The matrix equation $X+A^TX^{-1}A=Q$ and its application in nano research

Abstract

The matrix equation $X+A^TX^{-1}A=Q$ has been studied extensively when $A$ and $Q$ are real square matrices and $Q$ is symmetric positive definite. The equation has positive definite solutions under suitable conditions, and in that case the solution of interest is the maximal positive definite solution. The same matrix equation plays an important role in Green's function calculations in nano research, but the matrix $Q$ there is usually indefinite (so the matrix equation has no positive definite solutions) and one is interested in the case where the matrix equation has no positive definite solutions even when $Q$ is positive definite. The solution of interest in this nano application is a special weakly stabilizing complex symmetric solution. In this paper we show how a doubling algorithm can be used to find good approximations to the desired solution efficiently and reliably.