非線性矩陣方程在奈米研究上的應用

Abstract

在奈米材料的模擬研究中，非平衡態Green函數的計算手法是一重要的課題，而在 主要散色區的Green函數求解則歸結到求解如下特殊形態的非線性矩陣方程 X + ATX-1 A = Q，其中A是實方陣，Q是實對稱，且在一般參數下是對稱不定的矩陣。 在5函數捲積的引導下，我們有興趣是找到一類參數（有效的能源）使得上述非線性矩 陣方程存在複對稱的弱穩定解足。即其弱穩定解有不為零的虛部。此弱穩定解是擾動 方程X + ATX-1A = Q + iVI對稱值穩定解Xv的極限值。在[Guo/Lin, SISC, 2010] —文提出 一套可以有效地解出X。的乘冪法，此處的"〉0且非常小。如此提供了 Z*很好的近似 解，經由奈米領域中許多科學家的觀察，當0此近似解X。是存在的。在本計劃中， 我們將給出一個嚴格的數學分析方法來證明X。的存在性，我們也將證明Z*的虛部毛是 半正定的，且確定它的秩的大小取決於二次矩陣多項式A2AT -AQ + A在單位圓上的共軛 特徵對的個數。在數值方法上，基於+ S-1)—變換的技巧，我們將發展一套保結構的 _ A O ]「O I Q -1 _, _ AT O 漢米爾頓矩陣對，且有下列之形式（K, N)三(MJLt J + LJMtJ, LJLt J)。如此一來，再利 用保結構的雙保距正交變換求解(,N)的特徵值對其對應之特徵向量，進而求解(M,L) 所相應的穩定不變子空間所對應的一組基底。從而可以有結構性地求解非線性矩陣方程。

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.