Minimal residual methods for large scale Lyapunov equations

Projection methods have emerged as competitive techniques for solving large scale matrix Lyapunov equations. We explore the numerical solution of this class of linear matrix equations when a Minimal Residual (MR) condition is used during the projection step. We derive both a new direct method, and a preconditioned operator-oriented iterative solver based on CGLS, for solving the projected reduced least squares problem. Numerical experiments with benchmark problems show the effectiveness of an MR approach over a Galerkin procedure using the same approximation space.