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iv CONTENTS 12.2 Directions for further research ....................... 234 Bibliography 237 A Linearizing a 2-parameter eigenvalue problem 247 A.1 Linearization with respect to µ ...................... 247 A.2 Linearization with respect to λ ...................... 248 A.3 Example ................................... 250 B Polynomial test set 255 Summary 257 Samenvatting 261 Curriculum Vitae 265 List of Publications 267 List of Symbols and Abbreviations 269 List of Figures 270 List of Tables 272 Index 273
Acknowledgements It is a pleasure to finally thank those who made this thesis possible. Without the support of the people mentioned here it would have been impossible to complete this project. First of all, I want to thank my promotores and supervisors professor Ralf Peeters and professor Bernard Hanzon. Ralf supervised and supported me from the beginning, when he introduced me to computer algebra during my master thesis research, all the way to the final level of this thesis. The same holds for Bernard. He was always very helpful in our research no matter if he worked at the CWI in Amsterdam or at the UCC in Cork (Ireland). He always created a very welcoming atmosphere. Another word of thanks goes to my colleagues of the Department of Knowledge Engineering and especially to the people I shared a room with in the years 2004–2009, to the students I co-supervised, and of course to Jan Paredis, regarding the e-SAPAS project. It was always fun to work with all of you. Another word of thanks for another colleague, Dr. Michiel Hochstenbach, from the Technical University Eindhoven. You were always very enthusiastic about our research, however I wish we had spent more time together. A very special word of thanks goes to my friends, without giving any particular names. You know I feel endless love for all of you and this helped me through the good times, but also through the bad times I experienced during my PhD period. Last but not least, I offer my regards and blessings to Jennifer, the love of my life and my best friend ever, and also to my parents, parents-in-law, grand-parents, and family. I am proud of you all. Ivo Bleylevens Maastricht, December 2010 (ivo.bleylevens@gmail.com) 1
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Page 12 and 13: ii CONTENTS II Global Optimization
Page 17 and 18: Chapter 1 Introduction 1.1 Motivati
Page 19 and 20: 1.3. RESEARCH QUESTIONS 5 The goal
Page 21 and 22: 1.4. THESIS OUTLINE 7 Jacobi-Davids
Page 23: Part I General introduction and bac
Page 26 and 27: 12 CHAPTER 2. ALGEBRAIC BACKGROUND
Page 39 and 40: Chapter 3 Solving polynomial system
Page 41 and 42: 3.1. SOLVING POLYNOMIAL EQUATIONS I
Page 43 and 44: 3.2. METHODS FOR SOLVING POLYNOMIAL
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Page 51 and 52: 3.5. EXAMPLE 37 applying the Stette
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Page 57 and 58: Chapter 4 Global optimization of mu
Page 59 and 60: 4.1. THE GLOBAL MINIMUM OF A DOMINA
Page 61 and 62: 4.3. AN EXAMPLE 47 Using the matric
Page 63 and 64: 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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11.4. COMPUTING THE APPROXIMATION G
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11.5. EXAMPLE 223 AãN−2 (ρ 1 ,
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11.5. EXAMPLE 225 of both matrices
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11.5. EXAMPLE 227 of solutions (ρ
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Chapter 12 Conclusions & directions
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12.1. CONCLUSIONS 231 used first. T
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12.1. CONCLUSIONS 233 solver as des
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12.2. DIRECTIONS FOR FURTHER RESEAR
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Bibliography [1] W. Adams and P. Lo
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BIBLIOGRAPHY 239 [26] A. Cayley. On
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BIBLIOGRAPHY 241 [58] M.E. Hochsten
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BIBLIOGRAPHY 243 [90] S. Prajna, A.
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Appendix A A linearization techniqu
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A.2. LINEARIZATION WITH RESPECT TO
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A.3. EXAMPLE 251 The first step of
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A.3. EXAMPLE 253 where the eigenvec
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256 APPENDIX B. POLYNOMIAL TEST SET
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258 SUMMARY vector. This routine is
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260 SUMMARY globally optimal approx
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262 SAMENVATTING De matrix-vrije im
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264 SAMENVATTING eigenwaarde proble
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List of Publications [1] I.W.M. Ble
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List of Symbols and Abbreviations A
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LIST OF FIGURES 271 7.7 Residual no
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Index H 2 model-order reduction pro
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