On a family of low-rank algorithms for large-scale algebraic Riccati equations

In [3] it was shown that four seemingly different algorithms for computing low-rank approximate solutions $Xj$ to the solution $X$ of large-scale continuous-time algebraic Riccati equations (CAREs) $0 = \mathcal{R}(X) := A^{HX+XA+CHC-XBBHX} $ generate the same sequence $Xj$ when used with the same parameters. The Hermitian low-rank approximations $Xj$ are of the form $Xj = ZjYjZj^H,$ where $Zj$ is a matrix with only few columns and $Yj$ is a small square Hermitian matrix. Each $Xj$ generates a low-rank Riccati residual $\mathcal{R}(Xj)$ 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 $Xj$ 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.