A CLASS OF PETROV–GALERKIN KRYLOV METHODS FOR ALGEBRAIC RICCATI EQUATIONS∗

A class of (block) rational Krylov-subspace-based projection methods for solving the large-scale continuous-time algebraic Riccati equation (CARE) 0 = R(X):= AHX + XA + CHC − XBBHX 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 Kj spanned by blocks of the form (AH − skI)−1CH for some shifts sk, k = 1, . . ., 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 Kj. The resulting projected Riccati equation is solved for the small square Hermitian Yj. Then the Hermitian low-rank approximation Xj = ZjYjZjH to X is set up where the columns of Zj span Kj. The residual norm kR(Xj)kF can be computed efficiently via the norm of a readily available 2p × 2p matrix. We suggest reducing 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 Kj. This gives us a way to efficiently evaluate the norm of the resulting residual. Numerical examples are presented. © 2024, Kent State University.