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88 CHAPTER 6. ITERATIVE EIGENVALUE SOLVERS can potentially cause numerical difficulties. To overcome this problem the (natural) logarithmic transformation is applied to the entries of the eigenvector. Example 6.1. For n = 2 and d = 2 (hence m = 3) an eigenvector v is structured as (1, x 1 ,x 2 1,x 2 ,x 1 x 2 ,x 2 1x 2 ,x 2 2,x 1 x 2 2,x 2 1x 2 2) T . The logarithmic transformation applied to each of the entries of such a vector v with Stetter structure results in: ⎛ log ⎜ ⎝ ⎞ ⎛ 1 x 1 x 2 1 x 2 = log(x 1 ) ⎟ ⎜ ⎠ ⎝ x 1 x 2 x 2 1x 2 x 2 2 x 1 x 2 2 x 2 1x 2 2 0 1 2 0 1 2 0 1 2 ⎞ ⎛ + log(x 2 ) ⎟ ⎜ ⎠ ⎝ 0 0 0 1 1 1 2 2 2 ⎞ . (6.11) ⎟ ⎠ In general a Stetter vector v (where the first entry is normalized to 1) can be decomposed as follows using the same logarithmic transformation: ⎛ log(v) = ⎜ ⎝ log(v 1 ) . . log(v N ) ⎞ ⎟ ⎠ = log(x 1)w 1 + log(x 2 )w 2 + ...+ log(x n )w n (6.12) in which the vectors w 1 ,...,w n are fixed vectors, containing the integer values 0, 1, ..., m− 1 (each m n−1 times) in a specific ordering depending on the monomial ordering that is used. Remark 6.1. The right-hand side of Equation (6.12) can be extended without loss of generality with a term log(x 0 )w 0 , where w 0 =(1,...,1) T , to be able to deal with eigenvectors in which the first entry is not normalized to 1. The right-hand side of Equation (6.12) can be used to project a vector that has an approximate Stetter structure to the desired structure. Thus, we are looking for complex quantities x 1 ,...,x n such that: log(û) = log(x 1 )w 1 + log(x 2 )w 2 + ...+ log(x n )w n (6.13) is a good approximation of log(u). However, this logarithmic transformation is not straightforward when applying it to a complex or negative entry in the approximate eigenvector. To overcome this problem, all entries of a vector u, including complex and negative ones, are written in polar coordinates. This yields for a general vector z: log(z) = log(|z|)+i · arg(z), (6.14)
6.4. PROJECTION TO STETTER-STRUCTURE 89 so that the real part of z is well determined for z 0 whereas the imaginary part is determined only up to an integer multiple of 2π. Writing the entries of the vectors x =(x 1 ,...,x n ) T and u as follows: x j = r j e i·φj j =1,...,n (6.15) and u j = υ j e i·ψj j =1,...,N (6.16) where r j = |x j |,υ j = |u j |∈R + and −π
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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
Page 51 and 52: 3.5. EXAMPLE 37 applying the Stette
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Page 57 and 58: Chapter 4 Global optimization of mu
Page 59 and 60: 4.1. THE GLOBAL MINIMUM OF A DOMINA
Page 61 and 62: 4.3. AN EXAMPLE 47 Using the matric
Page 63 and 64: 4.3. AN EXAMPLE 49 the Stetter-Möl
Page 65 and 66: Chapter 5 An nD-systems approach in
Page 67 and 68: 5.1. THE ND-SYSTEM 53 cerned with t
Page 69 and 70: 5.1. THE ND-SYSTEM 55 Example 5.1.
Page 71 and 72: 5.1. THE ND-SYSTEM 57 Figure 5.2: T
Page 73 and 74: 5.3. EFFICIENCY OF THE ND-SYSTEMS A
Page 95 and 96: Chapter 6 Iterative eigenvalue solv
Page 97 and 98: 6.2. THE JACOBI-DAVIDSON METHOD 83
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Page 122 and 123: 108 CHAPTER 7. NUMERICAL EXPERIMENT
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138 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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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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