# 臺灣博碩士論文加值系統

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 AbstractThe purpose of this paper is to acquire model reductions of large-scale quadratic dynamical systems. The idea for solving the problem is based on the relation between a quadratic system and its related enlarged linear system. That is, we transfer the second-order system to an enlarged first-order systems and obtain the second-order model reduction by converting the first-order model reduction to a second-order one. Next, we give a brief introduction for numerical methods of the first-order model reduction. Particularly, these methods can be categorized to two groups: (a) SVD based methods; (b)Krylov based methods. Finally, some numerical experiments are performed. The results of numerical implementation show that some serious round-off errors may occur when the first-order model reduction converts to the second-order one.So, we conclude that the propose methods in this paper are eventually not good enough for the second-order reduction. However, a reliable and ecient method for the model reduction of a quadratic dynamical system is still under investigation.
 Contents1 Introduction 21.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Reduction via rst-order balance and truncate . . . . . . . . . 31.1.2 Transfer rst-order form in second-order form . . . . . . . . . 41.2 First-order form linear dynamical system . . . . . . . . . . . . . . . 51.3 Measures of the accuracy of the approximation . . . . . . . . . . . . 62 SVD based approximation methods 72.1 Proper orthogonal decomposition (POP) method . . . . . . . . . . . 92.2 Balanced truncation method . . . . . . . . . . . . . . . . . . . . . . 92.2.1 The minimal control and largest observation energies . . . . . 92.2.2 Hankel operator, Gramians and Lyapunov equations . . . . . 102.2.3 Balance and truncate . . . . . . . . . . . . . . . . . . . . . . 112.3 Optimal Hankel norm approximation method . . . . . . . . . . . . . 133 Projection based method 143.1 Approximation by moment matching . . . . . . . . . . . . . . . . . . 143.1.1 Lanczos procedure . . . . . . . . . . . . . . . . . . . . . . . . 163.1.2 Arnoldi procedure . . . . . . . . . . . . . . . . . . . . . . . . 173.1.3 Implicitly restarted Arnoldi and Lanczos methods . . . . . . 183.1.4 Rational Krylov method . . . . . . . . . . . . . . . . . . . . . 193.2 Krylov subspace method . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.1 Alternating direction implicity iteration (ADI) and LR-ADI . 213.2.2 Smith method . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.3 Low rank square root method (LRSRM) . . . . . . . . . . . . 223.2.4 Low rank Schur method (LRSM) . . . . . . . . . . . . . . . . 234 Some properties of the second-order system 244.1 Second-order transfer function . . . . . . . . . . . . . . . . . . . . . . 254.2 Second-order Gramian . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 Second-order singular values and balancing . . . . . . . . . . . . . . . 284.4 Direct second-order reduction method . . . . . . . . . . . . . . . . . 295 Numerical experiments 306 Conclusions and future work 36References 37
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Implicit application of polynomial lters in a k-step Arnoldimethod. SIAM J. Matrix Anal. Applic., 13:357-385, 1992.[34] E. Wachspress. The ADI minimax problem for complex spectra. In D. Kincaidand L. Hayes, editors, Iterative Methods for Large Linear Systems, pages 251-271. Academic Press, San Diego, 1990.[35] W.W. Lin and C.W. Wang Some numerical computations of model reductionsfor second-order systems . privily communicate.
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