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236 CHAPTER 12. CONCLUSIONS & DIRECTIONS FOR FURTHER RESEARCH k = 1 case, it is convenient to impose the technical condition that all the poles of the original given system H(s) are distinct. We have imposed a similar constraint for ease of exposition for all co-orders k in this thesis. Extensions may be developed in the future to handle the case with poles of multiplicities larger than 1 too. Moreover, the algebraic techniques presented in this part of the thesis may be combined with numerical local search techniques to arrive at practical algorithms, for instance to save on hardware requirements, computation time, or to deal with numerical accuracy issues. To deal with numerical conditioning issues is an important research topic, to be able to deal with problems that are of a somewhat larger size. A disadvantage of the co-order k = 2 and k = 3 case, from the perspective of computational efficiency, is that one can only decide about global optimality of a solution G(s) when all the solutions of the polynomial eigenvalue problem have been computed and further analyzed. Currently it is investigated for the co-order k =2 case whether the third order polynomial V H can be employed by iterative polynomial eigenproblem solvers to limit the number of eigenvalues that require computation. Possibly, one could modify the available Jacobi–Davidson methods for a polynomial eigenvalue problem in such a way that it iterates with the involved matrix but targets on small positive real values of the criterion function V H first (in the same spirit as the JDCOMM method). When we compute the Kronecker canonical forms of the singular matrix pencils in the co-order k = 2 and k = 3 case, we use exact arithmetic to split off the singular blocks of these pencils. This is only possible when the given transfer function H(s) is given in exact format. Exact computations are computationally hard and, moreover, they become useless when the transfer function H(s) is given in numerical format. To overcome this computational limitation it is advisable to use numerical algorithms to compute the Kronecker canonical forms of the singular matrix pencils as described for example in [34]. Finally, H 2 model-order reduction of the multi-input, multi-output case along similar lines as the approach followed here will lead to further generalizations too. Furthermore, the discrete time case can be handled too when using the isometric isomorphism mentioned in Remark 8.2 which transforms the continuous time case into the discrete time case.
Bibliography [1] W. Adams and P. Loustaunau. An Introduction to Gröbner Bases, Graduate Studies in Mathematics, volume 3. American Mathematical Society, 1994. [cited at p. 11] [2] P.R. Aigrain and E.M. Williams. Synthesis of n-reactance networks for desired transient response. Journal of Applied Physics, 20:597–600, 1949. [cited at p. 134] [3] E.L. Allgower and K. Georg. Numerical Continuation Methods: An Introduction. Springer, 1990. [cited at p. 30] [4] W.E. Arnoldi. The principle of minimized iteration in the solution of the matrix eigenvalue problem. Quarterly of Applied Mathematics, 9:17–29, 1951. [cited at p. 82] [5] S. Attasi. Modeling and recursive estimation for double indexed sequences. In R.K. Mehra and D.G. Lainiotis, editors, System Identification: Advances and Case Studies, pages 289–348, New York, 1976. Academic Press. [cited at p. 52] [6] Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, H. van der Vorst, and editors. Templates for the solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, 2000. [cited at p. 168] [7] L. Baratchart. Existence and generic properties of L 2 approximants for linear systems. IMA Journal of Mathematical Control and Information, 3:89–101, 1986. [cited at p. 134] [8] L. Baratchart, M. Cardelli, and M. Olivi. Identification and rational L 2 approximation: A gradient algorithm. Automatica, 27(2):413–417, 1991. [cited at p. 141] [9] L. Baratchart and M. Olivi. Index of critical points in rational L 2-approximation. Systems and Control Letters, 10(3):167–174, 1988. [cited at p. 141] [10] T. Becker, V. Weispfenning, and H. Kredel. Gröbner Bases: A Computational Approach to Commutative Algebra, volume 141. Springer, 1993. [cited at p. 11, 23] [11] M. Berhanu. The polynomial eigenvalue problem. Ph.D. Thesis, University of Manchester, Manchester, England, 2005. [cited at p. 166] [12] A.W.J. Bierfert. Polynomial optimization via recursions and parallel computing. Master Thesis, Maastricht University, MICC (Department Knowledge Engineering), 2007. [cited at p. 43] 237
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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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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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