  A class of (block) rational Krylov subspace based projection method for
solving large-scale continuous-time algebraic Riccati equation (CARE) $0 =
\mathcal{R}(X) := A^HX + XA + C^HC - XBB^HX$ with a large, sparse $A$ and $B$
and $C$ of full low rank is proposed. The CARE is projected onto a block
rational Krylov subspace $\mathcal{K}_j$ spanned by blocks of the form $(A^H+
s_kI)C^H$ for some shifts $s_k, k = 1, \ldots, j.$ The considered projections
do not need to be orthogonal and are built from the matrices appearing in the
block rational Arnoldi decomposition associated to $\mathcal{K}_j.$ The
resulting projected Riccati equation is solved for the small square Hermitian
$Y_j.$ Then the Hermitian low-rank approximation $X_j = Z_jY_jZ_j^H$ to $X$ is
set up where the columns of $Z_j$ span $\mathcal{K}_j.$ The residual norm
$\|R(X_j )\|_F$ can be computed efficiently via the norm of a readily available
$2p \times 2p$ matrix. We suggest to reduce the rank of the approximate
solution $X_j$ even further by truncating small eigenvalues from $X_j.$ This
truncated approximate solution can be interpreted as the solution of the
Riccati residual projected to a subspace of $\mathcal{K}_j.$ This gives us a
way to efficiently evaluate the norm of the resulting residual. Numerical
examples are presented.
