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264 SAMENVATTING eigenwaarde probleem. Om de eigenwaardes van deze singuliere matrix nauwkeurig te kunnen berekenen, worden de singuliere delen afgesplitst door de Kronecker canonieke vorm te berekenen. In Sectie 10.6 worden drie voorbeelden gegeven waar deze co-orde k = 2 aanpak met succes is toegepast om een globaal optimale benadering van orde N − 2 te verkrijgen. Het toepassen van de Stetter–Möller matrix methode voor het co-orde k = 3 geval in Hoofdstuk 11 leidt tot twee rationale matrices in twee parameters. Dit wordt herschreven als een twee–parameter polynomiaal eigenwaarde probleem dat twee matrices en één gemeenschappelijke eigenvector bevat. Beide matrices worden samengevoegd tot één rechthoekige en singuliere matrix. Hier zijn de ideeën van Hoofdstuk 10 met betrekking tot de Kronecker canonieke vorm berekeningen nuttig om de waardes van ρ 1 en ρ 2 , die beide matrices gelijktijdig singulier maken, te berekenen. Een belangrijk resultaat is beschreven in dit hoofdstuk: de oplossingen van het stelsel vergelijkingen worden bepaald door de waardes die de transformatie matrices, die voor de verkrijging van de Kronecker canonieke vorm gebruikt worden, singulier maken. Met behulp van deze waardes wordt de globaal optimale benadering G(s) van orde N − 3 berekend. Aan het einde van dit hoofdstuk wordt een voorbeeld uitgewerkt waarbij de co-orde k = 3 techniek een betere performance laat zien dan de co-orde k =1enk =2 technieken. Hoofdstuk 12 biedt concluderende opmerkingen en aanwijzingen voor verder onderzoek met betrekking tot de resultaten beschreven in de delen II en III van dit proefschrift.
Curriculum Vitae 1979 Born on March 12th 1979, Heerlen, the Netherlands. 1991–1997 Pre-university Education, Gymnasium Rolduc, Kerkrade. 1997–1998 Propaedeutic Chemical Technology, Hogeschool Limburg, Heerlen. 1998–2003 Master of Science in Knowledge Engineering, Math Major. Joint programme of the Maastricht University (the Netherlands) and the Limburgs Universitair Centrum (Belgium). Master thesis at the Department of Mathematics of the Maastricht University with the title ’Polynomial optimization using algebraic techniques with applications to system theory’. 2001–2002 Part-time position as software developer at MateUM B.V., Maastricht. 2001–2003 Part-time position as developer of e-SAPAS, an electronic self- and peer-assessment system, at Faculty of General Sciences and Faculty of Law, Maastricht University, Maastricht. 2003–2004 Developer of e-SAPAS at Faculty of General Sciences, Maastricht University, Maastricht. 2004–2010 PhD Student, Faculty of Humanities and Sciences, MICC, Department of Mathematics, Maastricht University, Maastricht. 2008–2009 Software Developer at IDEE (Instrument Development Engineering & Evaluation), Maastricht University, Maastricht. 2010–present Post-doctoral researcher at the BioMathematics and BioInformatics (BMI) group, Department of Knowledge Engineering, Maastricht University, Maastricht. 265
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Stellingen behorende bij het proefs
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ALGEBRAIC POLYNOMIAL SYSTEM SOLVING
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ALGEBRAIC POLYNOMIAL SYSTEM SOLVING
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For all those I love...
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ii CONTENTS II Global Optimization
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iv CONTENTS 12.2 Directions for fur
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Chapter 1 Introduction 1.1 Motivati
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1.3. RESEARCH QUESTIONS 5 The goal
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1.4. THESIS OUTLINE 7 Jacobi-Davids
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Part I General introduction and bac
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12 CHAPTER 2. ALGEBRAIC BACKGROUND
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Chapter 3 Solving polynomial system
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3.1. SOLVING POLYNOMIAL EQUATIONS I
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3.2. METHODS FOR SOLVING POLYNOMIAL
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3.3. THE STETTER-MÖLLER MATRIX MET
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3.5. EXAMPLE 37 applying the Stette
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3.5. EXAMPLE 39 be: I = 〈13x 2 2
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Chapter 4 Global optimization of mu
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4.1. THE GLOBAL MINIMUM OF A DOMINA
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4.3. AN EXAMPLE 47 Using the matric
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4.3. AN EXAMPLE 49 the Stetter-Möl
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Chapter 5 An nD-systems approach in
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5.1. THE ND-SYSTEM 53 cerned with t
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5.1. THE ND-SYSTEM 55 Example 5.1.
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5.1. THE ND-SYSTEM 57 Figure 5.2: T
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5.3. EFFICIENCY OF THE ND-SYSTEMS A
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Chapter 6 Iterative eigenvalue solv
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6.2. THE JACOBI-DAVIDSON METHOD 83
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6.2. THE JACOBI-DAVIDSON METHOD 85
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6.4. PROJECTION TO STETTER-STRUCTUR
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6.5. A JACOBI-DAVIDSON METHOD FOR C
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6.5. A JACOBI-DAVIDSON METHOD FOR C
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108 CHAPTER 7. NUMERICAL EXPERIMENT
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Part III H 2 Model-order Reduction
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136 CHAPTER 8. H 2 MODEL-ORDER REDU
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Chapter 10 H 2 Model-order reductio
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10.1. SOLVING THE SYSTEM OF QUADRAT
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10.1. SOLVING THE SYSTEM OF QUADRAT
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10.2. LINEARIZING A POLYNOMIAL EIGE
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10.3. THE KRONECKER CANONICAL FORM
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10.4. COMPUTING THE APPROXIMATION G
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10.6. EXAMPLES 181 5. Compute the n
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10.6. EXAMPLES 183 Figure 10.1: Spa
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10.6. EXAMPLES 185 Figure 10.3: The
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10.6. EXAMPLES 187 Figure 10.5: Imp
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10.6. EXAMPLES 189 Figure 10.7: Str
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10.6. EXAMPLES 191 Figure 10.8: Pol
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10.6. EXAMPLES 193 Table 10.1: Redu
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Chapter 11 H 2 Model-order reductio
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11.2. LINEARIZING THE TWO-PARAMETER
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11.3. KRONECKER CANONICAL FORM OF A
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Page 251 and 252: Bibliography [1] W. Adams and P. Lo
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Page 270 and 271: 256 APPENDIX B. POLYNOMIAL TEST SET
Page 272 and 273: 258 SUMMARY vector. This routine is
Page 274 and 275: 260 SUMMARY globally optimal approx
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Page 283 and 284: List of Symbols and Abbreviations A
Page 285 and 286: LIST OF FIGURES 271 7.7 Residual no
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