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142 CHAPTER 8. H 2 MODEL-ORDER REDUCTION One may now take the n coefficients of the monic polynomial a(s) together with the n coefficients of b(s) and the N − n coefficients of q(s) as the N + n optimization parameters in conjunction with the N + n equations associated with the terms in (8.18). This leads to a polynomial system of equations, which is structured in the sense that the coefficients of b(s) show up in a linear fashion, but which may still be quite hard to solve algebraically because the system is not in Gröbner basis form. By means of a suitable reparameterization this structure can be reworked. When the model-order is reduced by k from order N to order n = N − k, then the degree of q(s) is at most k − 1. We shall employ the notation: q(s) =q 0 + q 1 s + ...+ q k−1 s k−1 0. (8.19) A reparameterization is used which involves the N zeros δ 1 ,...,δ N of the given polynomial d(s), which upon substitution into Equation (8.18) enables the elimination of the polynomial b(s) from the problem. As stated before, we here require the technical assumption that the N zeros δ 1 ,...,δ N of d(s) are all distinct. We first focus on the generic situation where q 0 0. For i =0, 1,...,k − 1we define ρ i = qi q 0 and let the polynomial ρ(s) be: ρ(s) = q(s) =1+ρ 1 s + ρ 2 s 2 + ...+ ρ k−1 s k−1 . (8.20) q 0 Note that ρ(s) is of degree ≤ k − 1 and that ρ 0 = q0 q 0 =1. Then Equation (8.18) can be manipulated as follows. Multiplying both sides by q 0 and introducing: ã(s) :=q 0 a(−s) (8.21) yields: e(s)ã(−s) − q 0 b(s)d(s) =ã(s) 2 ρ(s). (8.22) Note that ã(s) is not monic, in contrast to a(s). Evaluation of Equation (8.22) at the N distinct zeros δ 1 ,...,δ N of the polynomial d(s) gives: e(δ i )ã(−δ i )=ã(δ i ) 2 ρ(δ i ), (i =1,...,N). (8.23) Note that in Equation (8.23) it holds that ã(δ i ) 0, and that the equations no longer involve the polynomial b(s). This equation constitutes a system of N equations in N unknowns. When studying the H 2 model-order reduction problem for co-order k, the polynomial ã(s) is, for future computational convenience, treated as a polynomial of degree ≤ N − 1, while its degree is actually N − k. Fork = 1 we write: ã(s) =ã N−1 s N−1 +ã N−2 s N−2 + ...+ã 1 s +ã 0 . (8.24) For k > 1 we use a similar notation, where we then have to add the additional constraints: ã N−1 = ···=ã N−k+1 =0. (8.25)
8.3. A REPARAMETERIZATION OF THE MODEL REDUCTION PROBLEM 143 This allows us to employ a joint framework in which to study the H 2 model-order reduction problem for various co-orders k. In the co-order k case equation (8.23) involves N equations and k − 1 additional constraints, in terms of N + k − 1 unknowns (the N coefficients of ã(s) and the k − 1 unknown parameters ρ 1 ,...,ρ k−1 in ρ(s)). We now reparameterize by introducing the N quantities x i := ã(δ i ), for i = 1,...,N, to characterize the polynomial ã(s). Note that a polynomial of degree N −1 has N degrees of freedom and is fully characterized by its values at N distinct points. These values are related to the coefficients of the polynomial ã(s) inalinear way, by means of a Vandermonde matrix involving the interpolation points. To be explicit, let V (β 1 ,β 2 ,...,β N ) denote a Vandermonde matrix in the variables β 1 ,...,β N : ⎛ V (β 1 ,β 2 ,...,β N )= ⎜ . . . ⎝ 1 β N βN 2 1 β 1 β 2 1 ... β N−1 1 1 β 2 β 2 2 ... β N−1 2 . ... βN−1 N ⎞ . (8.26) ⎟ ⎠ As in [50], note that the quantities x i := ã(δ i )=ã N−1 δ N−1 i + ...+ã 1 δ i +ã 0 (for i =1,...,N) depend linearly on the coefficients ã 0 ,...,ã N−1 through the fixed Vandermonde matrix V (δ 1 ,...,δ N ) in the following way: ⎛ ⎜ ⎝ x 1 x 2 . . x N ⎞ ⎛ ⎟ ⎠ = ⎜ ⎝ ã(δ 1 ) ã(δ 2 ) . ã(δ N ) ⎞ ⎛ ⎟ ⎠ = V (δ 1,...,δ N ) ⎜ ⎝ ã 0 ã 1 . . ã N−1 ⎞ ⎟ ⎠ . (8.27) Likewise the quantities ã(−δ i )=ã N−1 (−δ i ) N−1 + ...+ã 1 (−δ i )+ã 0 in Equation (8.23) depend linearly on the coefficients of ã(s) too. This relation can be expressed, by using the matrix V (−δ 1 ,...,−δ N ), as follows: ⎛ ⎜ ⎝ ã(−δ 1 ) ã(−δ 2 ) . ã(−δ N ) ⎞ ⎛ ⎟ ⎠ = V (−δ 1,...,−δ N ) ⎜ ⎝ ã 0 ã 1 . ã N−1 ⎞ ⎟ ⎠ . (8.28) In this expression existence of the inverse of the Vandermonde matrix V (δ 1 , ..., δ N ) is guaranteed by the poles δ 1 ,δ 2 , ...,δ N being all distinct, as its determinant is given by: det V (δ 1 ,...,δ N )= ∏ (δ i − δ j ). (8.29) i>j Combining Equations (8.27) and (8.28) allows to express the quantities ã(−δ i )in terms of x 1 ,...,x N through fixed Vandermonde matrices involving the zeros δ 1 ,...,δ N
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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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6.4. PROJECTION TO STETTER-STRUCTUR
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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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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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