  The approximate solution of large-scale algebraic Riccati equations is
considered. We are interested in approximate solutions which yield a Riccati
residual matrix of a particular small rank. It is assumed that such approximate
solutions can be written in factored form $ZYZ^*$ with a rectangular matrix $Z$
and a small quadratic matrix $Y$. We propose to choose $Z$ such that its
columns span a certain rational Krylov subspace. Conditions under which such an
approximate solution exists and is unique are determined. It is shown that the
proposed method can be interpreted as an oblique projection method. Two new
iterative procedures with efficient updates of the solution and the residual
factor are derived. With our approach complex system matrices can be handled,
realification is provided and parallelization is introduced.
