Solving linear and quadratic matrix equations on distributed memory parallel computers

We discuss the parallel implementation of numerical solution methods for linear and quadratic matrix equations occurring frequently in the control theory. In particular we consider equations related to analysis and synthesis of continuous-time, linear time-invariant control systems. These are the Sylvester equation, the Lyapunov equation, and the continuous-time algebraic Riccati equation. We assume the coefficient matrices to be dense and the state-space dimension to be roughly of order 10³-10⁴. For such problem classes, methods based on the sign function and related methods prove to be very efficient and usually outperform methods based on the QR or QZ algorithms even in sequential computing environments. We discuss the implementation of these methods on distributed memory parallel computers employing MPI and ScaLAPACK