Numerical solution of nonlinear matrix equations arising from Green's function calculations in nano research
| dc.contributor.author | Guo, Chun-Hua | |
| dc.contributor.author | Kuo, Yueh-Cheng | |
| dc.contributor.author | Lin, Wen-Wei | |
| dc.date.accessioned | 2014-04-27T22:12:02Z | |
| dc.date.available | 2014-04-27T22:12:02Z | |
| dc.date.issued | 2012 | |
| dc.description.abstract | 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 $X_{\eta}$. | en_US |
| dc.description.authorstatus | Faculty | en_US |
| dc.description.peerreview | yes | en_US |
| dc.description.sponsorship | NSERC, NSC (Taiwan), NCTS (Taiwan) | en_US |
| dc.identifier.citation | J. Comput. Appl. Math. | en_US |
| dc.identifier.uri | https://hdl.handle.net/10294/5261 | |
| dc.language.iso | en | en_US |
| dc.publisher | Elsevier | en_US |
| dc.title | Numerical solution of nonlinear matrix equations arising from Green's function calculations in nano research | en_US |
| dc.type | Article | en_US |