  In [3] it was shown that four seemingly different algorithms for computing
low-rank approximate solutions $X_j$ to the solution $X$ of large-scale
continuous-time algebraic Riccati equations (CAREs) $0 = \mathcal{R}(X) :=
A^HX+XA+C^HC-XBB^HX $ generate the same sequence $X_j$ when used with the same
parameters. The Hermitian low-rank approximations $X_j$ are of the form $X_j =
Z_jY_jZ_j^H,$ where $Z_j$ is a matrix with only few columns and $Y_j$ is a
small square Hermitian matrix. Each $X_j$ generates a low-rank Riccati residual
$\mathcal{R}(X_j)$ such that the norm of the residual can be evaluated easily
allowing for an efficient termination criterion. Here a new family of methods
to generate such low-rank approximate solutions $X_j$ of CAREs is proposed.
Each member of this family of algorithms proposed here generates the same
sequence of $X_j$ as the four previously known algorithms. The approach is
based on a block rational Arnoldi decomposition and an associated block
rational Krylov subspace spanned by $A^H$ and $C^H.$ Two specific versions of
the general algorithm will be considered; one will turn out to be a rediscovery
of the RADI algorithm, the other one allows for a slightly more efficient
implementation compared to the RADI algorithm (in case the
Sherman-Morrision-Woodbury formula and a direct solver is used to solve the
linear systems that occur). Moreover, our approach allows for adding more than
one shift at a time.
