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128 CHAPTER 7. NUMERICAL EXPERIMENTS Figure 7.11: Eigenvalue spectra of the matrices A p1 and A xi , i =1,...,4 Table 7.18: Details of the performance of JD and JDCOMM with mindim = 10 and maxdim = 100 Method MV A p1 MV A xi Flops ×10 9 Time (s) Global minimum JD 5941 0 7.5 139.9 −206.5 JDCOMM A x1 237 1607 0.4 10.1 −206.5 JDCOMM A x2 302 2322 0.5 7.6 −206.5 JDCOMM A x3 725 6975 1.2 20.8 −206.5 JDCOMM A x4 636 5996 1.1 20.7 −206.5 to compute the requested eigenvalue; mindim is 10 and maxdim is 100 (last row of Table 7.17). Figure 7.12 shows the plot of the norms of the residuals in the eigenvalue equation (at each JD outer step) against the number of matrix-vector products with A p1 . All methods compute the same global minimum and the identical location. To reveil the performance of a Krylov based eigenvalue solver, we also apply the Matlab eigs method, which is an implicitly restarted Arnoldi method [95], to this problem. This method computes the requested smallest real eigenvalue in 62 seconds as the 151th eigenvalue. We used no additional parameter settings here;
7.5. NUMERICAL EXPERIMENTS WITH JDCOMM SOFTWARE 129 Figure 7.12: Residual norms against MVs with A p1 for the JD and JDCOMM methods when using additional settings to target the required eigenvalue, eigs even requires more computation time. Comparing the performance of the JDCOMM method in this example with SOS- TOOLS and GloptiPoly, we find that both methods compute the same minimizer and require 42 and 39 seconds, respectively.
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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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Page 91 and 92: 5.3. EFFICIENCY OF THE ND-SYSTEMS A
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Page 95 and 96: Chapter 6 Iterative eigenvalue solv
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Page 101 and 102: 6.4. PROJECTION TO STETTER-STRUCTUR
Page 117 and 118: 6.5. A JACOBI-DAVIDSON METHOD FOR C
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Page 122 and 123: 108 CHAPTER 7. NUMERICAL EXPERIMENT
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Page 150 and 151: 136 CHAPTER 8. H 2 MODEL-ORDER REDU
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Page 177 and 178: 10.1. SOLVING THE SYSTEM OF QUADRAT
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Page 181 and 182: 10.2. LINEARIZING A POLYNOMIAL EIGE
Page 183 and 184: 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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