Browsing by Author "Chu, Eric King-wah"

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A structure-preserving doubling algorithm for quadratic eigenvalue problems arising from time-delay systems
(2010-02-15)
Li, Tie-Xiang
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Chu, Eric King-wah
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Lin, Wen-Wei
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應用數學系
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Department of Applied Mathematics
We propose a structure-preserving doubling algorithm for a quadratic eigenvalue problem arising from the stability analysis of time-delay systems. We are particularly interested in the eigenvalues oil the unit circle, which are difficult to estimate. The convergence and backward error of the algorithm are analyzed and three numerical examples are presented. Our experience shows that Our algorithm is efficient in comparison to the few existing approaches for small to medium size problems. (C) 2009 Elsevier B.V. All rights reserved.
Large-scale Stein and Lyapunov equations, Smith method, and applications
(2013-08-01)
Li, Tiexiang
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Weng, Peter Chang-Yi
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Chu, Eric King-wah
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Lin, Wen-Wei
;
應用數學系
We consider the solution of large-scale Lyapunov and Stein equations. For Stein equations, the well-known Smith method will be adapted, with A(k) = A(2k) not explicitly computed but in the recursive form A(k) = A(k-1)(2), and the fast growing but diminishing components in the approximate solutions truncated. Lyapunov equations will be first treated with the Cayley transform before the Smith method is applied. For algebraic equations with numerically low-ranked solutions of dimension n, the resulting algorithms are of an efficient O(n) computational complexity and memory requirement per iteration and converge essentially quadratically. An application in the estimation of a lower bound of the condition number for continuous-time algebraic Riccati equations is presented, as well as some numerical results.
PALINDROMIC EIGENVALUE PROBLEMS: A BRIEF SURVEY
(2010-06-01)
Chu, Eric King-wah
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Huang, Tsung-Ming
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Lin, Wen-Wei
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Wu, Chin-Tien
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應用數學系
The T-palindromic quadratic eigenvalue problem (lambda(2)B + lambda C + A)x = 0, with A, B,C is an element of C(nxn), C(T) = C and B(T) = A, governs the vibration behaviour of trains. Other palindromic eigenvalue problems, quadratic or higher order, arise from applications in surface acoustic wave filters, optimal control of discrete-time systems and crack modelling. Numerical solution of palindromic eigenvalue problems is challenging, with unacceptably low accuracy from the basic linearization approach. In this survey paper, we shall talk about the history of palindromic eigenvalue problems, in terms of their history, applications, numerical solution and generalization. We shall also speculate on some future directions of research.
Recent Advances in Numerical Methods for Eigenvalue Problems Preface
(2010-06-01)
Bai, Zhaojun
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Chu, Eric King-wah
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Lin, Wen-Wei
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Wang, Chern-Shuh
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Yang, Chao
Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model
(2011-01-01)
Li, Tie-Xiang
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Chu, Eric King-wah
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Juang, Jong
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Lin, Wen-Wei
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電子物理學系
For the steady-state solution of an integral-differential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B(-) - XF(-) - F(+)X + XB(+)X = 0, where F(+/-) equivalent to I - (s) over cap PD(+/-), B(-) equivalent to (b) over capl + (s) over capP)D(-))D(-) and B(+) equivalent to (b) over capl + (s) over capP)D(+))D(+) with a nonnegative matrix P, positive diagonal matrices D, and nonnegative parameters f, (b) over cap equivalent to(1 - f) and (s) over cap equivalent to (1 - f). We prove the existence of the minimal nonnegative solution X* under the physically reasonable assumption f + b + s parallel to P(D(+) + D-)parallel to(infinity) < 1, and study its numerical computation by fixed-point iteration, Newton's method and doubling. We shall also study several special cases; e.g. when = 0 and P is low-ranked, then X* = <(s)over cap>/2 UV is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results. (C) 2010 Elsevier Inc. All rights reserved.
Solving large-scale continuous-time algebraic Riccati equations by doubling
(2013-01-01)
Li, Tiexiang
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Chu, Eric King-wah
;
Lin, Wen-Wei
;
Weng, Peter Chang-Yi
;
應用數學系
We consider the solution of large-scale algebraic Riccati equations with numerically low-ranked solutions. For the discrete-time case, the structure-preserving doubling algorithm has been adapted, with the iterates for A not explicitly computed but in the recursive form A(k) = A(k-1)(2) - (DkSk-1)-S-(1)[D-k((2))](T), with D-k((1)) and D-k((2)) being low-ranked and S-k(-1) being small in dimension. For the continuous-time case, the algebraic Riccati equation will be first treated with the Cayley transform before doubling is applied. With n being the dimension of the algebraic equations, the resulting algorithms are of an efficient O(n) computational complexity per iteration, without the need for any inner iterations, and essentially converge quadratically. Some numerical results will be presented. For instance in Section 5.2, Example 3, of dimension n = 20 209 with 204 million variables in the solution X, was solved using MATLAB on a MacBook Pro within 45 s to a machine accuracy of O(10(-16)). (C) 2012 Elsevier B.V. All rights reserved.
The palindromic generalized eigenvalue problem A*x = lambda Ax: Numerical solution and applications
(2011-06-01)
Li, Tiexiang
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Chiang, Chun-Yueh
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Chu, Eric King-wah
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Lin, Wen-Wei
;
應用數學系
In this paper, we propose the palindromic doubling algorithm (PDA) for the palindromic generalized eigenvalue problem (PGEP) A*x = lambda Ax. We establish a complete convergence theory of the PDA for PGEPs without unimodular eigenvalues, or with unimodular eigenvalues of partial multiplicities two (one or two for eigenvalue 1). Some important applications from the vibration analysis and the optimal control for singular descriptor linear systems will be presented to illustrate the feasibility and efficiency of the PDA. (C) 2009 Elsevier Inc. All rights reserved.