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170 CHAPTER 10. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-2 For any W and V which bring B + ρ 1 C into such a form, the dimensions h, g, ε 1 , ..., ε s (when ordered), η 1 ,...,η t (when ordered) and n r are uniquely determined by the matrices B + ρ 1 C and so is the Jordan form of the regular part D r + ρ 1 E r . This Jordan form is given by: ⎛ ⎞ N 1 . .. N r1 J 1 D r + ρ 1 E r ∼ . .. , (10.20) J r2 R 1 ⎜ . ⎝ .. ⎟ ⎠ R r3 with: ⎛ N j = ⎜ ⎝ . .. 1 ρ 1 0 . .. ρ 1 0 1 ⎞ ⎟ ⎠ of dimension k j × k j (1 ≤ k 1 ≤···≤k r1 ), (10.21) ⎛ J j = ⎜ ⎝ ⎞ ρ 1 1 0 . .. . .. ⎟ 1 ⎠ of dimension l j × l j (1 ≤ l 1 ≤···≤l r2 ), (10.22) 0 ρ 1 and ⎛ R j = ⎜ ⎝ ⎞ ρ 1 − α j 1 0 . .. . .. ⎟ 1 ⎠ of dimension m j×m j (1 ≤ m 1 ≤···≤m r3 ). 0 ρ 1 − α j (10.23)
10.3. THE KRONECKER CANONICAL FORM OF A MATRIX PENCIL 171 Example 10.1. As an illustrative example, the Kronecker canonical form of a pencil B + ρ 1 C with h =2,g =1,ε 1 =2,ε 2 =3,η 1 = 1 and η 2 = 2 is shown below: ⎛ ⎞ 0 0 ρ 1 1 0 0 ρ 1 1 ρ 1 1 0 0 0 ρ 1 1 0 0 0 ρ 1 1 . (10.24) ρ 1 1 ρ 1 0 1 ρ 1 ⎜ ⎟ ⎝ 0 1 ⎠ D r + ρ 1 E r The empty blocks correspond to zero blocks of appropriate dimensions. The pencil D r + ρ 1 E r is the regular part of the pencil B + ρ 1 C, which can also be brought into a canonical form as in (10.20). Matrix pencils B + ρ 1 C and D + ρ 1 E, of dimension m × n, related by Equation (10.16), where W and V are square non-singular matrices independent of ρ 1 , are called strictly equivalent. The eigenvector ṽ in the eigenvalue problem (B + ρ 1 C)ṽ =0is transformed conformably to w = V −1 ṽ such that the following equation holds: W −1 (B + ρ 1 C)V (V −1 ṽ)=(D + ρ 1 E)w =0. (10.25) As shown in Theorem 10.3, the Kronecker Canonical form allows for a decomposition of a singular matrix pencil into a regular and a singular part. The singular part relates to all the so-called indeterminate eigenvalues. The regular part is what we are interested in since it relates to infinite and finite eigenvalues. For our purposes, the indeterminate, the infinite and the zero eigenvalues will not yield feasible solutions for our model-order reduction problem and should therefore all be excluded. The associated Jordan block structure for these eigenvalues can be obtained with exact arithmetic (if one starts from an exact matrix pencil B +ρ 1 C). In this way it is possible to arrive algebraically at a smaller sized exact regular matrix pencil in which both matrices are invertible. Its eigenvalues can then be computed reliably by standard numerical methods. In the next subsections the decomposition of a singular matrix pencil into a singular and a regular part and the deflation of a matrix pencil is discussed. Actually, the parts corresponding to indeterminate, zero and infinite eigenvalues are split off using exact arithmetic and the finite eigenvalues of the remaining (regular) part are computed using numerical arithmetic.
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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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Page 134 and 135: 120 CHAPTER 7. NUMERICAL EXPERIMENT
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Page 150 and 151: 136 CHAPTER 8. H 2 MODEL-ORDER REDU
Page 175 and 176: Chapter 10 H 2 Model-order reductio
Page 177 and 178: 10.1. SOLVING THE SYSTEM OF QUADRAT
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Page 181 and 182: 10.2. LINEARIZING A POLYNOMIAL EIGE
Page 183: 10.3. THE KRONECKER CANONICAL FORM
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Page 193 and 194: 10.4. COMPUTING THE APPROXIMATION G
Page 195 and 196: 10.6. EXAMPLES 181 5. Compute the n
Page 197 and 198: 10.6. EXAMPLES 183 Figure 10.1: Spa
Page 199 and 200: 10.6. EXAMPLES 185 Figure 10.3: The
Page 201 and 202: 10.6. EXAMPLES 187 Figure 10.5: Imp
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Page 207 and 208: 10.6. EXAMPLES 193 Table 10.1: Redu
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Page 211 and 212: 11.2. LINEARIZING THE TWO-PARAMETER
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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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