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176 CHAPTER 10. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-2 In the latter we have shown how to transform a pencil B + ρ 1 C into a form B 1 + ρ 1 C 1 containing one block L ε1 , once a solution z(ρ 1 ) for (B + ρ 1 C)z(ρ 1 )=0 of minimal degree ε 1 for ρ 1 is available. Subsequently, if B 1 + ρ 1 C 1 has a nonzero solution of minimal degree ε 2 , where ε 2 ≥ ε 1 , then it is possible to do one more transformation. Repeating until no more solutions z(ρ 1 ) are found, leads to a transformed pencil: ⎛ ⎜ ⎝ ⎞ 0 h×g L ε1 L ε2 . .. ⎟ L εs ⎠ D s + ρ 1 E s (10.38) with 0
10.3. THE KRONECKER CANONICAL FORM OF A MATRIX PENCIL 177 Note that the corresponding equations of L T η w 1 = 0 are: ⎧ ρ 1 w 1,1 =0 ⎪⎨ w 1,1 + ρ 1 w 1,2 =0 . w 1,ε−1 + ρ 1 w 1,ε =0 ⎪⎩ w 1,ε =0 (10.41) It is easy to see from (10.41) that only the trivial solution w 1 = 0 exists. The value of ρ 1 does not play a role in (10.41). The blocks L T η are said to generate indeterminate eigenvalues ρ 1 too. The quantities ε i and η i are called the minimal indices for, respectively, the columns and the rows of a pencil. One property of strictly equivalent pencils is that they have exactly the same minimal indices. During the transformation process, some special situations can occur: (i) if ε 1 + ε 2 +...+ε s = m, then the dimension of the pencil D s +ρ 1 E s is 0×0, (ii) if r = n, then there are no dependent columns and therefore the blocks L ε will be absent, (iii) if r = m, then there are no dependent rows and therefore the blocks L T η will be absent. Finally the pencil B + ρ 1 C is transformed into a quasi block diagonal matrix as in (10.17) containing a regular (hence square) block D r + ρ 1 E r . 10.3.2 The regular part of a pencil A regular pencil D r + ρ 1 E r is square, say of dimension n r × n r , such that its determinant does not vanish identically. In general, if a pencil is regular and the characteristic polynomial of degree k ≤ n r is equal to p k (ρ 1 )=det(D r + ρ 1 E r ), then the pencil has k finite eigenvalues (counting multiplicities) which are the zeros of p k (ρ 1 ) and n r − k infinite eigenvalues. The regular part D r + ρ 1 E r in (10.17) can be transformed into a Jordan canonical form as in (10.20), containing the Jordan blocks N 1 ,...,N r1 , J 1 ,...,J r2 and R 1 ,...,R r3 (see [34] and [42]). These blocks correspond with infinite, zero and nonzero finite eigenvalues, respectively. The sizes of the blocks N i , J i , and R i , the associated structure and the eigenvalues are all uniquely determined by the matrices B and C (but the order in which the blocks appear may vary). In the application to H 2 model-order reduction we do not compute all these regular blocks. Once the pencil is brought into the form (10.17), we remove the infinite and zero eigenvalues of the regular part by exact arithmetic and proceed with the remaining regular pencil, containing only finite non-zero eigenvalues, using conventional numerical methods. We will show in this subsection the canonical form of a regular matrix pencil, containing the Jordan blocks of infinite, zero and non-zero finite eigenvalues.
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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 140 and 141: 126 CHAPTER 7. NUMERICAL EXPERIMENT
Page 145: Part III H 2 Model-order Reduction
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
Page 179 and 180: 10.1. SOLVING THE SYSTEM OF QUADRAT
Page 181 and 182: 10.2. LINEARIZING A POLYNOMIAL EIGE
Page 183 and 184: 10.3. THE KRONECKER CANONICAL FORM
Page 189: 10.3. THE KRONECKER CANONICAL FORM
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
Page 203 and 204: 10.6. EXAMPLES 189 Figure 10.7: Str
Page 205 and 206: 10.6. EXAMPLES 191 Figure 10.8: Pol
Page 207 and 208: 10.6. EXAMPLES 193 Table 10.1: Redu
Page 209 and 210: Chapter 11 H 2 Model-order reductio
Page 211 and 212: 11.2. LINEARIZING THE TWO-PARAMETER
Page 213 and 214: 11.3. KRONECKER CANONICAL FORM OF A
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Page 237 and 238: 11.5. EXAMPLE 223 AãN−2 (ρ 1 ,
Page 239 and 240: 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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