### 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.