Heuristic approaches to the solution of very large sparse Lyapunov and algebraic Riccati equations

The authors present several algorithms that compute approximate solutions to the Lyapunov equation AX+XA^T+ BB^T=0 and the algebraic Riccati equation A ^TX+XA-XBR^-1 B ^TX+C^TC=0, where A is large and sparse and B and C are low rank. In particular, they test the Krylov subspace approximation and reduced rank integration for the Riccati equation. Although the algorithms are heuristically good, no convergence proofs are as yet available