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]表明，四种看似不同的算法在采用相同参数计算大规模连续时间代数Riccati方程（CARE）$0 = \mathcal{R}(X) := A^HX+XA+C^HC-XBB^HX $的解$X$的低秩近似解$X_j$时，会生成相同的序列$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$为仅含少量列的矩阵，$Y_j$为小型方阵埃尔米特矩阵。每个$X_j$生成低秩Riccati残差$\mathcal{R}(X_j)$，使得残差范数可简便计算，从而提供高效的终止准则。本文提出了一类生成此类CARE低秩近似解$X_j$的新方法族，该族中的每个新方法均能生成与四种已知算法相同的$X_j$序列。该方法基于块有理Arnoldi分解及由$A^H$和$C^H$张成的关联块有理Krylov子空间。本文考虑了该通用算法的两种具体形式：其一被证实是对RADI算法的再发现，另一种相比RADI算法可提供略微更高效的实现（当使用Sherman-Morrision-Woodbury公式和直接求解器处理线性方程组时）。此外，我们的方法允许同时添加多个位移。