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238 BIBLIOGRAPHY [13] I.W.M. Bleylevens, B. Hanzon, and R.L.M. Peeters. A multidimensional systems approach to polynomial optimization. Proceedings of the MTNS, Leuven, Belgium, 2004. [cited at p. 4, 26, 54] [14] I.W.M. Bleylevens, M.E. Hochstenbach, and R.L.M. Peeters. Polynomial optimization and a Jacobi–Davidson type method for commuting matrices. In preparation, 2009. [cited at p. 4] [15] I.W.M. Bleylevens, R.L.M. Peeters, and B. Hanzon. A multidimensional systems approach to algebraic optimization. Proceedings of the HPOPT – the 8th International Workshop on High Performance Optimization Techniques: Optimization and Polynomials, Amsterdam, pages 12–13, 2004. [cited at p. 4, 5] [16] I.W.M. Bleylevens, R.L.M. Peeters, and B. Hanzon. Efficiency improvement in an nD–systems approach to polynomial optimization. Proceedings of the MEGA 2005: Effective Methods in Algebraic Geometry Computing in and with algebraic geometry: Theory, Algorithms, Implementations, Applications, Alghero, Sardinia, Italy, 2005. [cited at p. 4, 26] [17] I.W.M. Bleylevens, R.L.M. Peeters, and B. Hanzon. Efficient polynomial optimization with an nD–systems approach. Proceedings of the 24th Benelux Meeting on Systems and Control, Houffalize, Belgium, page 46, 2005. [cited at p. 4] [18] I.W.M. Bleylevens, R.L.M. Peeters, and B. Hanzon. An nD–systems approach to global polynomial optimization with an application to H 2 model-order reduction. Proceedings of the 44th IEEE CDC and the ECC, Seville, Spain, pages 5107–5112, 2005. [cited at p. 4, 5, 153] [19] I.W.M. Bleylevens, R.L.M. Peeters, and B. Hanzon. Optimal H 2 model reduction from order N to N–2. Proceedings of the 13th ILAS Conference, Amsterdam, pages 76–77, 2006. [cited at p. 5] [20] I.W.M. Bleylevens, R.L.M. Peeters, and B. Hanzon. Efficiency improvement in an nD–systems approach to polynomial optimization. Journal of Symbolic Computation, 42(1–2):30–53, 2007. [cited at p. 4, 26, 35, 36, 46, 114] [21] I.W.M. Bleylevens, R.L.M. Peeters, and B. Hanzon. Optimal H 2 model reduction from order N to N–2. Internal Report M07-01, Department of Mathematics, Maastricht University, 2007. [cited at p. 5] [22] I.W.M. Bleylevens, R.L.M. Peeters, and B. Hanzon. A generalized eigenvalue approach to H 2 model-order reduction. Proceedings of the 27th Benelux Meeting on Systems and Control, Heeze, page 98, 2008. [cited at p. 5] [23] B. Buchberger. Gröbner bases: An algorithmic method in polynomial ideal theory. In N.K. Bose, editor, Multidimensional Systems Theory, Mathematics and Its Applications, pages 184–232. D. Reidel Publishing Company, 1985. [cited at p. 11, 19, 34] [24] J. Canny and I. Emiris. Efficient incremental algorithms for the sparse resultant and the mixed volume. Journal of Symbolic Computation, 20:117Ű–149, 1995. [cited at p. 28] [25] J. Canny, E. Kaltofen, and Y. Lakshman. Solving systems of non-linear polynomial equations faster. In Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation 1989, pages 121–128, New York, 1989. ACM Press. [cited at p. 29]
BIBLIOGRAPHY 239 [26] A. Cayley. On the theory of elimination. Cambridge and Dublin Mathematical Journal, 3:210–270, 1865. [cited at p. 29] [27] T.Y. Chen and J. Demmel. Balancing sparse matrices for computing eigenvalues. Linear Algebra and Its Applications, 309(1–3):261–287, 2000. [cited at p. 110, 158] [28] D. Cox, J. Little, and D. O’Shea. Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, 2nd edition, 1997. [cited at p. 11, 17, 18, 19, 20, 23, 34] [29] D. Cox, J. Little, and D. O’Shea. Using algebraic geometry. Springer, 1998. [cited at p. 35] [30] E.R. Davidson. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. Journal of Computational Physics, 17:87Ű–94, 1975. [cited at p. 82] [31] B.L. Van der Waerden. Algebra: Volume I. Springer-Verlag, 6th edition, 1964. [cited at p. 26] [32] E.W. Dijkstra. A note on two problems in connexion with graphs. Numerische Mathematik, 1:269–271, 1959. [cited at p. 52] [33] A.L. Dixon. The elimination of three quantics in two independent variables. In Proceedings of London Mathematical Society, volume 6, pages 468–478, 1908. [cited at p. 29] [34] P. Van Dooren. The computation of Kronecker’s canonical form of a singular pencil. Linear Algebra and Its Applications, 27:103–140, 1979. [cited at p. 166, 168, 172, 177, 236] [35] D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels. Mathematical Computations with Macaulay 2 (http://www.math.uiuc.edu/Macaulay2), volume 8. Springer Verlag, 2001. [cited at p. 25] [36] I. Emiris and J. Canny. A practical method for the sparse resultant. In M. Bronstein, editor, Proceedings of the 1993 international symposium on Symbolic and algebraic computation ISSAC ’93, pages 183–192, New York, 1993. ACM Press. [cited at p. 29] [37] G. Farin. Curves and surfaces for computer aided geometric design, A practical guide. Academic Press, 1990. [cited at p. 32] [38] R.W. Floyd. Algorithm 97: Shortest Path. Communications of the ACM, 5(6):345, 1962. [cited at p. 52] [39] D.R. Fokkema, G.L.G. Sleijpen, and H.A. van der Vorst. Jacobi–Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM Journal on Scientific Computing, 20(1):94–125, 1998. [cited at p. 51, 83, 112] [40] E. Fornasini, P. Rocha, and S. Zampieri. State space realization of 2–D finite– dimensional behaviours. SIAM Journal on Control and Optimization, 31(6):1502–1517, 1993. [cited at p. 52] [41] P. Fulcheri and M. Olivi. Matrix rational H 2 approximation: a gradient algorithm based on Schur analysis. SIAM Journal on Control and Optimization, 36(6):2103– 2127, 1998. [cited at p. 134]
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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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Page 251: 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
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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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