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72 CHAPTER 5. ND-SYSTEMS APPROACH IN POLYNOMIAL OPTIMIZATION (a) (b) (c) Figure 5.11: Non-minimal stable patterns (green layers) allowing for shifts along one time axis: (a) t 1 time axis, (b) t 2 time axis, (c) t 3 time axis method has the effect that the calculated points are first directed towards the nearest axis and then to the origin. (v) A slightly more sophisticated fifth method has been designed on the basis of insights gained from the test experiments with the first four methods introduced above. It is discussed later in this section and will be called the ‘least-increments method’. For the two-dimensional (n = 2) and three-dimensional (n = 3) cases the paths to a few particular points have been computed using the first four described methods. The results are collected in the Tables 5.1 and 5.2. For each method the total number
5.3. EFFICIENCY OF THE ND-SYSTEMS APPROACH 73 of all stored points required for computing the requested point (i.e., the size of the path), the total number of floating point operations (flops) and the actual running time (in ms. obtained on a PC platform with an Intel Pentium PIV 2.8GHz processor and 512MB internal memory) are given for each requested point. Table 5.1: Two-dimensional case (2 × 2 initial state w 0,0 ) Requested point Linear Diagonal Equalizing Axis (0,10) Stored points 36 91 46 34 Flops 100 265 167 118 Cpu time (ms) 7 6 2 2 (0,50) Stored points 156 1431 546 154 Flops 460 4285 2181 718 Cpu time (ms) 32 23 25 8 (0,100) Stored points 306 5356 1921 304 Flops 910 16060 7697 1468 Cpu time (ms) 16 40 55 20 (10,10) Stored points 66 276 64 143 Flops 209 820 239 598 Cpu time (ms) 30 20 1 20 (50,50) Stored points 306 5356 304 1723 Flops 1009 16060 1199 7918 Cpu time (ms) 40 40 10 60 (100,100) Stored points 606 20706 604 5948 Flops 2009 62110 2399 27758 Cpu time (ms) 20 120 50 220 Table 5.2: Three-dimensional case (2 × 2 × 2 initial state w 0,0,0 ) Requested point Linear Diagonal Equalizing Axis (0,0,10) Stored points 120 560 159 88 Flops 344 2216 832 399 Cpu time (ms) 12 18 4 1 (0,0,50) Stored points 520 27720 4825 408 Flops 1544 110856 32328 2199 Cpu time (ms) 23 193 89 18 (0,0,100) Stored points 1020 192920 29808 808 Flops 3044 771656 214696 4449 Cpu time (ms) 32 1017 1875 33 (10,10,10) Stored points 320 7140 248 1868 Flops 1040 28536 1379 10565 Cpu time (ms) 20 40 1 20 (50,50,50) Stored points 1520 620620 1208 55348 Flops 5040 2482456 7079 351861 Cpu time (ms) 80 2774 60 2614 (100,100,100) Stored points 3020 4728720 2408 281948 Flops 10040 18914856 14204 1892839 Cpu time (ms) 50 265008 150 781646 From this experiment we may conclude the following: • The linear method indeed exhibits a linear complexity with respect to |t|. But in higher dimensions (e.g., n>10) the linear method may become inefficient because the simplex entirely covering the hypercube of initial values becomes very large. Although it constitutes a stable pattern associated with shifts in all
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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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Page 36 and 37: 22 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
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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
Page 59 and 60: 4.1. THE GLOBAL MINIMUM OF A DOMINA
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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
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Page 71 and 72: 5.1. THE ND-SYSTEM 57 Figure 5.2: T
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Page 95 and 96: Chapter 6 Iterative eigenvalue solv
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122 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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