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168 CHAPTER 10. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-2 structure, because the vectors v are structured too as being Stetter vectors (with a structure corresponding to the chosen monomial basis). The matrices B and C in this set-up have dimensions (N − 1) 2 N × (N − 1) 2 N . This gives rise to (N − 1) 2 N generalized eigenvalues. Because spurious eigenvalues can be introduced by making the matrix AãN−1 (ρ 1 ) T polynomial in ρ 1 , one has to check which eigenvalues (which can be numerical values, indeterminate or infinite) are eigenvalues of the problem (10.10) too. Recall that the roots −1/δ i for all i =1,...,N of the terms (1+ρ 1 δ 1 ), (1+ρ 1 δ 2 ),...,(1+ρ 1 δ N ) are the quantities that may be introduced as spurious eigenvalues when constructing the polynomial matrix Ãã N−1 (ρ 1 ) T . The equivalence between the polynomial eigenvalue problem and the generalized eigenvalue problem, is expressed by the fact that if one has computed a solution of the generalized eigenvalue problem (10.13), then this yields a solution of the polynomial eigenvalue problem (10.10) too, and vice versa. Once the eigenvalues ρ 1 and eigenvectors ṽ of generalized eigenvalue problem (10.13) are found and the appropriate ones have been selected, the values x 1 ,...,x N can be read off from the vectors ṽ. This provides a method to find all the solutions of the polynomial eigenvalue problem (10.10) and, therefore, to find all the solutions of the system of equations (10.2) that satisfy the constraint ã N−1 = 0 in (10.3). 10.3 The Kronecker canonical form of a matrix pencil A matrix pencil B + ρ 1 C of dimensions m × n is regular when the involved matrices are square (m = n) and the determinant is not identically zero. When a pencil is singular, then m n, or m = n and the determinant of the matrix is ≡ 0. Therefore, when a pencil is rectangular it is singular by definition. The matrix pencil B +ρ 1 C of size (N −1) 2 N ×(N −1) 2 N in (10.13) of the previous section will in general be singular, although it involves square matrices. Singular generalized eigenvalue problems may have indeterminate, infinite, zero and non-zero finite eigenvalues and are well-known to be numerically highly ill-conditioned. The eigenvalues which cause this singularity of the eigenvalue problem are unwanted as they do not have any meaning for our H 2 model-order reduction problem. A solution here is to decompose, using exact arithmetic, the matrix pencil B + ρ 1 C into a singular part, which corresponds to indeterminate eigenvalues, and a regular part, which corresponds to infinite and finite eigenvalues, and to split off the unwanted eigenvalues. This can be achieved by using the computation of the Kronecker canonical form (see [6], [34] and [42]). In the conventional square case, the matrix pencil A − ρ 1 I n can be conveniently characterized by means of its Jordan canonical form. The sizes of the Jordan blocks for the various eigenvalues of the matrix A provide geometrical information on the associated eigenspaces, both for right eigenvectors and for left eigenvectors. The Kronecker canonical form is a generalization of the Jordan canonical form to the possible rectangular and singular case, and applies to arbitrary m×n matrix pencils B +ρ 1 C.
10.3. THE KRONECKER CANONICAL FORM OF A MATRIX PENCIL 169 By computing such a decomposition into the Kronecker canonical form it is possible to split off the unwanted eigenvalues of the generalized eigenvalue problem. In the procedure to compute the Kronecker Canonical form of a pencil B + ρ 1 C of dimension m × n, one transforms this pencil into another convenient pencil D + ρ 1 E with an additional block structure, as stated in the theorem below. Theorem 10.3. Let B + ρ 1 C be an m × n complex-valued matrix pencil. Then there exists an m × m invertible matrix W and an n × n invertible matrix V for which the matrix pencil D + ρ 1 E obtained as: W −1 (B + ρ 1 C)V = D + ρ 1 E (10.16) attains the form: ⎛ D + ρ 1 E = ⎜ ⎝ 0 h×g L ε1 . .. L εs L T η 1 . .. L T η t D r + ρ 1 E r ⎞ , (10.17) ⎟ ⎠ where the blocks 0 h×g , L ε1 ,...,L εs and L T η 1 ,...,L T η t correspond to indeterminate eigenvalues. It holds that g = dim ( ( ) ( ) B ), ( B T ), ker h = dim ker 0 C C T
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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 132 and 133: 118 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: 10.2. LINEARIZING A POLYNOMIAL EIGE
Page 185 and 186: 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
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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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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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