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100 CHAPTER 6. ITERATIVE EIGENVALUE SOLVERS AU − θU to solve Equation (6.49), can be used. When minimizing the norm ∣ ∣ ( AU − θU ) c ∣ ∣ 2 , the deviation of u = Uc from the 2 Stetter structure can be taken into account too, as follows: ∣ ∣ ∣ ( AU − θU ) c ∣ ∣ ∣ ∣ 2 2 + α ∣ ∣ ∣ ∣ ( Uc− û ) ∣ ∣ ∣ ∣ 2 2 . (6.50) Note that the only unknown in this equation is c. Equation (6.50) can be written as: ∣ ∣ ∣ Bc − v ∣ ∣ ∣ ∣ 2 2 where ⎛ B = ⎜ ⎝ AU − θU αU ⎞ ⎟ ⎠ ⎛ and v = ⎜ ⎝ 0 α û ⎞ . ⎟ ⎠ (6.51) The remaining problem that has to be solved in this Refined Rayleigh-Ritz extraction method for projection, where the deviation from the Stetter structure is taken into account, is: ĉ = argmin c∈C k ,||c||=1 ∣ ∣ ∣∣ Bc − v 2 2 (6.52) involving the tall rectangular matrix B and the vector v. The problem of computing the minimum of ∣ ∣ Bc ∣ ∣ 2 2 where c ∈ Ck and ||c|| =1 can be solved by using the Courant-Fischer Theorem as mentioned in [72]. Here the additional vector −v makes the situation more difficult and as a result this theorem can not be applied directly here. A solution would be to extend this theorem or to use the method of Lagrange Multipliers [46] as shown below. If no projection is used the matrix B = AU − θU in Equation (6.52) is square since α = 0 and also v = 0. Then a singular value decomposition of the matrix (AU − θU) yields solutions for ĉ as in the original implementation of the Jacobi– Davidson method. When the involved matrix B is not square, the problem (6.52) is solved using the method of Lagrange Multipliers. To apply such a technique the system of equations Bc − v has to be transformed first into a more convenient form. Applying a singular value decomposition to the (possibly) complex valued matrix B yields B = WDV ∗ with unitary matrices W and V . The matrix D is a rectangular matrix containing the singular values d 1 ,...,d k of the matrix B on its main diagonal. The problem
6.4. PROJECTION TO STETTER-STRUCTURE 101 (6.52) is rewritten as: ĉ = argmin c∈C k ,||c||=1 ⎛ W ⎜ ⎝ ∣∣ d 1 . .. d k 0 . 0 ⎞ V ∗ c − v ⎟ ⎠ ∣∣ 2 2 (6.53) and ĉ = argmin c∈C k ,||c||=1 ⎛ ⎜ ⎝ ∣∣ d 1 . .. d k 0 . 0 Which can be simplified, using ˜c = V ∗ c and ṽ = W ∗ v, to: argmin ˜c∈C k ,||˜c||=1 ⎞ 2 V ∗ c − W ∗ v . (6.54) ⎟ ⎠ ∣∣ ⎛ ⎞ 2 d 1 ⎜ ⎝ . .. ⎟ ⎠ ˜c − ṽ . (6.55) ∣∣ d ∣∣ k Note that the size of the vectors ˜c and ṽ is decreased such that the dimensions match to the matrix diag((d 1 ,...,d k ) T ). Note furthermore that ||c|| = ||˜c||. Now the problem can be written as: ⎧ k∑ ˜c opt = argmin f(˜c) where f(˜c) = (d i˜c i − ṽ i ) 2 ⎪⎨ ⎪⎩ (6.56) subject to g(˜c) =˜c 2 1 + ...+˜c 2 k − 1=0 Now the theory of Lagrange multipliers can be applied because we are searching for the minimum ˜c opt of a multivariate function f(˜c) with continuous first derivatives subject to one constraint g(˜c). The theory of Lagrange multipliers is well-posed when the objective function and constraints are real-valued functions. In our case we are dealing with complex functions f(˜c) and g(˜c). Variational calculus can be easily extended to cope with this complication (see [52] and [93] for details). To prevent for complicated notation we will below present shortly how the Lagrange multiplier methods works for real-valued functions f(˜c) and g(˜c). i=1 2 2
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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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Page 65 and 66: Chapter 5 An nD-systems approach in
Page 67 and 68: 5.1. THE ND-SYSTEM 53 cerned with t
Page 69 and 70: 5.1. THE ND-SYSTEM 55 Example 5.1.
Page 71 and 72: 5.1. THE ND-SYSTEM 57 Figure 5.2: T
Page 73 and 74: 5.3. EFFICIENCY OF THE ND-SYSTEMS A
Page 95 and 96: Chapter 6 Iterative eigenvalue solv
Page 97 and 98: 6.2. THE JACOBI-DAVIDSON METHOD 83
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Page 117 and 118: 6.5. A JACOBI-DAVIDSON METHOD FOR C
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Page 122 and 123: 108 CHAPTER 7. NUMERICAL EXPERIMENT
Page 145: Part III H 2 Model-order Reduction
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150 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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10.3. THE KRONECKER CANONICAL FORM
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10.4. COMPUTING THE APPROXIMATION G
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10.6. EXAMPLES 181 5. Compute the n
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10.6. EXAMPLES 183 Figure 10.1: Spa
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10.6. EXAMPLES 185 Figure 10.3: The
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10.6. EXAMPLES 187 Figure 10.5: Imp
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10.6. EXAMPLES 189 Figure 10.7: Str
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10.6. EXAMPLES 191 Figure 10.8: Pol
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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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