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274 INDEX Lagrange Multiplier, 100 Lanczos method, 82 Laplace transform, 139 Laurent series, 138 layer, 68 least-increments method, 76 left generalized eigenvalue problem, 167 left-half plane, 133 linear complexity, 73 linear least-squares method, 96 linear matrix inequality, 111 linear method, 71 linear operator, 46 LMI, 111 local minimum, 49, 152 local search method, 110 Lyapunov equation, 138 Lyapunov-Stein equation, 139 matrix pencil, 166 matrix solution, 46, 148 McMillan degree, 133 MIMO, 135 minimal index, 177 minimal stable pattern, 63, 67, 74 Minkowski, 44 model approximation problem, 133 model-order reduction problem, 133 monomial basis, 45 monomial ordering, 45 multidimensional system, 53 multidimensional time instant, 53 multidimensional time series, 53 nodes, 60 one-parameter problem, 202 orthant, 62, 66 Padé approximation, 134 parallel, 61 Parseval’s identity, 139 Perron-Frobenius, 110 polar coordinates, 88 polynomial eigenvalue, 137, 165 polynomial eigenvalue problem, 137, 165 power method, 82 power set, 61 preconditioner, 84, 110 refined Ritz vector, 85 regular matrix pencil, 168 relaxation of shortest path problem, 60, 61 reversed lexicographic, 45 RGS, 86 Ritz vector, 85 RSPP, 60, 61 saddle point, 49, 152 SDP, 111 SeDuMi, 111 semi-definite program, 111 shift operator, 52, 53 shortest path problem, 52, 60 similarity transform, 110 simplex, 73 singular matrix pencil, 168 SISO, 135 SOS, 111 SOSTOOLS, 52, 111 sparse, 51, 62 SPP, 60 stable pattern, 60, 62, 66, 67, 73 state transition, 60 state vector, 53 Stetter vector, 110 strictly equivalent, 171 suboptimal, 70 subspace expansion, 83 subspace extraction, 84 sum of squares, 111 SYNAPS, 52, 111 terminal node, 60 theory of moments, 111 time instant, 53 total degree, 44, 45 trace operator, 141 translation, 62, 66 tree, 61 unit disk, 133 upper bound, 60 variety, 45 Weak Hilbert Nullstellensatz, 34 weight function, 60 quotient space, 45, 54 QZ-algorithm, 166 Rayleigh-Ritz approach, 84 recursion, 58
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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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11.4. COMPUTING THE APPROXIMATION G
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Page 251 and 252: Bibliography [1] W. Adams and P. Lo
Page 253 and 254: BIBLIOGRAPHY 239 [26] A. Cayley. On
Page 255 and 256: BIBLIOGRAPHY 241 [58] M.E. Hochsten
Page 257: BIBLIOGRAPHY 243 [90] S. Prajna, A.
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Page 270 and 271: 256 APPENDIX B. POLYNOMIAL TEST SET
Page 272 and 273: 258 SUMMARY vector. This routine is
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Page 278 and 279: 264 SAMENVATTING eigenwaarde proble
Page 281 and 282: List of Publications [1] I.W.M. Ble
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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