Decoupled low-rank iterative methods for a large-scale system of nonlinear matrix equations arising from electron transport of nano materials

Abstract

To describe the electron transport in nano materials more accurately, the single layer after the material surface has been generalized to multi-periodic layers and the computation of the corresponding Green function in this case requires to solve a system of nonlinear matrix equations (SNME). A sufficient condition of existence of the stabilizing solution pair of the SNME is first proposed and the decoupled doubling algorithm is then developed into the decoupled tripling version. By the use of the discretized low-rank structure of coefficient matrices, the decoupled low-rank doubling algorithm (DLDA) and tripling algorithm (DLTA) are presented subsequently and the evaluation of the residual is redesigned for the large-scale SNME. Analysis of iteration formats shows that the DLTA shares the same pre-processing complexity with that of the DLDA, but fortunately being able to attain a lower residual level within less iterations via additionally neglected iteration time. Numerical experiments show that the DLTA is highly efficient to compute the stabilizing solution pair of the large-scale SNME.