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 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 equivalent to the RADI algorithm, the other one allows for a slightly more efficient implementation compared to the RADI algorithm. Moreover, our approach allows for adding more than one shift at a time.

翻译：文献[3]表明，用于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 $X_j$ 的四种看似不同的算法，在使用相同参数时生成相同的序列 $X_j$。埃尔米特低秩近似解 $X_j$ 的形式为 $X_j = Z_jY_jZ_j^H,$ 其中 $Z_j$ 是仅有少量列的矩阵，$Y_j$ 是小型方阵埃尔米特矩阵。每个 $X_j$ 生成低秩Riccati残差 $\mathcal{R}(X_j)$，使得残差范数易于评估，从而实现高效的终止准则。本文提出了一类生成此类CARE低秩近似解 $X_j$ 的新型方法族。该算法族的每个成员都能生成与前述四种算法相同的 $X_j$ 序列。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。将考虑该通用算法的两种具体版本：一种被证明等价于RADI算法，另一种则允许比RADI算法略高效的实现。此外，我们的方法允许一次添加多个移位。