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230 CHAPTER 12. CONCLUSIONS & DIRECTIONS FOR FURTHER RESEARCH SPP too. A central role in this approach is played by the notion of stable patterns, which are shifted along the 2D-grid. The same approach leads to partial results in the 3D-case (and higher dimensions). The results of Section 5.3.2 also underlie the design of the heuristic methods discussed in Section 5.3.3. These heuristic procedures are developed to arrive cheaply at suboptimal paths with acceptable performance. We have implemented five of these heuristics and compared their performance. Another way to improve the efficiency of an nD-system is to apply parallel computing techniques as described in Section 5.3.4. Here the ‘least-increments’ method is implemented in such a way that it computes each path in the nD-system on a separate processor. The parallel nD-system is applied in combination with the JDQZ and the JD methods. However, it turns out that parallelization is not useful in this application of the nD-system. The use of iterative eigenvalue solvers is described in Chapter 6. The iterative eigenvalue solver implementations we used in this thesis are JDQR, JDQZ and JD. An advantage of such an iterative eigenvalue solver, is that it is able to focus on a subset of eigenvalues of the involved matrix. Here the most interesting eigenvalue is the smallest real eigenvalue. Focusing on the smallest real eigenvalue of a matrix is not a usual parameter setting for an iterative eigenvalue solver. Therefore this option is developed and implemented as described in Section 6.3: the selection criterion is adapted such that it is no longer necessary to compute all the eigenvalues of the matrix but that it becomes possible to focus on the smallest real eigenvalues. An important observation is that the eigenvectors of the matrix at hand exhibit a large amount of internal structure, called the Stetter structure. In Section 6.4 some procedures are studied to project an approximate eigenvector to a close-by vector with Stetter structure to speed up the convergence process of the iterative solver. The main bottleneck was the existence of negative and complex entries in the approximate eigenvectors in combination with the use of a logarithmic transformation. Subsequently, two methods are given to embed such a projection method in the Jacobi–Davidson implementation. The development of a Jacobi–Davidson eigenvalue solver for commuting matrices is described in Section 6.5. This Jacobi–Davidson method is called the JDCOMM method and is an extended version of the JD method. Its most important newly implemented feature is that it computes the eigenvalues of the matrix A p in the outer loop while iterating with a much sparser (but commuting) matrix A xi in the inner loop. Most of the computation time is spent in this inner loop, while the outer loop is only used occasionally, which results in a speed up in computation time and a decrease in the amount of required floating point operations. Chapter 7 presents the results of the numerical experiments in which the global minima of various Minkowski dominated polynomials are computed using the approaches and techniques mentioned in Part II of this thesis. Section 7.1 describes the computations on a polynomial of order eight in four variables. To compute the global minimum of this polynomial conventional methods are
12.1. CONCLUSIONS 231 used first. Then the nD-systems approach is used in combination with the iterative eigenproblem solvers JDQR, JDQZ, and Eigs. Finally, the outcomes of these computations are compared with those of the software packages SOSTOOLS, GloptiPoly and SYNAPS. The JDQZ method computes the global minimum and its minimizer faster than the JDQR and Eigs method. However, the methods SOSTOOLS, GloptiPoly and SYNAPS are faster than the methods using our nD-systems approach. But where these methods appear to be faster, they are not as accurate as the iterative eigenvalue solvers. This may be due to the default tolerance settings used in these software packages. In Section 7.2 the nD-systems approach is used to compute the global minima of 22 distinct polynomials. These 22 polynomials vary in the number of variables and in the total degree and therefore they also vary in the size of the involved linear operators and in their numerical conditioning. Here the target selection as described in Section 6.3 is used to let the iterative solvers JD and JDQZ focus on the smallest real eigenvalues first. To put the performance of these methods into perspective they are compared with the performance of the SOSTOOLS package. In the majority of the test cases the SOSTOOLS software approach is more efficient than the nD-system implementation presented here. In particular it appears that in the SOSTOOLS approach the processing time increases less rapidly with increasing complexity. However, it should be noted that the error in the SOSTOOLS approach is higher in some cases than the error in the nDsystems implementation. Some test cases can not accurately and reliably be solved by the SOSTOOLS software. These large errors can limit the actual application of this software on large problem instances. Furthermore there are multivariate polynomials that are positive but can not be written as a sum of squares [92]. The global minimum of such polynomials can not be found directly using the SOSTOOLS approach. Such a limitation does not affect the Stetter-Möller matrix method. Section 7.3 shows the result of applying the projection method of Section 6.4. Projection to Stetter structure using a logarithmic transformation can be performed if the effects on complex and negative numbers are taken into account. The JD eigenvalue solver with JD expansion no longer converges to the desired eigenvector if projection of the absolute values is used. When the expansion is changed to Rayleigh Quotient Iteration (RQI) and the projection is applied, the number of matrix-vector computations is comparable to the numbers required in the original JD implementation in most cases and lower for some particular cases. This indicates that projection can indeed be used to increase the convergence speed of the iterative eigenvalue solvers at least for certain examples. Section 7.4 shows the results of applying the parallel approach of Section 5.3.4 on the set of 22 polynomials. The number of processors used in parallel here is 4. It turns out that the computing time is longer but the required number of matrixvector operations is roughly identical in this parallel approach in comparison with a sequential approach. Therefore, we conclude that parallelization is not appropriate for this type of problems. This is caused by the high density of short paths that, for an efficient calculation, require communication between the different path computations.
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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
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4.1. THE GLOBAL MINIMUM OF A DOMINA
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4.3. AN EXAMPLE 47 Using the matric
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4.3. AN EXAMPLE 49 the Stetter-Möl
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Chapter 5 An nD-systems approach in
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5.1. THE ND-SYSTEM 53 cerned with t
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5.1. THE ND-SYSTEM 55 Example 5.1.
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5.1. THE ND-SYSTEM 57 Figure 5.2: T
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5.3. EFFICIENCY OF THE ND-SYSTEMS A
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Chapter 6 Iterative eigenvalue solv
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6.2. THE JACOBI-DAVIDSON METHOD 83
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6.2. THE JACOBI-DAVIDSON METHOD 85
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6.4. PROJECTION TO STETTER-STRUCTUR
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6.5. A JACOBI-DAVIDSON METHOD FOR C
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6.5. A JACOBI-DAVIDSON METHOD FOR C
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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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Page 193 and 194: 10.4. COMPUTING THE APPROXIMATION G
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Page 247 and 248: 12.1. CONCLUSIONS 233 solver as des
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Page 251 and 252: Bibliography [1] W. Adams and P. Lo
Page 253 and 254: BIBLIOGRAPHY 239 [26] A. Cayley. On
Page 255 and 256: BIBLIOGRAPHY 241 [58] M.E. Hochsten
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Page 265 and 266: A.3. EXAMPLE 251 The first step of
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Page 270 and 271: 256 APPENDIX B. POLYNOMIAL TEST SET
Page 272 and 273: 258 SUMMARY vector. This routine is
Page 274 and 275: 260 SUMMARY globally optimal approx
Page 276 and 277: 262 SAMENVATTING De matrix-vrije im
Page 278 and 279: 264 SAMENVATTING eigenwaarde proble
Page 281 and 282: List of Publications [1] I.W.M. Ble
Page 283 and 284: List of Symbols and Abbreviations A
Page 285 and 286: LIST OF FIGURES 271 7.7 Residual no
Page 287 and 288: Index H 2 model-order reduction pro
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