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166 CHAPTER 10. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-2 10.2 Linearizing a polynomial eigenvalue problem Polynomial eigenvalue problems have been well studied in the literature and numerical algorithms for its solution are available [55]. These problems occur for example in structural mechanics, molecular dynamics, gyroscopic systems and constrained leastsquares problems (see, e.g., [11] and [72]). Usually the problem is first linearized with respect to ρ 1 , yielding a generalized eigenvalue problem with an associated matrix pencil. This linearization is the subject of this section. The standard tool for the numerical solution of a generalized eigenvalue problem is the QZ-algorithm, which however is notoriously ill-conditioned if the matrix pencil has a singular part [42]. To remove any singular part that may be present, we may proceed algebraically as in the algorithm for the computation of the Kronecker canonical form of a matrix pencil (see [34] and [42]). This can be implemented with exact and symbolic arithmetic when required. Once the singular part and the associated ‘indeterminate’ eigenvalues are removed from the matrix pencil, the square regular part remains for which the (finite and infinite) eigenvalues can subsequently be computed by standard means. Computing the Kronecker canonical form of a singular matrix pencil and deflating the pencil to get rid of the unwanted eigenvalues, is the subject of Section 10.3. First we introduce the generalized eigenvalue problem and some of its features. Let B and C be two m 1 × m 2 matrices with complex-valued entries. The associated matrix pencil is the one-parameter family of matrices B + ρ 1 C, where ρ 1 ranges over all complex values. The generalized eigenvalue problem for B +ρ 1 C consists of finding values for ρ 1 and associated non-zero vectors w ∈ C m2 such that: (B + ρ 1 C)w =0. (10.11) Note that the conventional eigenvalue problem for a given square m 2 × m 2 matrix A occurs when choosing m 1 = m 2 , B = A and C = −I m2 , producing the associated equation (A−ρ 1 I m2 )w = 0. Because B and C in general are allowed to be rectangular matrices rather than square matrices, the generalized eigenvalue problem may exhibit a few features that do not occur for conventional eigenvalue problems. Another issue that may occur in the square case m 1 = m 2 , is that the matrix C may be singular, so that the generalized eigenvalue problem can not be transformed directly into an equivalent conventional eigenvalue problem upon premultiplication by C −1 . Obviously, a solution to the generalized eigenvalue problem (B + ρ 1 C)w = 0 requires a value of ρ 1 , which makes B + ρ 1 C into an m 1 × m 2 matrix having a set of linearly dependent columns. When m 1
10.2. LINEARIZING A POLYNOMIAL EIGENVALUE PROBLEM 167 generalized eigenvalue problem (B + ρ 1 C)w = 0 one may also consider the left generalized eigenvalue problem u ∗ (B + ρ 1 C)=0 T . Thus, linear dependence of the rows of B+ρ 1 C for various values of ρ 1 (including infinity) can be given similar meaning too. As demonstrated in the previous section, the problem of finding solutions of the system of equations (10.2) subject to (10.3) can be cast in the form of a polynomial eigenvalue problem involving the square polynomial matrix Ãã N−1 (ρ 1 ) T . The problem of computing all the values ρ 1 and corresponding non-trivial vectors v that satisfy Ãã N−1 (ρ 1 ) T v = 0 can be approached by rewriting it as a generalized eigenvalue problem as follows. First, expand the polynomial matrix Ãã N−1 (ρ 1 ) T as a matrix polynomial in ρ 1 of degree N − 1, by: ÃãN−1 (ρ 1 ) T = A 0 + ρ 1 A 1 + ...+ ρ N−1 1 A N−1 . (10.12) Applying a linearization technique as mentioned, e.g., in [55], [56] or [72], makes the polynomial eigenvalue problem (10.10), where the involved matrix is written as (10.12), equivalent to a generalized eigenvalue problem: (A 0 + ρ 1 A 1 + ...+ ρ N−1 1 A N−1 ) v =0 ⇐⇒ (B + ρ 1 C)ṽ = 0 (10.13) where ρ 1 occurs linearly and where the matrices B and C are (large) block-partitioned square matrices with the following structure: ⎛ ⎞ 0 I 0 . B = ⎜ . .. ⎟ ⎝ 0 0 I ⎠ , A 0 A 1 ... A N−2 ⎛ C = ⎜ ⎝ (10.14) ⎞ −I 0 0 . .. . ⎟ 0 −I 0 ⎠ . 0 ... 0 A N−1 The eigenvectors ṽ of the generalized eigenvalue problem (B+ρ 1 C)ṽ = 0 in (10.13) have a built-in structure, which is given by: ⎛ ṽ = ⎜ ⎝ v ρ 1 v . ρ N−2 1 v ⎞ ⎟ ⎠ . (10.15) The block vectors v inside such an eigenvector ṽ are eigenvectors of the original polynomial eigenvalue problem (10.10). Note that the eigenvectors ṽ have additional
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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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11.3. KRONECKER CANONICAL FORM OF A
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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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