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126 CHAPTER 7. NUMERICAL EXPERIMENTS Table 7.16: Sparsity of the matrices A p1 and A x1 ,A x2 ,A x3 ,A x4 Matrix of dimension 6561 × 6561 Number of non-zeros %filled A p1 1255231 2.92 A x1 45717 0.11 A x2 59756 0.14 A x3 41129 0.10 A x4 41712 0.10 real eigenvalue immediately, but then the JDCOMM methods become (almost) just as fast as the conventional JD method. Respectively, the JD and JDCOMM method with the matrices A x1 , A x2 , A x3 , A x4 , and A x5 need 3.5, 3.7, 3.5, 3.8, 3.7, and 3.6 seconds and 2.1×10 8 ,2.2×10 8 ,2.2×10 8 ,2.2×10 8 ,2.2×10 8 , and 2.2×10 8 flops. To compute the same global minimizer including its location, SOSTOOLS and GloptiPoly require as much as 9.2 and 8.7 seconds. 7.5.4 Experiment 4 In this final experiment we study an optimization problem of the same dimensions as the previous one: n =4,d = 5, and β = 1 with the following Minkowski-dominated polynomial: p 1 (x 1 ,x 2 ,x 3 ,x 4 )=(x 10 1 + x 10 2 + x 10 3 + x 10 4 )+3x 5 1x 2 x 3 x 4 + +x 5 1x 3 x 2 4 +7x 5 1x 3 x 4 − 5x 3 1x 2 3x 4 + −7x 2 1x 2 x 2 3x 2 4 − 5x 1 x 3 3x 3 4 − 5. (7.9) The involved quotient space and thus also the matrices A p1 , A x1 , A x2 , A x3 , and A x4 are of dimension 6561 × 6561. Table 7.16 shows the differences in the number of nonzero elements between those matrices. See Figure 7.10 for a representation of the sparsity structure of these matrices. The global minimizer, the smallest real eigenvalue of the matrix A p1 , we are looking for has value −206.5 and is located at the point with coordinates x 1 =1.79, x 2 =1.43, x 3 = −1.54, and x 4 =1.54, which are the corresponding eigenvalues of the matrices A x1 , A x2 , A x3 , and A x4 . The location of these eigenvalues with respect to the entire eigenvalue spectrum of each matrix is depicted with red dots in Figure 7.11. The smallest real eigenvalue of the matrix A p1 is not located very close to the exterior of the spectrum: there are 34 eigenvalues with a real part smaller than −206.5. See the top-left subfigure of Figure 7.11. The smallest real eigenvalue is computed with the JD and JDCOMM method using various search space parameters. The minimal (mindim) and maximal (maxdim) dimensions of the search space are varied between 5 and 100. See Table 7.17 for the results. Blanks in this table indicate that the corresponding method with these settings is unable to converge to the requested eigenvalue using a maximum of 1000 outer iterations and 10 inner iterations with GMRES. Note here that the standard JD
7.5. NUMERICAL EXPERIMENTS WITH JDCOMM SOFTWARE 127 Figure 7.10: Sparsity structure of the matrices A p1 and A xi , i =1,...,4 Table 7.17: Comparison between performance of JD and JDCOMM using various search space dimensions Value Value Time (s) mindim maxdim A p1 A x1 A x2 A x3 A x4 5 30 11.6 10 15 10.0 10 25 15.4 10 50 10.7 6.1 10 75 6.8 41.5 10 100 139.9 10.1 7.7 20.8 20.7 method on the matrix A p1 computes the requested eigenvalue only in the case when the maximum search space dimension maxdim is 100, whereas the JDCOMM method using the matrix A x2 works in all cases with a superior performance. Note also that the increase of maxdim from 25 to 50 more than halves the computation time of the JDCOMM method on the matrix A x2 . Moreover, this is the fastest computation shown in Table 7.17. In Table 7.18 we give the more detailed results (with respect to required matrixvector products and flops) of the situation where all the eigenvalue methods succeed
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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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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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