Preconditioned Krylov Subspace Methods for Lyapunov Matrix Equations

Abstract

The authors study the iterative solution of Lyapunov matrix equations \[ AX + XA^T = - D^T D \] by reconditioned Krylov subspace methods. These solution techniques are of interest for problems leading to large and sparse matrices A as those arising from certain applications in large space structure control theory. We show how conjugate gradient (CG)-type methods for nonsymmetric linear systems can be applied to this type of equation utilizing the special structure when computing matrix-vector and inner products. In contrast to recently developed methods for such matrix equations based on Krylov subspaces associated with A, the authors implicitly work with the equivalent system of linear equations involving the Kronecker sum $M = A \otimes I_N + I_N \otimes A$. Motivation for this new approach comes from the observation that it allows the straightforward incorporation of preconditioners. Several preconditioners for such problems are presented and analyzed. In particular, since the solution matrix X is known to be symmetric, it is of interest to know which of the methods produces symmetric iterates in each step. It is proven that this is the case for alternating direction implicit (ADI)-type and (point) symmetric successive overrelaxation (SSOR) preconditioning in association with the quasiminimal residual (QMR) method. As numerical results show, it is essential to use reconditioning in association with Krylov subspace methods. Several preconditioners for such problems are presented and analyzed. Finally numerical examples are presented where the different preconditioners are compared.