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268 LIST OF PUBLICATIONS [9] I.W.M. Bleylevens, B. Hanzon, and R.L.M. Peeters. A multidimensional systems approach to polynomial optimization. Proceedings of the MTNS, Leuven, Belgium, 2004. [10] I.W.M. Bleylevens, R.L.M. Peeters, and B. Hanzon. A multidimensional systems approach to algebraic optimization. Proceedings of the HPOPT - the 8th International Workshop on High Performance Optimization Techniques: Optimization and Polynomials, Amsterdam, 2004. [11] R.L.M. Peeters, I.W.M. Bleylevens, and B. Hanzon. A mixed algebraic-numerical global optimization approach to H 2 model reduction. Proceedings of the 22th Benelux Meeting on Systems and Control, Lommel, Belgium, 2003. [12] I.W.M. Bleylevens. De berekening van de globale minima van een polynomiaal criterium. Master Thesis (in Dutch), Department of Mathematics, Maastricht University, 2003.
List of Symbols and Abbreviations Abbreviation Description Definition ARPACK Arnoldi algorithm page 51 Eig Direct eigenvalue solver in Matlab page 109 Eigensystem Eigensystem solver in Mathematica page 109 Eigs Iterative eigenvalue solver in Matlab page 110 Flops Floating point operations page 60 GloptiPoly Multivariable polynomial optimization problem solver page 52 in Matlab JD Jacobi–Davidson method by M. Hochstenbach page 81 JDCOMM Jacobi–Davidson method for commuting matrices page 81 JDQR Jacobi–Davidson QR-method page 81 JDQZ Jacobi–Davidson QZ-method page 81 LMI Linear matrix inequality page 111 MIMO Multiple-input, multiple-output page 135 NSolve Solver for polynomial systems of equations in Mathematica page 108 RGS Repeated Gram-Schmidt page 86 RSPP Relaxation of the shortest path problem page 60 SDP Semi Definite Program page 111 SeDuMi Self-dual-minimization / optimization package page 111 SISO Single-input, single-output page 135 SOS Sum of squares page 111 SOSTOOLS Matlab toolbox for formulating and solving sum of page 25 square problems SPP Shortest path problem page 60 SYNAPS C++ solver for polynomial systems page 52 269
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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 231 and 232: 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
Page 276 and 277: 262 SAMENVATTING De matrix-vrije im
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Page 287 and 288: Index H 2 model-order reduction pro
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