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92 CHAPTER 6. ITERATIVE EIGENVALUE SOLVERS this matrix W as follows: P = W (W T W ) −1 W T = ⎛ ⎞ 0 0 0 0 0 0 0 0 0 5 5 0 − 1 1 7 − 1 − 1 1 48 24 16 24 48 8 48 12 5 5 0 − 1 1 7 − 1 − 1 1 24 12 8 12 24 4 24 6 0 − 1 − 1 5 1 − 1 5 7 1 16 8 48 24 48 24 48 12 1 1 1 1 1 1 1 1 0 24 12 24 12 8 12 8 6 . 7 7 0 − 1 1 13 − 1 5 1 48 24 48 8 48 24 48 4 0 − 1 − 1 5 1 − 1 5 7 1 8 4 24 12 24 12 24 6 ⎜ ⎝ 0 − 1 − 1 7 1 5 7 13 1 ⎟ 48 24 48 8 48 24 48 4 ⎠ 0 1 12 1 6 1 12 1 6 1 4 1 6 1 4 1 3 (6.32) Note that the explicit matrix P is constructed here for analytical purposes, but is avoided in numerical computations. The vectors log(ˆυ) and ˆυ are computed as: ⎛ log(ˆυ) =P · log(υ) = ⎜ ⎝ 0.000 1.675 3.349 2.532 4.207 5.881 5.064 6.738 8.413 ⎞ ⎟ ⎠ ⎛ and ˆυ = e log(ˆυ) = ⎜ ⎝ 1.000 5.337 28.499 12.576 67.120 358.234 158.149 844.083 4505.076 ⎞ . (6.33) ⎟ ⎠ For a particular choice of k =(k 1 ,...,k N )=(0, 0, 1, 0, 0, 1, −1, 0, 0) the projected vector ˆψ is computed as: ⎛ ˆψ = Pψ+2π (P − I) k = ⎜ ⎝ 0.000 3.903 1.523 −2.424 1.479 −0.901 1.435 −0.945 2.958 ⎞ . (6.34) ⎟ ⎠ The entries in the projected vector log(ˆυ) satisfy the following relations in order to exhibit Stetter structure with respect to addition (because it is in the logarithmic
6.4. PROJECTION TO STETTER-STRUCTURE 93 domain): ⎧ ⎪⎨ ⎪⎩ log(ˆυ 1 )= 0 log(ˆυ 2 ) = log(ˆυ 2 ) log(ˆυ 3 ) = 2 log(ˆυ 2 ) log(ˆυ 4 ) = log(ˆυ 4 ) log(ˆυ 5 ) = log(ˆυ 2 ) + log(ˆυ 4 ) log(ˆυ 6 ) = 2 log(ˆυ 2 ) + log(ˆυ 4 ) log(ˆυ 7 ) = 2 log(ˆυ 4 ) log(ˆυ 8 ) = log(ˆυ 2 ) + 2 log(ˆυ 4 ) log(ˆυ 9 ) = 2 log(ˆυ 2 ) + 2 log(ˆυ 4 ) (6.35) The entries in the vector ˆυ satisfy the following relations in order to exhibit Stetter structure with respect to multiplication: ⎧ ⎪⎨ ⎪⎩ ˆυ 1 = 1 ˆυ 2 = ˆυ 2 ˆυ 3 = ˆυ 2 2 ˆυ 4 = ˆυ 4 ˆυ 5 = ˆυ 2 · ˆυ 4 ˆυ 6 = ˆυ 2 2 · ˆυ 4 ˆυ 7 = ˆυ 4 2 ˆυ 8 = ˆυ 2 · ˆυ 4 2 ˆυ 9 = ˆυ 2 2 · ˆυ 4 2 (6.36) Also the projected vector ˆψ exhibits Stetter structure with respect to addition (because this vector is also in the logarithmic domain). To show this, we use the leastsquares solution for φ in Equation (6.24) denoted by φ LS . We know from (6.25) that the following holds: ˆψ = Wφ LS − 2π k. (6.37) The vector φ LS contains the two elements φ LS 1 and φ LS 2 which yields the following for ˆψ: ⎛ ˆψ = ⎜ ⎝ 0 0 1 0 2 0 0 1 1 1 2 1 0 2 1 2 2 2 ⎞ ⎛ ⎝ ⎟ ⎠ φ LS 1 φ LS 2 ⎞ ⎠ − 2π ⎛ ⎜ ⎝ k 1 k 2 k 3 k 4 k 5 k 6 k 7 k 8 k 9 ⎞ . (6.38) ⎟ ⎠ From the equation above we can derive the relations φ LS 1 = ˆψ 2 +2πk 2 and φ LS 2 = ˆψ 4 +2πk 4 . Substituting this back into the latter equation yields the structure in the
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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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Page 67 and 68: 5.1. THE ND-SYSTEM 53 cerned with t
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Page 71 and 72: 5.1. THE ND-SYSTEM 57 Figure 5.2: T
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Page 97 and 98: 6.2. THE JACOBI-DAVIDSON METHOD 83
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142 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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