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252 APPENDIX A. LINEARIZING A 2-PARAMETER EIGENVALUE PROBLEM Using the eigenvector w of size 9n × 1 with a structure: ⎛ ⎞ v µv ⎛ ⎞ µ 2 v z ⎜ ⎟ λv w = ⎝ λz ⎠ = µλv λ 2 z µ 2 λv λ 2 v ⎜ ⎝ µλ 2 ⎟ v ⎠ µ 2 λ 2 v yields the following eigenvalue problem which is linear in µ and λ: ⎛⎛ ⎜⎜ ⎝⎝ (P 0 + µP 1 + λP 2 )w = ⎞ 0 I 0 ⎟ 0 0 I ⎠ + µ K 0 K 1 K 2 ⎛ ⎜ λ ⎝ ⎞ −I 0 0 ⎟ 0 −I 0 ⎠ 0 0 K 3 ⎛ ⎜ ⎝ 0 0 0 0 0 0 L 0 0 0 ⎞ ⎛ ⎟ ⎜ ⎠ w = ⎝ 0 0 0 ⎞ ⎞ ⎟ ⎠ . ⎟ ⎠ + (A.24) (A.25) where every block I and 0 has dimensions 3n × 3n. The dimensions of the matrices P 0 ,P 1 , and P 2 are (m 2 n) × (m 2 n)=9n × 9n. Note that the last equation represented by (A.25) satisfies Equation (A.22) times the eigenvector w and that the other rows are required to capture the structure of the eigenvector w. Remark A.2. The existence of another possible approach to linearize a matrix is demonstrated below. The matrix M(µ, λ) in this example with polynomial coefficients in µ and λ with a total degree 3, can also be linearized in one step as follows: ⎛⎛ ⎜⎜ ⎝⎝ ⎞ ⎛ 0 I 0 0 0 0 0 0 I 0 0 0 0 0 0 I 0 0 + µ 0 0 0 0 I 0 ⎟ ⎜ 0 0 0 0 0 I ⎠ ⎝ N 0 N 2 N 1 N 4 N 5 N 3 ⎛ λ ⎜ ⎝ −I 0 0 0 0 0 0 0 0 0 0 0 0 −I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 N 9 N 7 0 ⎞ 0 0 0 0 0 0 −I 0 0 0 0 0 0 0 0 0 0 0 0 −I 0 0 0 0 ⎟ 0 0 −I 0 0 0 ⎠ 0 0 0 0 N 8 N 6 ⎞⎞ ⎟⎟ ⎠⎠ ⎛ ˜w = ⎜ ⎝ 0 0 0 0 0 0 ⎞ ⎟ ⎠ + (A.26)
A.3. EXAMPLE 253 where the eigenvector ˜w is structured as: ⎛ ˜w = ⎜ ⎝ v λv µv λ 2 v λµv µ 2 v ⎞ . (A.27) ⎟ ⎠ This problem is of dimension 6n × 6n and the eigenvector ˜w only incorporates powers of λ and µ up till a total degree of m − 1 = 2. Note that the last equation represented by (A.26) equals M(µ, λ)v = 0 with the matrix M(µ, λ) as in Equation (A.16).
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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
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.
Page 261 and 262: Appendix A A linearization techniqu
Page 263 and 264: A.2. LINEARIZATION WITH RESPECT TO
Page 265: A.3. EXAMPLE 251 The first step of
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
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
Page 287 and 288: Index H 2 model-order reduction pro
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