A class of (block) rational Krylov subspace based projection method for solving large-scale continuous-time algebraic Riccati equation (CARE) $0 = \mathcal{R}(X) := A^HX + XA + C^HC - XBB^HX$ with a large, sparse $A$ and $B$ and $C$ of full low rank is proposed. The CARE is projected onto a block rational Krylov subspace $\mathcal{K}_j$ spanned by blocks of the form $(A^H+ s_kI)C^H$ for some shifts $s_k, k = 1, \ldots, j.$ The considered projections do not need to be orthogonal and are built from the matrices appearing in the block rational Arnoldi decomposition associated to $\mathcal{K}_j.$ The resulting projected Riccati equation is solved for the small square Hermitian $Y_j.$ Then the Hermitian low-rank approximation $X_j = Z_jY_jZ_j^H$ to $X$ is set up where the columns of $Z_j$ span $\mathcal{K}_j.$ The residual norm $\|R(X_j )\|_F$ can be computed efficiently via the norm of a readily available $2p \times 2p$ matrix. We suggest to reduce the rank of the approximate solution $X_j$ even further by truncating small eigenvalues from $X_j.$ This truncated approximate solution can be interpreted as the solution of the Riccati residual projected to a subspace of $\mathcal{K}_j.$ This gives us a way to efficiently evaluate the norm of the resulting residual. Numerical examples are presented.

翻译：本文提出一类基于(分块)有理Krylov子空间的投影方法，用于求解大规模连续时间代数Riccati方程(CARE) $0 = \mathcal{R}(X) := A^HX + XA + C^HC - XBB^HX$，其中$A$为大规模稀疏矩阵，$B$和$C$为满低秩矩阵。该CARE被投影到由形如$(A^H+ s_kI)C^H$的块（其中$s_k, k = 1, \ldots, j$为位移参数）张成的分块有理Krylov子空间$\mathcal{K}_j$上。所考虑的投影无需正交，而是基于与$\mathcal{K}_j$相关联的分块有理Arnoldi分解中出现的矩阵构建。求解得到的投影Riccati方程，可得小型Hermite方阵$Y_j$，进而建立低秩Hermite近似解$X_j = Z_jY_jZ_j^H$（其中$Z_j$的列张成$\mathcal{K}_j$）以逼近$X$。残差范数$\|R(X_j )\|_F$可通过一个现成的$2p \times 2p$矩阵的范数高效计算。我们建议通过截断$X_j$中的小特征值进一步降低近似解$X_j$的秩。该截断近似解可解释为将Riccati残差投影至$\mathcal{K}_j子空间后的解，从而提供一种高效评估最终残差范数的途径。文中给出了数值算例。