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ii CONTENTS II Global Optimization 41 4 Global optimization of multivariate polynomials 43 4.1 The global minimum of a dominated polynomial ............ 44 4.2 The Stetter-Möller matrix method for global optimization ....... 46 4.3 An example ................................. 47 5 nD-systems approach in polynomial optimization 51 5.1 The nD-system ............................... 52 5.2 Usage of the nD-system in polynomial optimization ........... 58 5.3 Efficiency of the nD-systems approach .................. 59 5.3.1 A linear complexity result for computing y t1,...,t n ........ 59 5.3.2 Formulation as a shortest path problem ............. 60 5.3.3 Some heuristic procedures for the shortest path problem .... 69 5.3.4 Parallel computations in the nD-systems approach ....... 79 6 Iterative eigenvalue solvers 81 6.1 Iterative eigenvalue solvers ......................... 82 6.2 The Jacobi–Davidson method ....................... 83 6.2.1 Subspace expansion ........................ 83 6.2.2 Subspace extraction ........................ 84 6.2.3 Remaining important issues .................... 85 6.2.4 Pseudocode of the Jacobi–Davidson method . . ......... 86 6.3 Target selection: the smallest real eigenvalue .............. 86 6.4 Projection to Stetter-structure . ..................... 87 6.4.1 Projecting an approximate eigenvector .............. 87 6.4.2 Embedding projection in a Jacobi–Davidson method ...... 98 6.5 A Jacobi–Davidson method for commuting matrices .......... 102 6.5.1 The method JDCOMM ........................ 102 6.5.2 The JDCOMM expansion phase ................... 103 6.5.3 Pseudocode of the JDCOMM method ................ 104 7 Numerical experiments 107 7.1 Computing the minimum of a polynomial of order 8 .......... 108 7.2 Computing the global minimum using target selection ......... 112 7.3 Projection to Stetter structure ...................... 115 7.4 Parallel computations in the nD-systems approach ........... 117 7.5 Numerical experiments with JDCOMM software ............ 117 7.5.1 Experiment 1 ............................ 118 7.5.2 Experiment 2 ............................ 121 7.5.3 Experiment 3 ............................ 123 7.5.4 Experiment 4 ............................ 126
CONTENTS iii III H 2 Model-order Reduction 131 8 H 2 model-order reduction 135 8.1 Introduction ................................. 135 8.2 The H 2 model-order reduction problem ................. 137 8.3 A reparameterization of the model reduction problem ......... 141 8.4 The H 2 -approximation criterion ...................... 146 8.5 Stetter-Möller for H 2 model-order reduction ............... 147 8.6 Computing the approximation G(s) ................... 149 9 H 2 Model-order reduction from order N to N-1 151 9.1 Solving the system of quadratic equations ................ 151 9.2 Computing the approximation G(s) ................... 153 9.3 Example ................................... 154 10 H 2 Model-order reduction from order N to N-2 161 10.1 Solving the system of quadratic equations ................ 163 10.2 Linearizing a polynomial eigenvalue problem . . . ............ 166 10.3 The Kronecker canonical form of a matrix pencil ............ 168 10.3.1 The singular part of a pencil ................... 172 10.3.2 The regular part of a pencil .................... 177 10.3.3 Deflation of a singular pencil ................... 179 10.4 Computing the approximation G(s) ................... 179 10.5 Algorithm for H 2 model-order reduction for the co-order k = 2 case . . 180 10.6 Examples . ................................. 181 10.6.1 Example 1 .............................. 182 10.6.2 Example 2 .............................. 186 10.6.3 Example 3 .............................. 190 11 H 2 Model-order reduction from order N to N-3 195 11.1 Solving the system of quadratic equations ................ 196 11.2 Linearizing the two-parameter eigenvalue problem ........... 197 11.3 Kronecker canonical form of a two-parameter pencil . ......... 198 11.3.1 Transformation into the Kronecker canonical form ....... 199 11.3.2 Transformation matrices for linear 2-parameter pencil ..... 203 11.3.3 Transformation matrices for non-linear 2-parameter pencil . . . 209 11.3.4 A second approach for a non-linear 2-parameter pencil ..... 215 11.4 Computing the approximation G(s) ................... 221 11.5 Example ................................... 222 11.5.1 Model-order reduction from 4 to 1 ................ 222 11.5.2 Comparing co-order 3 technique with co-order 1 and 2 ..... 228 12 Conclusions & directions for further research 229 12.1 Conclusions ................................. 229
Page 3 and 4: Stellingen behorende bij het proefs
Page 5 and 6: ALGEBRAIC POLYNOMIAL SYSTEM SOLVING
Page 7 and 8: ALGEBRAIC POLYNOMIAL SYSTEM SOLVING
Page 9: For all those I love...
Page 14 and 15: iv CONTENTS 12.2 Directions for fur
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
Page 49 and 50: 3.3. THE STETTER-MÖLLER MATRIX MET
Page 51 and 52: 3.5. EXAMPLE 37 applying the Stette
Page 53: 3.5. EXAMPLE 39 be: I = 〈13x 2 2
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
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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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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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