标题: 非线性矩阵方程在奈米研究上的应用
Nonlinear Matrix Equations and Its Applications in Nano Research
作者: 林文伟 
国立交通大学应用数学系(所) 
关键字: 非线性矩阵方程;Green函数;复对称弱稳定解;固定点迭代;保结构演算法 ;nonlinear matrix equation;Green’s function;complex symmetric weakly stabilizing solution;fixed-point iteration;structure-preserving algorithm 
公开日期: 2012
摘要: 在奈米材料的模拟研究中,非平衡态Green函数的计算手法是一重要的课题,而在 主要散色区的Green函数求解则归结到求解如下特殊形态的非线性矩阵方程 X + ATX-1 A = Q,其中A是实方阵,Q是实对称,且在一般参数下是对称不定的矩阵。 在5函数卷积的引导下,我们有兴趣是找到一类参数(有效的能源)使得上述非线性矩 阵方程存在复对称的弱稳定解足。即其弱稳定解有不为零的虚部。此弱稳定解是扰动 方程X + ATX-1A = Q + iVI对称值稳定解Xv的极限值。在[Guo/Lin, SISC, 2010] —文提出 一套可以有效地解出X。的乘幂法,此处的"〉0且非常小。如此提供了 Z*很好的近似 解,经由奈米领域中许多科学家的观察,当0此近似解X。是存在的。在本计划中, 我们将给出一个严格的数学分析方法来证明X。的存在性,我们也将证明Z*的虚部毛是 半正定的,且确定它的秩的大小取决于二次矩阵多项式A2AT -AQ + A在单位圆上的共轭 特征对的个数。在数值方法上,基于+ S-1)—变换的技巧,我们将发展一套保结构的 _ A O ]“O I Q -1 _, _ AT O 汉米尔顿矩阵对,且有下列之形式(K, N)三(MJLt J + LJMtJ, LJLt J)。如此一来,再利 用保结构的双保距正交变换求解(,N)的特征值对其对应之特征向量,进而求解(M,L) 所相应的稳定不变子空间所对应的一组基底。从而可以有结构性地求解非线性矩阵方程。 
In the simulation of nano research, the non-equilibrium Green’s function calculation provides a powerful conceptual and computational framework for quantum transport in nanodevices. The main task of Green’s function calculation can be focused on a nonlinear matrix equation of the form X + ATX—1A = Q corresponding to the scattering region, where A is a real square matrix and Q is a real symmetric matrix dependent on a parameter and is usually indefinite. In particular one is mainly interested in those values of the parameter (efficient energy) 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” of the perturbed equation X + ATX—1A = Q + i”I as ” ^ 0+ . It has been shown that a doubling algorithm [Guo/Lin,SISC,2010] can be used to compute X” sufficiently even for very small values of ”,thus providing a good approximation X*. It has been observed by nano scientists that X” exists for ”〉0 sufficiently small. We will provide a rigorous analysis to show the unique existence of X”, for ”〉0. We also show that the imaginary part Xi of the matrix X* is positive semidefinite and determine the rank of XI in terms of the number of eigenvalues on T of the quadratic pencil A2AT — AQ + A . Based on the (S + S—1) -transform, we will develop a structure algorithm that is applied directly to the symplectic pair O AT A Q O —I (M, L): and transform it to a skew-Hamiltonian pair of the form O O —I . We use the structured bi-isotropic O (K, N) EE (MJLT J + LJMtJ, LJLt J) ,where J - orthogonal transformations to compute all desired eigenpairs of (K, N), and then compute the associated basis for the weakly stable invariant subspace of (M, L). In doing so, we work real arithmetic most of the time and solve the nonlinear matrix equation structurally. 
官方说明文件#: NSC100-2115-M009-010-MY3 
URI: https://www.grb.gov.tw/search/planDetail?id=2377518&docId=376525
http://hdl.handle.net/11536/132095
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