Browsing by Author "Chiang, Chun-Yueh"

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A NOTE ON GOLDEN MEANS, NONLINEAR MATRIX EQUATIONS AND STRUCTURED DOUBLING ALGORITHMS
(2010-06-01)
Chiang, Chun-Yueh
;
Chu, Eric King-Wah
;
Lin, Wen-Wei
;
應用數學系
;
Department of Applied Mathematics
Several beautiful formulae for the solutions of some nonlinear matrix equations were proposed by Yongdo Lim in 2007, in terms of the matrix golden means. Numerically, these formulae will not be applicable when some matrices involved are ill-conditioned. In this note, we propose to partially fill in this gap of applicability with the structured doubling algorithm, under some favourable conditions. We also discuss how some pre-processing or scaling procedures can be applied to the matrix equations, to improve their condition. More generally, we also explore the possibility of computing the matrix golden mean using structured doubling algorithms. Some numerical examples will be presented for illustrative purposes.
A STRUCTURED DOUBLING ALGORITHM FOR DISCRETE-TIME ALGEBRAIC RICCATI EQUATIONS WITH SINGULAR CONTROL WEIGHTING MATRICES
(2010-06-01)
Chiang, Chun-Yueh
;
Fan, Hung-Yuan
;
Lin, Wen-Wei
;
應用數學系
;
Department of Applied Mathematics
In this paper we propose a structured doubling algorithm for solving discrete-time algebraic Riccati equations without the invertibility of control weighting matrices. In addition, we prove that the convergence of the SDA algorithm is linear with ratio less than or equal 1/2 when all unimodular eigenvalues of the closed-loop matrix are semi-simple. Numerical examples are shown to illustrate the feasibility and efficiency of the proposed algorithm.
CONVERGENCE ANALYSIS OF THE DOUBLING ALGORITHM FOR SEVERAL NONLINEAR MATRIX EQUATIONS IN THE CRITICAL CASE
(2009)
Chiang, Chun-Yueh
;
Chu, Eric King-Wah
;
Guo, Chun-Hua
;
Huang, Tsung-Ming
;
Lin, Wen-Wei
In this paper, we review two types of doubling algorithm and some techniques for analyzing them. We then use the techniques to study the doubling algorithm for three different nonlinear matrix equations in the critical case. We show that the convergence of the doubling algorithm is at least linear with rate 1/2. As compared to earlier work on this topic, the results we present here are more general, and the analysis here is much simpler.
On the star-Sylvester equation AX +/- X-star B-star = C
(2012-05-01)
Chiang, Chun-Yueh
;
Chu, Eric King-Wah
;
Lin, Wen-Wei
;
應用數學系
;
Department of Applied Mathematics
We consider the solution of the star-Sylvester equations AX +/- (XB star)-B-star = C, for star = T, H and A, B, is an element of C-nxn, and the related linear matrix equations AXB(star) +/- X-star = C, AXB(star) +/- (CXD star)-D-star = E and AX +/- X(star)A(star) = C. Solvability conditions and numerical methods are considered, in terms of the (generalized and periodic) Schur and QR decompositions. We emphasize the square cases where m = n but the rectangular cases will be considered. (C) 2012 Elsevier Inc. All rights reserved.
The palindromic generalized eigenvalue problem A*x = lambda Ax: Numerical solution and applications
(2011-06-01)
Li, Tiexiang
;
Chiang, Chun-Yueh
;
Chu, Eric King-wah
;
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.