Numerical solution of nonlinear matrix equations arising from Green%26apos;s function calculations in nano research

作者：Guo Chun Hua^*; Kuo Yueh Cheng; Lin Wen Wei

来源：Journal of Computational and Applied Mathematics, 2012, 236(17): 4166-4180.

DOI：10.1016/j.cam.2012.05.012

• Nonlinear matrix equation
• Weakly stabilizing solution
• Structure-preserving algorithm
• Green%26apos
• s function

The Green%26apos;s function approach for treating quantum transport in nano devices requires the solution of nonlinear matrix equations of the form X + (C* + i eta D*)X-1(C + i eta D) = R+i eta P. where R and P are Hermitian, P + lambda D* + lambda D-1 is positive definite for all lambda on the unit circle, and eta -%26gt; 0(+). For each fixed eta %26gt; 0, we show that the required solution is the unique stabilizing solution X-eta. Then X-center dot = lim(eta -%26gt; 0+) X-eta is a particular weakly stabilizing solution of the matrix equation X + C*X-1C = R. In nano applications, the matrices R and C are dependent on a parameter, which is the system energy E. In practice one is mainly interested in those values of g for which the equation X + C*X-1C = R has no stabilizing solutions or, equivalently, the quadratic matrix polynomial P(lambda) = lambda C-2* - 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 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 exploited to drastically reduce the complexity of the doubling algorithm for computing X-eta.

• 出版日期2012-11