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242 BIBLIOGRAPHY [73] C. Lanczos. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Research of the National Bureau of Standards, 45:255Ű–282, 1950. [cited at p. 82] [74] J.B. Lasserre. Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization, 11(3):796–817, 2001. [cited at p. 111] [75] J.B. Lasserre. An explicit equivalent positive semidefinite program for nonlinear 0–1 programs. SIAM Journal on Optimization, 12(3):756–769, 2002. [cited at p. 111] [76] D. Lazard. Resolution des systemes d’equation algebriques. Theoretical Computer Science, 15:77–110, 1981. [cited at p. 29] [77] R.B. Lehoucq, D.C. Sorensen, and C. Yang. ARPACK Users’ Guide: Solution of Large–Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM Publications, 1998. [cited at p. 51, 82] [78] D. Lemonnier and P. van Dooren. Balancing regular matrix pencils. Journal on Matrix Analysis and Applications, 28(1):253–263, 2006. [cited at p. 184] [79] F.S. Macaulay. Algebraic theory of modular systems. Cambridge Tracts in Mathematics and Mathematical physics, no. 19, 1916. [cited at p. 28, 29] [80] L. Meier and D. Luenberger. Approximation of linear constant systems. IEEE Transactions on Automatic Control, 12(5):585–588, 1967. [cited at p. 141] [81] H.M. Möller and H.J. Stetter. Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems. Numerische Mathematik, 70:311–329, 1995. [cited at p. 3, 26, 35, 36, 229, 257, 261] [82] H.M. Möller and R. Tenberg. Multivariate polynomial system solving using intersections of eigenspaces. Journal of Symbolic Computation, 32:513–531, 2001. [cited at p. 35] [83] A.P. Morgan. Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems. Prentice Hall, 1987. [cited at p. 30] [84] B. Mourrain. Symbolic-numeric methods for solving polynomial equations and applications. In A. Dickenstein, editor, Lecture notes of the 1st Latin American School on Polynomial Systems, CIMPA, 2003. [cited at p. 28, 37] [85] B. Mourrain and J.P. Pavone. Subdivision methods for solving polynomial equations. Journal of Symbolic Computation, 44:292–306, 2009. [cited at p. 32] [86] P.A. Parrilo. Semidefinite programming relaxations for semialgebraic problems. Mathematical Programming, 96(2):293–320, 2003. [cited at p. 111] [87] P.A. Parrilo and B. Sturmfels. Minimizing Polynomial Functions. Algorithmic and quantitative real algebraic geometry, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 60:83–100, 2003. [cited at p. 111] [88] R.L.M. Peeters, I.W.M. Bleylevens, and B. Hanzon. A mixed algebraic-numerical global optimization approach to H 2 model reduction. Proceedings of the 22th Benelux Meeting on Systems and Control, Lommel, Belgium, 2003. [cited at p. 4] [89] R.L.M. Peeters, B. Hanzon, and D. Jibetean. Optimal H 2 model reduction in statespace: a case study. Proceedings of the European Control Conference ECC 2003, Cambridge, UK, 2003. [cited at p. 222]
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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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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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Page 243 and 244: Chapter 12 Conclusions & directions
Page 245 and 246: 12.1. CONCLUSIONS 231 used first. T
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Page 251 and 252: Bibliography [1] W. Adams and P. Lo
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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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