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70 CHAPTER 5. ND-SYSTEMS APPROACH IN POLYNOMIAL OPTIMIZATION Figure 5.9: Computation of points with total time 4 using the three different recursions (associated with f (1) , f (2) and f (3) ) involving points with total time 3 Note that in our application we need to solve the SPP only once (before the iterative eigenproblem solver is started), and that this solution can then be used in every iteration of the iterative eigenvalue solver (because the required path is the same in every iteration and already known). For the overall efficiency of the approach the number of iterations determines whether it is worthwhile to set up and solve the SPP, or if it is advisable to avoid this and to employ cheaper but suboptimal heuristic methods instead. Using the insights from the previous section, five heuristic methods are developed to calculate the value y t1,...,t n that appears in a state vector w t1,t 2,...,t n with suboptimal but acceptable computational effort. The performance of these methods is
5.3. EFFICIENCY OF THE ND-SYSTEMS APPROACH 71 Figure 5.10: A non-minimal stable pattern for n = 3 and m = 3 allowing for shifts in three time directions compared for a number of situations involving full recursions and uniform costs. The heuristic methods under investigation are the following. (i) The linear method starts by computing the values of y t1,...,t n at all the points on and below the hyperplane which corresponds to the total time n(m − 1), thereby just covering the area (hypercube) of initial values. Then this stable pattern of values N (see Proposition 5.7) is shifted step by step, with each step involving a single shift in one of the axis directions, until the requested location is reached. The sequence of shifts may depend on the multidimensional time instant at the requested location, but this does not affect the total amount of points encountered in this procedure. (ii) The diagonal method proceeds by first computing y t1,...,t n for all the time instants with constant total time |t| = 1, then all those with total time |t| = 2 and so on, until the requested location is reached. (iii) The equalizing method proceeds by computing a value y t1,...,t n by employing that recursion of the system (5.3) which reduces the largest possible coordinate of the time instant y t1,...,t n (which is larger than 2d − 2, to exclude negative time indices). It uses a lexicographic ordering in case there is more than one largest coordinate. The path between the initial state w 0,...,0 and the requested location is determined in a backwards fashion, i.e., starting from the requested location. This method has the effect that the calculated points are first directed towards the diagonal and then to the origin. (iv) The axis method uses the same methodology as the equalizing method with the difference that it applies that recursion of the system (5.3) which reduces the smallest possible coordinate of the time instant y t1,...,t n which is larger than 2d − 2. This
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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 34 and 35: 20 CHAPTER 2. ALGEBRAIC BACKGROUND
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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 67 and 68: 5.1. THE ND-SYSTEM 53 cerned with t
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120 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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