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96 CHAPTER 6. ITERATIVE EIGENVALUE SOLVERS A second approach for Stetter projection, which works for negative values as well as for complex values in the approximate eigenvector, is described next. The idea is to compare the structured eigenvector v and the multiplication ṽ of the eigenvector with a variable x i : ṽ = x i v. Then the vectors ṽ and v have entries in common which describe the relations between the separate entries in v itself. If an approximate vector is indeed a ‘true’ eigenvector, then the values within the eigenvector should satisfy these relations too. The relations can be expressed in the form of P (x 1 ,...,x n ) T −u = 0 for all the variables x i . A linear least-squares method can compute the solution (x 1 ,...,x n ) T with the smallest error e = P (x 1 ,...,x n ) T − u. This approach is illustrated with an example. Example 6.3. As an example the situation with n =2,d = 2 and m = 3 is observed again where the eigenvector v =(v 1 ,...,v 9 ) is structured as (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 . When the vector v is multiplied with the variable x 1 and when the vector v is indeed an eigenvector, the following relations hold with respect to x 1 : v 1 x 1 = v 2 , v 2 x 1 = v 3 , v 4 x 1 = v 5 , v 5 x 1 = v 6 , v 7 x 1 = v 8 and v 8 x 1 = v 9 . With respect to x 2 the following holds: v 1 x 2 = v 4 , v 2 x 2 = v 5 , v 3 x 2 = v 6 , v 4 x 2 = v 7 , v 5 x 2 = v 8 , and v 6 x 2 = v 9 . These relations can be expressed in matrix-vector form as: ⎛ ⎜ ⎝ ⎞ v 1 0 v 2 0 v 4 0 v 5 0 v 7 0 v 8 0 0 v 1 0 v 2 0 v 3 0 v 4 ⎟ 0 v 5 ⎠ 0 v 6 ( x1 x 2 ) ⎛ − ⎜ ⎝ v 2 v 3 v 5 v 6 v 8 v 9 v 4 v 5 v 6 v 7 v 8 v 9 ⎞ =0. (6.43) ⎟ ⎠ By introducing a matrix P and a vector u, Equation (6.43) can also be written as P (x 1 ,x 2 ) T − u = 0. Using a linear least-squares method, the values for x 1 and x 2 , which minimize the error e = P (x 1 ,x 2 ) T − u, are computed. Suppose for example the approximate (normalized) eigenvector is u =(u 1 , ..., u 9 )=(1, 4, 6, 2, 18, 43, 27, 76, 224) T (which has much resemblance with the Stetter vector (1, 3, 9, 5, 15, 45, 25, 75, 225) T ). A linear least-squares method yields the values x 1 =2.907 and x 2 =5.106 as the values which produce the smallest error e. When these values are substituted back into the symbolic structured eigenvector v, this yields the projected vector û =(û 1 ,...,û 9 ) which exhibits Stetter structure: û =(1, 2.91, 8.45, 5.11, 14.84, 43.15, 26.07, 75.79, 220.33) T . To illustrate this Stetter structure in û the following relations hold for example: û 5 =û 2 û 4 and û 9 =û 2 2 û 2 4. Remark 6.2. Note that the quantities x 1 and x 2 in Equation (6.43) can be computed independently from each other from the following:
6.4. PROJECTION TO STETTER-STRUCTURE 97 min x 1 min x 2 ∣∣ ⎛ ∣∣∣∣∣∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣∣∣∣∣∣ ⎜ ⎝ ∣∣ ⎛ ∣∣∣∣∣∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣∣∣∣∣∣ ⎜ ⎝ v 1 v 2 v 4 v 5 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 ⎞ ⎛ x 1 − ⎟ ⎜ ⎠ ⎝ ⎞ ⎛ x 2 − ⎟ ⎜ ⎠ ⎝ v 2 v 3 v 5 v 6 v 8 v 9 v 4 v 5 v 6 v 7 v 8 v 9 ⎞ ⎟ ⎠ ∣∣ 2 2 ⎞ 2 . ⎟ ⎠ ∣∣ 2 and (6.44) When in (6.44) the involved vectors are denoted by q 1 and q 2 , the linear leastsquares method yields x i = qT 1 q2 q as a solution for min 1 T q1 x i ||q 1 x i − q 2 || 2 2 for i =1, 2. In this example this yields x 1 = 9957 3425 =2.907 and x 2 = 5693 1115 =5.106. Example 6.2 (continued). Suppose one wants to project the vector u from example 6.2 using the latter projection method. The vector u is defined as (1, −4, −5+29i, −13 − 5i, 5+72i, 255 − 283i, 16 + 140i, 507 − 640i, −4613 + 543i) T . The projected vector û is computed as: ⎛ û = ⎜ ⎝ 1.000 −4.030 − 4.016i 0.118 + 32.366i −9.152 − 8.053i 4.548 + 69.202i 259.551 − 297.153i 18.911 + 147.389i 515.626 − 669.935i −4768.156 + 629.428i ⎞ ⎟ ⎠ (6.45) and is structured like (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 and x 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
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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146 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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