Chun-Hua Guo

Permanent URI for this collectionhttps://hdl.handle.net/10294/5255

Office: CW 307.10
E-mail: Chun-Hua.Guo@uregina.ca
Phone: 306-585-4423
Website: http://uregina.ca/~chguo/

Current Classes
Math 103 (Calculus for Social Science & Management)

Research Interests
Matrix analysis, scientific computing, applications

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Now showing 1 - 7 of 7

Open Access
Complex symmetric stabilizing solution of the matrix equation $X+A^{T}X^{-1}A=Q$
(Elsevier, 2011) Guo, Chun-Hua; Kuo, Yueh-Cheng; Lin, Wen-Wei
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.
Open Access
Convergence Analysis of the Doubling Algorithm for Several Nonlinear Matrix Equations in the Critical Case
(SIAM, 2009) Chiang, Chun-Yueh; Chu, Eric King-Wah; Guo, Chun-Hua; Huang, Tsung-Ming; Lin, Wen-Wei; Xu, Shu-Fang
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.
Open Access
Convergence rates of some iterative methods for nonsymmetric algebraic Riccati equations arising in transport theory
(Elsevier, 2010) Guo, Chun-Hua; Lin, Wen-Wei
We determine and compare the convergence rates of various fixed-point iterations for finding the minimal positive solution of a class of nonsymmetric algebraic Riccati equations arising in transport theory.
Open Access
The matrix equation $X+A^TX^{-1}A=Q$ and its application in nano research
(SIAM, 2010) Guo, Chun-Hua; Lin, Wen-Wei
The matrix equation $X+A^TX^{-1}A=Q$ 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.
Open Access
Numerical solution of nonlinear matrix equations arising from Green's function calculations in nano research
(Elsevier, 2012) Guo, Chun-Hua; Kuo, Yueh-Cheng; Lin, Wen-Wei
The Green's function approach for treating quantum transport in nano devices requires the solution of nonlinear matrix equations of the form $X+(C^*+{\rm i} \eta D^*)X^{-1}(C+{\rm i} \eta D)=R+{\rm i}\eta P$, where $R$ and $P$ are Hermitian, $P+\lambda D^*+\lambda^{-1} D$ is positive definite for all $\lambda$ on the unit circle, and $\eta \to 0^+$. For each fixed $\eta>0$, we show that the required solution is the unique stabilizing solution $X_{\eta}$. Then $X_*=\lim_{\eta\to 0^+} X_{\eta}$ is a particular weakly stabilizing solution of the matrix equation $X+C^*X^{-1}C=R$. In nano applications, the matrices $R$ and $C$ are dependent on a parameter, which is the system energy $\mathcal E$. In practice one is mainly interested in those values of $\mathcal E$ for which the equation $X+C^*X^{-1}C=R$ has no stabilizing solutions or, equivalently, the quadratic matrix polynomial $P(\lambda)=\lambda^2 C^*-\lambda R+ C$ has eigenvalues on the unit circle. We point out that a doubling algorithm can be used to compute $X_{\eta}$ efficiently even for very small values of $\eta$, thus providing good approximations to $X_*$. We also explain how the solution $X_*$ can be computed directly using subspace methods such as the QZ algorithm by determining which unimodular eigenvalues of $P(\lambda)$ should be included in the computation. In some applications the matrices $C, D, R, P$ have very special sparsity structures. We show how these special structures can be expoited to drastically reduce the complexity of the doubling algorithm for computing $X_{\eta}$.
Open Access
On a nonlinear matrix equation arising in nano research
(SIAM, 2012) Guo, Chun-Hua; Kuo, Yueh-Cheng; Lin, Wen-Wei
The matrix equation $X+A^{T}X^{-1}A=Q$ arises in Green's function calculations in nano research, where $A$ is a real square matrix and $Q$ is a real symmetric matrix dependent on a parameter and is usually indefinite. In practice one is mainly interested in those values of the parameter for which the matrix equation has no stabilizing solutions. The solution of interest in this case is a special weakly stabilizing complex symmetric solution $X_*$, which is the limit of the unique stabilizing solution $X_{\eta}$ of the perturbed equation $X+A^{T}X^{-1}A=Q+i\eta I$, as $\eta\to 0^+$. It has been shown that a doubling algorithm can be used to compute $X_{\eta}$ efficiently even for very small values of $\eta$, thus providing good approximations to $X_*$. It has been observed by nano scientists that a modified fixed-point method can sometimes be quite useful, particularly for computing $X_{\eta}$ for many different values of the parameter. We provide a rigorous analysis of this modified fixed-point method and its variant, and of their generalizations. We also show that the imaginary part $X_I$ of the matrix $X_*$ is positive semi-definite and determine the rank of $X_I$ in terms of the number of unimodular eigenvalues of the quadratic pencil $\lambda^2 A^{T}-\lambda Q+A$. Finally we present a new structure-preserving algorithm that is applied directly on the equation $X+A^{T}X^{-1}A=Q$. In doing so, we work with real arithmetic most of the time.
Open Access
Solving a structured quadratic eigenvalue problem by a structure-preserving doubling algorithm
(SIAM, 2010) Guo, Chun-Hua; Lin, Wen-Wei
In studying the vibration of fast trains, we encounter a palindromic quadratic eigenvalue problem (QEP) $(\lambda^2 A^T + \lambda Q + A)z = 0$, where $A, Q \in \mathbb{C}^{n \times n}$ and $Q^T = Q$. Moreover, the matrix $Q$ is block tridiagonal and block Toeplitz, and the matrix $A$ has only one nonzero block in the upper-right corner. So most of the eigenvalues of the QEP are zero or infinity. In a linearization approach, one typically starts with deflating these known eigenvalues, for the sake of efficiency. However, this initial deflation process involves the inverses of two potentially ill-conditioned matrices. As a result, large error might be introduced into the data for the reduced problem. In this paper we propose using the solvent approach directly on the original QEP, without any deflation process. We apply a structure-preserving doubling algorithm to compute the stabilizing solution of the matrix equation $X+A^TX^{-1}A=Q$, whose existence is guaranteed by a result on the Wiener--Hopf factorization of rational matrix functions associated with semi-infinite block Toeplitz matrices and a generalization of Bendixson's theorem to bounded linear operators on Hilbert spaces. The doubling algorithm is shown to be well defined and quadratically convergent. The complexity of the doubling algorithm is drastically reduced by using the Sherman--Morrison--Woodbury formula and the special structures of the problem. Once the stabilizing solution is obtained, all nonzero finite eigenvalues of the QEP can be found efficiently and with the automatic reciprocal relationship, while the known eigenvalues at zero or infinity remain intact.

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