On a nonlinear matrix equation arising in nano research

Abstract

The matrix equation $X+A^{T}X^{-1}A=Q$ arises in Green's function calculations in nano research, where $A$ is a real square matrix and $Q$ is a real symmetric matrix dependent on a parameter and is usually indefinite. In practice one is mainly interested in those values of the parameter 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 $X_{\eta}$ of the perturbed equation $X+A^{T}X^{-1}A=Q+i\eta I$, as $\eta\to 0^+$. It has been shown that a doubling algorithm can be used to compute $X_{\eta}$ efficiently even for very small values of $\eta$, thus providing good approximations to $X_*$. It has been observed by nano scientists that a modified fixed-point method can sometimes be quite useful, particularly for computing $X_{\eta}$ for many different values of the parameter. We provide a rigorous analysis of this modified fixed-point method and its variant, and of their generalizations. We also show that the imaginary part $X_I$ of the matrix $X_*$ is positive semi-definite and determine the rank of $X_I$ in terms of the number of unimodular eigenvalues of the quadratic pencil $\lambda^2 A^{T}-\lambda Q+A$. Finally we present a new structure-preserving algorithm that is applied directly on the equation $X+A^{T}X^{-1}A=Q$. In doing so, we work with real arithmetic most of the time.