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.

翻译：文献[3]指出，四种看似不同的算法在相同参数下计算大规模连续时间代数Riccati方程(CAREs) $0 = \mathcal{R}(X) := A^HX+XA+C^HC-XBB^HX $ 的低秩近似解 $X_j$ 时，会产生相同的序列 $X_j$。这些Hermite低秩近似 $X_j$ 具有形式 $X_j = Z_jY_jZ_j^H$，其中 $Z_j$ 为列数较少的矩阵，$Y_j$ 为小型方Hermite矩阵。每个 $X_j$ 生成低秩Riccati残差 $\mathcal{R}(X_j)$，可轻松评估残差范数以实现高效终止准则。本文提出了一类生成此类CARE低秩近似解 $X_j$ 的新方法族。该族算法中每个成员生成的 $X_j$ 序列与四种已知算法完全一致。该方法基于块有理Arnoldi分解，以及由 $A^H$ 和 $C^H$ 张成的关联块有理Krylov子空间。本文将考虑通用算法的两个具体版本：其中一个版本实为RADI算法的重新发现，另一个版本则允许比RADI算法略高效地实现（在使用Sherman-Morrision-Woodbury公式及直接求解器处理线性系统时）。此外，本方法允许一次添加多个位移。