A rational Krylov subspace based projection method for solving large-scale algebraic Riccati equations via low-rank approximations

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 $(A^H+ skI)C^H$ for some shifts $sk, k = 1, \ldots, j.$ 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 $Yj.$ Then the Hermitian low-rank approximation $Xj = ZjYjZj^H$ to $X$ is set up where the columns of $Zj$ span $\mathcal{K}j.$ The residual norm $|R(Xj )|F$ 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 $Xj$ even further by truncating small eigenvalues from $Xj.$ 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.