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Solving large-scale nonlinear matrix equations by doubling

LINEAR ALGEBRA AND ITS APPLICATIONS 2013年第4期439卷 914-932页

作者： Weng, Peter Chang-Yi Chu, Eric King-Wah Kuo, Yueh-Cheng Lin, Wen-Wei Monash Univ Sch Math Sci Clayton Vic 3800 Australia Natl Univ Kaohsiung Dept Appl Math Kaohsiung 811 Taiwan Natl Chiao Tung Univ Dept Appl Math Hsinchu 300 Taiwan

We consider the solution of the large-scale nonlinear matrix equation X + BX-1 A - Q = 0, with A, B, Q, X is an element of C-nxn, and in some applications B = A(star) (star = T or H). The matrix Q is assumed to be nonsingular and sparse with its structure allowing the solution of the corresponding linear system Qv = r in O(n) computational complexity. Furthermore, B and A are respectively of ranks ra, rb << n. The type 2 structure-preserving doubling algorithm by Lin and Xu (2006) [241 is adapted, with the appropriate applications of the Sherman-Morrison-Woodbury formula and the lowrank updates of various iterates. Two resulting large-scale doubling algorithms have an O((r(a) + r(b))(3)) computational complexity per iteration, after some pre-processing of data in O(n) computational complexity and memory requirement, and converge quadratically. These are illustrated by the numerical examples. (C) 2012 Elsevier Inc. All rights reserved.

关键词： doubling algorithm Green's function Krylov subspace Leaky surface wave Nano research Nonlinear matrix equation Surface acoustic wave

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Low-rank approximation to the solution of a nonsymmetric algebraic Riccati equation from transport theory

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APPLIED MATHEMATICS AND COMPUTATION 2012年第2期219卷 729-740页

作者： Weng, Peter Chang-Yi Fan, Hung-Yuan Chu, Eric King-Wah Natl Taiwan Normal Univ Dept Math Taipei 116 Taiwan Monash Univ Sch Math Sci Clayton Vic 3800 Australia

We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX - XD - AX + B = 0 from transport theory (Juang 1995), with M equivalent to [D, -C;-B, A] is an element of R-2nx2n being a nonsingular M-matrix. In addition, A, D are rank-1 updates of diagonal matrices, with the products A(-1)u, A(-T)u, D-1 v and D-T v computable in O(n) complexity, for some vectors u and v, and B, C are rank 1. The structure-preserving doubling algorithm by Guo et al. (2006) is adapted, with the appropriate applications of the Sherman-Morrison-Woodbury formula and the sparse-plus-low-rank representations of various iterates. The resulting large-scale doubling algorithm has an O(n) computational complexity and memory requirement per iteration and converges essentially quadratically, as illustrated by the numerical examples. (C) 2012 Elsevier Inc. All rights reserved.

关键词： doubling algorithm Krylov subspace M-matrix Nonsymmetric algebraic Riccati equation

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Solution of a nonsymmetric algebraic Riccati equation from a one-dimensional multistate transport model

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IMA JOURNAL OF NUMERICAL ANALYSIS 2011年第4期31卷 1453-1467页

作者： Li, Tiexiang Chu, Eric King-Wah Juang, Jong Lin, Wen-Wei Southeast Univ Dept Math Nanjing 211189 Peoples R China Monash Univ Sch Math Sci Clayton Vic 3800 Australia Natl Chiao Tung Univ Dept Appl Math Hsinchu 300 Taiwan Natl Taiwan Univ Dept Math Taipei 10617 Taiwan Natl Taiwan Univ Ctr Math Modelling & Sci Comp Ctr Theoret Sci Taipei 10617 Taiwan

For the steady-state solution of a differential equation from a one-dimensional multistate model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B- -XF- -F+ X + XB+X = 0, where F-+/- = (I -F) D-+/- and B-+/- = BD +/- with positive diagonal matrices D-+/- and possibly low-ranked matrices F and B. We prove the existence of the minimal positive solution X* under a set of physically reasonable assumptions and study its numerical computation by fixed-point iteration, Newton's method and the doubling algorithm. We shall also study several special cases. For example when B and F are low ranked then X* = Gamma circle(Sigma(i=1UiViT)-U-r) with low-ranked U-i and V-i that can be computed using more efficient iterative processes. Numerical examples will be given to illustrate our theoretical results.

关键词： algebraic Riccati equation doubling algorithm fixed-point iteration Newton's method reflection transport theory

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Complex symmetric stabilizing solution of the matrix equation X plus A^TX^-1A = Q

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LINEAR ALGEBRA AND ITS APPLICATIONS 2011年第6期435卷 1187-1192页

作者： Guo, Chun-Hua Kuo, Yueh-Cheng Lin, Wen-Wei Natl Taiwan Univ Dept Math Taipei 106 Taiwan Univ Regina Dept Math & Stat Regina SK S4S 0A2 Canada Natl Univ Kaohsiung Dept Appl Math Kaohsiung 811 Taiwan Natl Chiao Tung Univ Ctr Math Modelling & Sci Comp Hsinchu 300 Taiwan

We study the matrix equation X + A(T)X(-1)A = Q, where A is a complex square matrix and Q is complex symmetric. Special cases of this equation appear in Green's function calculation in nano research and also in the vibration analysis of fast trains. In those applications, the existence of a unique complex symmetric stabilizing solution has been proved using advanced results on linear operators. The stabilizing solution is the solution of practical interest. In this paper we provide an elementary proof of the existence for the general matrix equation, under an assumption that is satisfied for the two special applications. Moreover, our new approach here reveals that the unique complex symmetric stabilizing solution has a positive definite imaginary part. The unique stabilizing solution can be computed efficiently by the doubling algorithm. (C) 2011 Elsevier Inc. All rights reserved.

关键词： Nonlinear matrix equation Complex symmetric solution Stabilizing solution doubling algorithm

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Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model

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LINEAR ALGEBRA AND ITS APPLICATIONS 2011年第1期434卷 201-214页

作者： Li, Tie-Xiang Chu, Eric King-wah Juang, Jong Lin, Wen-Wei Southeast Univ Dept Math Nanjing 211189 Peoples R China Monash Univ Sch Math Sci Clayton Vic 3800 Australia Natl Chiao Tung Univ Dept Appl Math Hsinchu 300 Taiwan Natl Chiao Tung Univ CMMSC Hsinchu 300 Taiwan Natl Chiao Tung Univ NCTS Dept Appl Math Hsinchu 300 Taiwan

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.

关键词： Algebraic Riccati equation doubling algorithm Fixed-point iteration Newton's method Reflection kernel Transport theory

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The palindromic generalized eigenvalue problem A*x = λAx: Numerical solution and applications

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LINEAR ALGEBRA AND ITS APPLICATIONS 2011年第11期434卷 2269-2284页

作者： Li, Tiexiang Chiang, Chun-Yueh Chu, Eric King-wah Lin, Wen-Wei Southeast Univ Dept Math Nanjing 211189 Peoples R China Natl Formosa Univ Ctr Gen Educ Huwei 632 Taiwan Monash Univ Sch Math Sci Clayton Vic 3800 Australia Natl Chiao Tung Univ CMMSC Hsinchu 300 Taiwan Natl Chiao Tung Univ Dept Appl Math NCTS Hsinchu 300 Taiwan

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.

关键词： Palindromic generalized eigenvalue problem doubling algorithm Singular descriptor system

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ARE-type Iterations for Rational Riccati Equations Arising in Stochastic Control

ARE-type Iterations for Rational Riccati Equations Arising i...

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23rd Chinese Control and Decision Conference

作者： Chu, Eric King-wah Li, Tiexiang Lin, Wen-Wei Monash Univ Sch Math Sci Clayton Vic 3800 Australia Southeast Univ Dept Math Nanjing 211189 Jiangsu Peoples R China Natl Taiwan Univ CTS CMMSC Taipei 10617 Taiwan Natl Taiwan Univ Dept Math Taipei 10617 Taiwan

ISBN: (纸本)9781424487363

We consider the solution of the rational matrix equations, or generalized algebraic Riccati equations with rational terms, arising in stochastic optimal control in continuous- and discrete-time. The modified Newton's methods, the DARE- and CARE-type iterations for continuous- and discrete-time rational Riccati equations respectively, will be considered. In particular, the convergence of these new modified Newton's method will be proved.

关键词： algebraic Riccati equation doubling algorithm rational Riccati equation stochastic control system

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On the doubling algorithm for a (shifted) nonsymmetric algebraic Riccati equation

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2007年第4期29卷 1083-1100页

作者： Guo, Chun-Hua Iannazzo, Bruno Meini, Beatrice Univ Regina Dept Math & Stat Regina SK S4S 0A2 Canada Univ Pisa Dipartimento Matemat I-56127 Pisa Italy

Nonsymmetric algebraic Riccati equations for which the four coefficient matrices form an irreducible M-matrix M are considered. The emphasis is on the case where M is an irreducible singular M-matrix, which arises in the study of Markov models. The doubling algorithm is considered for finding the minimal nonnegative solution, the one of practical interest. The algorithm has been recently studied by others for the case where M is a nonsingular M-matrix. A shift technique is proposed to transform the original Riccati equation into a new Riccati equation for which the four coefficient matrices form a nonsingular matrix. The convergence of the doubling algorithm is accelerated when it is applied to the shifted Riccati equation.

关键词： nonsymmetric algebraic Riccati equation minimal nonnegative solution doubling algorithm convergence acceleration shift technique

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ARE-type Iterations for Rational Riccati Equations Arising in Stochastic Control

ARE-type Iterations for Rational Riccati Equations Arising i...

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2011 China Control and Decision Conference(2011中国控制与决策会议 CCDC)

作者： Eric King-wah Chu Tiexiang Li Wen-Wei Lin School of Mathematical Sciences Building28 Monash University VIC3800 Australia Department of Mathematics Southeast University Nanjing 211189 People’s Republic of China. This w CMMSC CTS and Department of MathematicsNational Taiwan University No. 1 Sec. 4 Roosevelt Road

We consider the solution of the rational matrix equations, or generalized algebraic Riccati equations with rational terms, arising in stochastic optimal control in continuous-and discrete-time. The modified Newton's methods, the DAREand CARE-type iterations for continuous-and discrete-time rational Riccati equations respectively, will be considered. In particular, the convergence of these new modified Newton's method will be proved.

关键词： Algebraic Riccati equation doubling algorithm rational Riccati equation stochastic control system

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THE MATRIX EQUATION X + A^TX^-1 A = Q AND ITS APPLICATION IN NANO RESEARCH

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2010年第5期32卷 3020-3038页

作者： Guo, Chun-Hua Lin, Wen-Wei Univ Regina Dept Math & Stat Regina SK S4S 0A2 Canada Natl Chiao Tung Univ Dept Appl Math Hsinchu 300 Taiwan

The matrix equation X + A(T)X(-1) A = Q has been studied extensively when A and Q are real square matrices and Q is symmetric positive definite. The equation has positive definite solutions under suitable conditions, and in that case the solution of interest is the maximal positive definite solution. The same matrix equation plays an important role in Green's function calculations in nano research, but the matrix Q there is usually indefinite (so the matrix equation has no positive definite solutions), and one is interested in the case where the matrix equation has no positive definite solutions even when Q is positive definite. The solution of interest in this nano application is a special weakly stabilizing complex symmetric solution. In this paper we show how a doubling algorithm can be used to find good approximations to the desired solution efficiently and reliably.

关键词： nonlinear matrix equation complex symmetric solution stable solution fixed-point iteration doubling algorithm Newton's method Green's function

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