link to my thesis

More documents

Recommendations

Info

94 CHAPTER 6. ITERATIVE EIGENVALUE SOLVERS vector ˆψ as follows: ⎛ ˆψ = ⎜ ⎝ 0 0 1 0 2 0 0 1 1 1 2 1 0 2 1 2 2 2 ⎞ ( ) ˆψ2 ˆψ 4 ⎟ ⎠ ⎛ + 2π ⎜ ⎝ 0 0 1 0 2 0 0 1 1 1 2 1 0 2 1 2 2 2 ⎞ ⎟ ⎠ ( k2 k 4 ) − 2π ⎛ ⎜ ⎝ k 1 k 2 k 3 k 4 k 5 k 6 k 7 k 8 k 9 ⎞ . (6.39) ⎟ ⎠ This results in the following relations for the chosen values k =(0, 0, 1, 0, 0, 1, −1, 0, 0): ⎧ ⎪⎨ ⎪⎩ (6.40) + 2 4 ˆψ 1 = 0 ˆψ 2 = ˆψ2 ˆψ 3 = 2 ˆψ 2 −2π ˆψ 4 = ˆψ4 ˆψ 5 = ˆψ2 + ˆψ4 ˆψ 9 = 2 ˆψ 2 ˆψ 6 = 2 ˆψ 2 + ˆψ4 −2π ˆψ 7 = 2 ˆψ 4 +2π ˆψ 8 = ˆψ2 + 2 ˆψ 4 The projected vector is finally computed as log(ˆυ)+i · ˆψ and transformed back into the non-logarithmic domain to yield the vector û: ⎛ û = e log(ˆυ)+i· ˆψ = ⎜ ⎝ 1.000 − 3.862 − 3.684i 1.350 + 28.454i − 9.476 − 8.268i 6.142 + 66.838i 222.471 − 280.782i 21.421 + 156.693i 494.436 − 684.113i −4429.619 + 821.089i ⎞ . (6.41) ⎟ ⎠ The vector û exhibits the Stetter structure with respect to multiplication since it
6.4. PROJECTION TO STETTER-STRUCTURE 95 satisfies the following relations: ⎧ ⎪⎨ ⎪⎩ û 1 = 1 û 2 = û 2 û 3 = û 2 2 û 4 = û 4 û 5 = û 2 · û 4 û 6 = û 2 2 · û 4 û 7 = û 2 4 û 8 = û 2 · û 2 4 û 9 = û 2 2 · û 2 4 (6.42) Note that there are other possible choices for the values of k =(k 1 ,...,k 9 ) T in Equation (6.34) which yield the same projected vector û as in (6.41). For example, the choices k =(0, −2, −2, −1, −2, −1, −2, −1, 0), k =(0, −3, −3, −1, −3, −3, −1, −2, −3) or k =(0, −2, −1, 0, −1, −1, −1, −2, −1) yield all the same vector û (this is because for these values the vectors e i· ˆψ in (6.41) are identical). The 2-norm of the difference between the original vector u and the projected vector û is 338.443. This norm is larger than the 2-norm of the difference between the original vector u and the Stetter vector corresponding to (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 , with given values x 1 =(−4 − 4i) and x 2 =(−9 − 8i), which is computed as 13.342. The large norm of u−û is caused by the errors in the last entries of the vector u − û. These errors are relatively small but have a large influence on the computation of the norm. That the errors in the last entries of the vector u − û are larger than in the first entries is explained by the fact that this projection method operates in the logarithmic domain to regulate the influence of large dominating powers of the Stetter structure. When looking at the values of x 1 and x 2 before and after the projection we see that they are projected, respectively, from −4 and (−13 − 5i) in the vector u to (−3.862 − 3.684i) and (−9.476 − 8.268i) in the projected vector û. The latter values for x 1 and x 2 are close to the given values of x 1 =(−4 − 4i) and x 2 =(−9 − 8i). A drawback of the above projection method is that the optimal values for k in Equation (6.26) are not directly known: there exist various choices for k. Because of this, it is required to search for the optimal values for k before this projection method can be applied. This amounts in determining the vector k =(k 1 ,k 2 ,...,k N ) T such that the current estimate of the eigenpair improves. Currently, no efficient method to calculate k is implemented other than a brute force search for a fixed maximum number of iterations. However, this requires a large amount of matrix-vector multiplications, which is contradictory to the initial goal of projection. To overcome this, the projection to Stetter structure discussed here is limited to the situation where the imaginary part of the approximate eigenvector is small enough to be discarded. This approach is used in the implementations of the Jacobi–Davidson methods and in the experiments of Chapter 7.
Page 3 and 4:
Stellingen behorende bij het proefs
Page 5 and 6:
ALGEBRAIC POLYNOMIAL SYSTEM SOLVING
Page 7 and 8:
ALGEBRAIC POLYNOMIAL SYSTEM SOLVING
Page 9:
For all those I love...
Page 12 and 13:
ii CONTENTS II Global Optimization
Page 14 and 15:
iv CONTENTS 12.2 Directions for fur
Page 17 and 18:
Chapter 1 Introduction 1.1 Motivati
Page 19 and 20:
1.3. RESEARCH QUESTIONS 5 The goal
Page 21 and 22:
1.4. THESIS OUTLINE 7 Jacobi-Davids
Page 23:
Part I General introduction and bac
Page 26 and 27:
12 CHAPTER 2. ALGEBRAIC BACKGROUND
Page 28 and 29:
Page 30 and 31:
Page 32 and 33:
Page 34 and 35:
Page 36 and 37:
Page 39 and 40:
Chapter 3 Solving polynomial system
Page 41 and 42:
3.1. SOLVING POLYNOMIAL EQUATIONS I
Page 43 and 44:
3.2. METHODS FOR SOLVING POLYNOMIAL
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
3.3. THE STETTER-MÖLLER MATRIX MET
Page 51 and 52:
3.5. EXAMPLE 37 applying the Stette
Page 53:
3.5. EXAMPLE 39 be: I = 〈13x 2 2
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
Page 99 and 100: 6.2. THE JACOBI-DAVIDSON METHOD 85
Page 101 and 102: 6.4. PROJECTION TO STETTER-STRUCTUR
Page 107: 6.4. PROJECTION TO STETTER-STRUCTUR
Page 117 and 118: 6.5. A JACOBI-DAVIDSON METHOD FOR C
Page 119: 6.5. A JACOBI-DAVIDSON METHOD FOR C
Page 122 and 123: 108 CHAPTER 7. NUMERICAL EXPERIMENT
Page 145: Part III H 2 Model-order Reduction
Page 150 and 151: 136 CHAPTER 8. H 2 MODEL-ORDER REDU
Page 158 and 159:
144 CHAPTER 8. H 2 MODEL-ORDER REDU
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 175 and 176:
Chapter 10 H 2 Model-order reductio
Page 177 and 178:
10.1. SOLVING THE SYSTEM OF QUADRAT
Page 179 and 180:
10.1. SOLVING THE SYSTEM OF QUADRAT
Page 181 and 182:
10.2. LINEARIZING A POLYNOMIAL EIGE
Page 183 and 184:
10.3. THE KRONECKER CANONICAL FORM
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
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
Page 203 and 204:
10.6. EXAMPLES 189 Figure 10.7: Str
Page 205 and 206:
10.6. EXAMPLES 191 Figure 10.8: Pol
Page 207 and 208:
10.6. EXAMPLES 193 Table 10.1: Redu
Page 209 and 210:
Chapter 11 H 2 Model-order reductio
Page 211 and 212:
11.2. LINEARIZING THE TWO-PARAMETER
Page 213 and 214:
11.3. KRONECKER CANONICAL FORM OF A
Page 215 and 216:
Page 217 and 218:
Page 219 and 220:
Page 221 and 222:
Page 223 and 224:
Page 225 and 226:
Page 227 and 228:
Page 229 and 230:
Page 231 and 232:
Page 233 and 234:
Page 235 and 236:
11.4. COMPUTING THE APPROXIMATION G
Page 237 and 238:
11.5. EXAMPLE 223 AãN−2 (ρ 1 ,
Page 239 and 240:
11.5. EXAMPLE 225 of both matrices
Page 241 and 242:
11.5. EXAMPLE 227 of solutions (ρ
Page 243 and 244:
Chapter 12 Conclusions & directions
Page 245 and 246:
12.1. CONCLUSIONS 231 used first. T
Page 247 and 248:
12.1. CONCLUSIONS 233 solver as des
Page 249 and 250:
12.2. DIRECTIONS FOR FURTHER RESEAR
Page 251 and 252:
Bibliography [1] W. Adams and P. Lo
Page 253 and 254:
BIBLIOGRAPHY 239 [26] A. Cayley. On
Page 255 and 256:
BIBLIOGRAPHY 241 [58] M.E. Hochsten
Page 257:
BIBLIOGRAPHY 243 [90] S. Prajna, A.
Page 261 and 262:
Appendix A A linearization techniqu
Page 263 and 264:
A.2. LINEARIZATION WITH RESPECT TO
Page 265 and 266:
A.3. EXAMPLE 251 The first step of
Page 267:
A.3. EXAMPLE 253 where the eigenvec
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
show all

link to my thesis

Create successful ePaper yourself

Delete template?

Save as template?