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78 CHAPTER 5. ND-SYSTEMS APPROACH IN POLYNOMIAL OPTIMIZATION Figure 5.17: Points required by the linear, diagonal, equalizing and the axis method for computing state vectors at time instants (0, 500), (125, 375), (250, 250), (375, 125), and (500, 0) with a 3 × 3 initial state zoomed in on a specific region Figure 5.18: Points required by the least-increments method with a 3 × 3 initial state holds for points along the time axes.
5.3. EFFICIENCY OF THE ND-SYSTEMS APPROACH 79 Table 5.4: Increase in the number of stored points along the axis Increase factor Diagonal method Linear/Axis method Equalizing method n =2/ n =3 n =2and n =3 n =2/ n =3 2 3.7 / 7.0 2.0 3.5 / 6.2 5 15.7 / 49.5 4.5 11.9 / 30.3 10 58.9 / 344.5 8.8 41.8 / 187.5 5.3.4 Parallel computations in the nD-systems approach Suppose one wants to compute the state vectors at the time instants (0, 500), (125, 375), (250, 250), (375, 125), and (500, 0). Figure 5.18 shows that the least-increments method determines five paths for this purpose. In every iteration of the eigenvalue solver the values of w 0,500 , w 125,375 , w 250,250 ,w 375,125 , and w 500,0 have to be computed following these paths given the values of the initial state w 0,0 . Because these five paths are well separated from each other (except in the very beginning near the initial state), one could think of determining and computing these paths of the least-increments method in parallel, i.e., each path is considered by one processor. In such a situation a parallel approach is reasonable, but it has no effect when using an nD-systems approach for global polynomial optimization. This is a conclusion from the parallel numerical examples described in Section 7.4. The same results are described in [53]. The reason that parallelization does not give any benefit in an nD-systems approach for global polynomial optimization is the absence of long paths. Such long paths as displayed in Figure 5.18 can not occur: the smallest real eigenvalue of the operator A T p λ (x 1,...,x n) is of importance as it yields the value of the global minimum of p λ . The total time of the state vectors w t1,...,t n , needed to compute the action of A T p λ (x 1,...,x n) is smaller than or equal to 2d, since the order of the polynomial p λ is 2d (see the Equations (4.1) and (5.10)). The maximum total time of a time instant y t1,...,t n in the initial state w 0,...,0 is 2d − 2. As a result the occurring paths contain time instants with a total time increasing from 2d − 2to2d and therefore these paths are clearly shorter as the paths denoted in Figure 5.18. The state vectors required to compute the action of the operator A T p λ (x , 1,...,x n) among which also the initial state w 0,...,0 , have the dimensions (2d − 1) × (2d − 1) × ···×(2d − 1) and they all lie inside the region of points with maximum total time 2d, thereby forming a ‘closed’ region. These ’closed’ regions are displayed for the 2- dimensional and 3-dimensional case in the Figures 5.19(a) and 5.19(b), respectively, by a solid line. The number of paths is obviously given by the number of monomial terms in the polynomial p λ . Because of the fact that all these short paths start from the initial state w 0,...,0 , there is much overlap especially near this initial state. Remark 5.1. Note that even when the degree of p λ increases the paths remain short because only the initial state, containing the given values, becomes larger and still only a few steps in the nD-system are required to follow one path.
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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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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
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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
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Page 122 and 123: 108 CHAPTER 7. NUMERICAL EXPERIMENT
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128 CHAPTER 7. NUMERICAL EXPERIMENT
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Part III H 2 Model-order Reduction
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134 CHAPTER 7. NUMERICAL EXPERIMENT
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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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