Convergence Analysis of Projection Methods for the Numerical Solution of Large Lyapunov Equations

The numerical solution of large-scale continuous-time Lyapunov matrix equations is of great importance in many application areas. Assuming that the coefficient matrix is positive definite, but not necessarily symmetric, in this paper we analyze the convergence of projection type methods for approximating the solution matrix. Under suitable hypotheses on the coefficient matrix, we provide new asymptotic estimates for the error matrix when a Galerkin method is used in a Krylov subspace. Numerical experiments confirm the good behavior of our upper bounds when linear convergence of the solver is observed. 1. The problem. We are interested in the approximate solution of the following Lyapunov matrix equation: AX +XA = BB, (1.1) with A a real matrix of large dimension and B a real tall matrix. Here A indicates the transpose of A. We assume that the n× n matrix A is either symmetric and positive definite, or nonsymmetric with positive definite symmetric part, that is, (A+A)/2 is positive definite. In the following we mostly deal with the case of B having a single column, that is B = b and we assume that b has unit Euclidean norm, that is ‖b‖ = 1. Nonetheless, our results can be extended to the multiple vector case. This problem arises in a large variety of applications, such as signal processing and system and control theory. The symmetric solution X carries important information on the stability and energy of an associated dynamical linear system and on the feasibility of order reduction techniques [2], [6], [8]. The analytic solution of (1.1) can be written as