-11 Skapad: 2006-10-11 Senast uppdaterad: 2019-06-26Bibliografiskt granskad
8. A Web Computing Environment for the SLICOT Library
Öppna denna publikation i ny flik eller fönster >>A Web Computing Environment for the SLICOT Library
2001 (Engelska)Ingår i: The Third NICONET Workshop on Numerical Control Software, s. 53-61Artikel i tidskrift (Refereegranskat) 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.

Nationell ämneskategori
Datavetenskap (datalogi)
Forskningsämne
administrativ databehandling
Identifikatorer
urn:nbn:se:umu:diva-5406 (URN)
Tillgänglig från: 2006-10-11 Skapad: 2006-10-11 Senast uppdaterad: 2019-06-26
9. Semi-automatic generation of web-based computing environments for software libraries
Öppna denna publikation i ny flik eller fönster >>Semi-automatic generation of web-based computing environments for software libraries
2002 (Engelska)Ingår i: Lecture Notes in Computer Science, ISSN 0302-9743, E-ISSN 1611-3349, Vol. 2329, s. 872-880Artikel i tidskrift (Refereegranskat) Published
Identifikatorer
urn:nbn:se:umu:diva-5407 (URN)10.1007/3-540-46043-8 (DOI)
Tillgänglig från: 2006-10-11 Skapad: 2006-10-11 Senast uppdaterad: 2019-06-26Bibliografiskt granskad

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