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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Browsing Chun-Hua Guo by Author "Kuo, Yueh-Cheng"

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    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.
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    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 $X_{\eta}$.
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    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.