It is well known that surface Green’s function g(ω) arising from simulation of quantum transport is characterized by the nonlinear matrix equation X + AᵀX⁻¹A = Q,in which both the matrices A, Q depend on ω, under non-orthogonal basis set. Starting from this matrix equation, we will investigate the asympotics of g(ω) = X⁻¹ and the self-energy Σ(ω) = Aᵀ(ω)g(ω)A(ω), as ω → ∞. In this talk, we will see some

unexpected and interesting connections between this topic and classical linear/nonlinear matrix, e.g. discrete-time Lyapunov equation, discrete-time algebraic Riccati equation and others. Then, based on these results, the algorithm to calculate g(ω) + g(−ω), Σ(ω) + Σ(−ω) with ω → ∞ and g(ω + ε) − g(ω − ε) with |ε| ≪ ω, free from catastrophic cancellation will be proposed. Finally, some open questions and challenges will be discussed.