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56 CHAPTER 5. ND-SYSTEMS APPROACH IN POLYNOMIAL OPTIMIZATION Figure 5.1: Relationship between time series in Equation (5.16) and the initial state w 0,0 (red square) the t 1 -axis. Thus there is an overlap of six two-dimensional time instants between the two arrays, which makes that only the additional computation of y 3,0 , y 3,1 and y 3,2 is required. The values of y 3,0 and y 3,1 are readily obtained from the given values in the initial state through application of the first difference equation in (5.16). However, computation of y 3,2 requires prior computation of y 0,3 . The latter can only be achieved through application of the second difference equation of (5.16). Similarly, the action of the matrix A T x 2 on w 0,0 yields the vector A T x 2 w 0,0 = w 0,1 =(y 0,1 ,y 1,1 ,y 2,1 ,y 0,2 ,y 1,2 ,y 2,2 ,y 0,3 ,y 1,3 ,y 2,3 ) T , which can be computed in a similar fashion. (ii) More generally, the action of any matrix r(A T x 1 ,A T x 2 )onw 0,0 , where r(x 1 ,x 2 )is some arbitrary polynomial, can be computed as a linear combination of the values of the two-dimensional time series y t1,t 2 at various 3×3 square arrays of two-dimensional time instants. For instance, the action of the matrix A T p = p 1(x 1,x 2) 1(A T x 1 ,A T x 2 )on w 0,0 can be obtained by a linear combination of the individual actions of the matrices (A T x 1 ) 4 ,(A T x 2 ) 4 ,(A T x 1 ) 3 ,(A T x 1 ) 2 and A T x 1 A T x 2 on w 0,0 . In Figure 5.2 the locations of the time instants of these five actions on w 0,0 are visualized. Here it is noted that the action of any matrix of the form (A T x 1 ) α1 (A T x 2 ) α2 on w 0,0 yields the vector (A T x 1 ) α1 (A T x 2 ) α2 w 0,0 = (y α1,α 2 ,y α1+1,α 2 ,y α1+2,α 2 ,y α1,α 2+1, y α1+1,α 2+1, y α1+2,α 2+1, y α1,α 2+2, y α1+1,α 2+2, y α1+2,α 2+2) T . A systematic procedure to arrive at the value of y t1,t 2 at an arbitrary 2D-time instant (t 1 ,t 2 ), is to compute
5.1. THE ND-SYSTEM 57 Figure 5.2: Time instants to compute the action of A T p 1 the values along the (finite) diagonals for which t 1 + t 2 is constant, for increasing values of this constant. One last remark concerns the computation of all the coordinates of a stationary point from the eigenvector of a single matrix such as A T x 1 . Note that if v denotes an eigenvector of the matrix A T x 1 corresponding to an eigenvalue ξ 1 , then the vector v = (y 0,0 ,y 1,0 ,y 2,0 ,y 0,1 ,y 1,1 ,y 2,1 ,y 0,2 ,y 1,2 ,y 2,2 ) T , and the vector A T x 1 v =(y 1,0 ,y 2,0 ,y 3,0 , y 1,1 ,y 2,1 ,y 3,1 ,y 1,2 ,y 2,2 ,y 3,2 ) T differ by the scalar factor ξ 1 . This implies that the columns of the matrix ⎛ y 0,0 y 1,0 y 2,0 y 3,0 ⎜ y 0,1 y 1,1 y 2,1 y 3,1 ⎟ (5.17) ⎝ ⎠ y 0,2 y 1,2 y 2,2 y 3,2 ⎞ are all dependent: each column is equal to ξ 1 times its preceding column. If the vector v also happens to be an eigenvector of the matrix A T x 2 corresponding to an eigenvalue ξ 2 (the theory of the previous chapter makes clear that such vectors v always exist and in case of distinct eigenvalues of A x1 will automatically have this property) then the rows of the matrix exhibit a similar pattern: each row is equal
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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
Page 19 and 20: 1.3. RESEARCH QUESTIONS 5 The goal
Page 21 and 22: 1.4. THESIS OUTLINE 7 Jacobi-Davids
Page 23: Part I General introduction and bac
Page 26 and 27: 12 CHAPTER 2. ALGEBRAIC BACKGROUND
Page 39 and 40: Chapter 3 Solving polynomial system
Page 41 and 42: 3.1. SOLVING POLYNOMIAL EQUATIONS I
Page 43 and 44: 3.2. METHODS FOR SOLVING POLYNOMIAL
Page 49 and 50: 3.3. THE STETTER-MÖLLER MATRIX MET
Page 51 and 52: 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: 5.1. THE ND-SYSTEM 55 Example 5.1.
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Page 95 and 96: Chapter 6 Iterative eigenvalue solv
Page 97 and 98: 6.2. THE JACOBI-DAVIDSON METHOD 83
Page 99 and 100: 6.2. THE JACOBI-DAVIDSON METHOD 85
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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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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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