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90 CHAPTER 6. ITERATIVE EIGENVALUE SOLVERS Also Equation (6.18) can be written in matrix-vector form as: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ φ 1 ψ 1 k 1 φ 2 W ⎜ . ⎟ ⎝ . ⎠ ≈ ψ 2 ⎜ ⎟ ⎝ . ⎠ +2π k 2 ⎜ . ⎟ ⎝ . ⎠ φ n ψ N k N (6.23) in which we need to solve for φ =(φ 1 ,φ 2 ,...,φ n ) T and for an integer vector k = (k 1 ,k 2 ,...,k N ) T . Note that φ satisfies: −π
6.4. PROJECTION TO STETTER-STRUCTURE 91 This vector u has much resemblance with a Stetter vector with a structure corresponding to (1, x 1 ,x 2 1,x 2 ,x 1 x 2 ,x 2 1x 2 ,x 2 2,x 1 x 2 2,x 2 1x 2 2) T where x 1 = −4 − 4i and x 2 = −9 − 8i. In polar coordinates the entries of the vector u can be written as u j = υ j e i·ψj for j =1,...,N. This yields in this example: ⎛ υ = ⎜ ⎝ 1.000 4.000 29.428 13.928 72.173 380.938 140.911 816.486 4644.849 ⎞ ⎛ , ψ = ⎟ ⎜ ⎠ ⎝ 0.000 3.142 1.742 −2.774 1.502 −0.837 1.457 −0.901 3.024 ⎞ . (6.29) ⎟ ⎠ To project the vectors υ and ψ, the matrix W and the projection matrix P is required. To construct the matrix W , the logarithmic transformation is applied to the vector (1, x 1 ,x 2 1,x 2 ,x 1 x 2 ,x 2 1x 2 ,x 2 2,x 1 x 2 2,x 2 1x 2 2) T . This yields: ⎛ log ⎜ ⎝ ⎞ ⎛ 1 x 1 x 2 1 x 2 = log(x 1 )w 1 + log(x 2 )w 2 = log(x 1 ) ⎟ ⎜ ⎠ ⎝ x 1 x 2 x 2 1x 2 x 2 2 x 1 x 2 2 x 2 1x 2 2 0 1 2 0 1 2 0 1 2 ⎞ ⎛ + log(x 2 ) ⎟ ⎜ ⎠ ⎝ 0 0 0 1 1 1 2 2 2 ⎞ . (6.30) ⎟ ⎠ The vectors w 1 and w 2 above constitute the matrix W of dimensions N × n =9× 2: ⎛ W = ⎜ ⎝ 0 0 1 0 2 0 0 1 1 1 2 1 0 2 1 2 2 2 ⎞ . (6.31) ⎟ ⎠ The symmetric projection matrix P of dimension N × N =9× 9 is constructed using
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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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Page 57 and 58: Chapter 4 Global optimization of mu
Page 59 and 60: 4.1. THE GLOBAL MINIMUM OF A DOMINA
Page 61 and 62: 4.3. AN EXAMPLE 47 Using the matric
Page 63 and 64: 4.3. AN EXAMPLE 49 the Stetter-Möl
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 122 and 123: 108 CHAPTER 7. NUMERICAL EXPERIMENT
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140 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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