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158 CHAPTER 9. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-1 (9.4% filled), i =1,...,N. The matrix A T V H contains 550100 non-zeros (52.5% filled). See Figure 9.3 for a representation of the sparsity structure of the matrices A T x 1 , A T x 2 and of the matrix A T V H , respectively. Figure 9.3: Sparsity representation of the matrices A T x 1 (9.4% filled), A T x 2 (9.4% filled) and A T V H (52.5% filled) To compute only the smallest non-zero real eigenvalue/eigenvector of the operator A T V H using an nD-systems approach, the iterative eigenvalue solvers JDQR, JDQZ and Eigs are used. These solvers are introduced and discussed in Chapter 6. Because the condition numbers of the operators involved are very large, a similar balancing technique is used as described in [27], and employed before in Section 7.1. The smallest real non-zero eigenvalue computed matrix-free using the operator, is equal to the eigenvalue computed before, with the only exception that the computation only takes about 200 seconds and the memory requirements are minimal because the matrix is never constructed explicitly. The memory requirements for this
9.3. EXAMPLE 159 matrix-free approach stay below 100 MB. Therefore, one may conclude that using an iterative eigenvalue solver in combination with an nD-system is the most efficient approach to compute the smallest real eigenvalue of such a sparse matrix A T V H . In practice, the algebraic techniques presented here 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. How to deal with numerical ill-conditioning is an important research topic when one wants to solve problems that are of a somewhat larger size. Such implementation issues are still under investigation.
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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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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
Page 193 and 194: 10.4. COMPUTING THE APPROXIMATION G
Page 195 and 196: 10.6. EXAMPLES 181 5. Compute the n
Page 197 and 198: 10.6. EXAMPLES 183 Figure 10.1: Spa
Page 199 and 200: 10.6. EXAMPLES 185 Figure 10.3: The
Page 201 and 202: 10.6. EXAMPLES 187 Figure 10.5: Imp
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Page 205 and 206: 10.6. EXAMPLES 191 Figure 10.8: Pol
Page 207 and 208: 10.6. EXAMPLES 193 Table 10.1: Redu
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Page 211 and 212: 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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