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220 CHAPTER 11. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-3 31m × 6n: ⎛ M 1,2 = ⎜ ⎝ ⎞ B 0 0 0 0 0 0 B 1 B 0 0 0 0 0 B 2 B 1 B 0 0 0 0 B 3 B 2 B 1 0 0 0 0 B 3 B 2 0 0 0 0 0 B 3 0 0 0 C 1,0 0 0 B 0 0 0 C 2,0 0 0 0 0 0 C 1,1 C 1,0 0 B 1 B 0 0 C 3,0 0 0 0 0 0 C 2,1 C 2,0 0 0 0 0 C 1,2 C 1,1 C 1,0 B 2 B 1 B 0 0 C 3,0 0 0 0 0 0 C 2,1 C 2,0 0 0 0 0 C 1,2 C 1,1 B 3 B 2 B 1 0 0 C 3,0 0 0 0 0 0 C 2,1 0 0 0 0 0 C 1,2 0 B 3 B 2 0 0 0 0 0 B 3 0 0 0 C 1,0 0 0 0 0 0 C 2,0 0 0 0 0 0 C 1,1 C 1,0 0 0 0 0 C 3,0 0 0 0 0 0 C 2,1 C 2,0 0 0 0 0 C 1,2 C 1,1 C 1,0 0 0 0 0 C 3,0 0 0 0 0 0 C 2,1 C 2,0 0 0 0 0 C 1,2 C 1,1 0 0 0 0 0 C 3,0 ⎟ 0 0 0 0 0 C 2,1 ⎠ 0 0 0 0 0 C 1,2 . (11.73) If the rank of this matrix M 1,2 is smaller than 6n, one can compute solutions z(ρ 1 ,ρ 2 )ofA(ρ 1 ,ρ 2 ) T z(ρ 1 ,ρ 2 ) = 0, by computing non-zero vectors in the kernel of this matrix. From the vectors z 0,0 ,z 0,1 ,z 0,2 ,z 1,0 ,z 1,1 ,z 1,2 , which are parts of a vector in the kernel of M 1,2 , one can construct the solution z as z(ρ 1 ,ρ 2 )=(z 0,0 + ρ 2 z 0,1 + ρ 2 2 z 0,2 ) − ρ 1 (z 1,0 + ρ 2 z 1,1 + ρ 2 2 z 1,2 ). The vectors z(ρ 1 ,ρ 2 ) are used to split off singular parts of the singular matrix pencil A(ρ 1 ,ρ 2 ) T = B(ρ 2 )+ρ 1 C(ρ 1 ,ρ 2 ): first construct transformation ( matrices V (ρ 2 ) and W (ρ 2 ), second, perform the transformation W (ρ 2 ) −1 B(ρ 2 )+ρ 1 C(ρ 1 ,ρ 2 )V (ρ 2 ) and finally split off one or more singular parts of dimensions (η 1 +1)× η 1 with the same structure as in (11.14).
11.4. COMPUTING THE APPROXIMATION G(S) 221 The difference here with respect to the previous approach lies in the construction of the transformation matrices V and W . The first columns of the matrices V are chosen, as usual, as the η 1 + 1 vectors z 0 (ρ 2 ),...,z η1 (ρ 2 ) which makes this a polynomial matrix in ρ 2 . The first columns of the matrix W are chosen as the η 1 vectors B(ρ 2 ) z 1 (ρ 2 ),...,B(ρ 2 ) z η1 (ρ 2 ). In this approach both the transformation matrices are polynomial in the variable ρ 2 , whereas in the previous approach the matrix W was polynomial in both ρ 1 and ρ 2 . This makes it easier to compute the inverse of the transformation matrix W . After all the singular parts of the matrix pencil B(ρ 2 )+ρ 1 C(ρ 1 ,ρ 2 ) are split off, there should not exist a regular part which admits other solutions than the trivial solution. Because the existence of a regular part is impossible (for the same reasons mentioned in Section 11.3.1), the solutions for ρ 1 and ρ 2 correspond to the values which make the transformation matrices V (ρ 2 ) and W (ρ 2 ) singular as in (11.20) of Proposition 11.2. Remark 11.7. The value of Ñ can change (increase or decrease), during the process of splitting off singular parts of the matrix pencil. This has a direct effect on the decomposition of the matrix pencil (11.59) in (11.60) and (11.61). 11.4 Computing the approximation G(s) Let x 1 ,...,x N and ρ 1 and ρ 2 be a solution of the system of equations (11.1) which also satisfies the additional constraints ã N−1 = 0 and ã N−2 = 0 in (11.2). Using such a solution an approximation of order N −3 can be computed as described in Theorem 8.3 for k =3. From the set of solutions to the quadratic system of equations (11.1) that satisfy the constraints ã N−1 = 0 and ã N−2 = 0 in (11.2), it is straightforward to select those that are feasible, i.e., which give rise to a real stable approximation G(s) of order N − 3. It is then possible to select the globally optimal approximation by computing the H 2 -norm of the difference H(s)−G(s) for every feasible solution G(s) and selecting the one for which this criterion is minimal. This H 2 -norm of H(s)−G(s) is, according to Theorem 8.2 for k = 3, given by: V H (x 1 ,x 2 ,...,x N ,ρ 1 ,ρ 2 ) = e(s) ∣∣d(s) − b(s) a(s) ∣∣ = N∑ i=1 2 H 2 (1 + ρ 1 δ i + ρ 2 δi 2)2 (1 − ρ 1 δ i + ρ 2 δi 2) e(δ i )d ′ x 3 i . (δ i )d(−δ i ) (11.74)
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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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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.
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Page 265 and 266: A.3. EXAMPLE 251 The first step of
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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
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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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