Krylov Subspace Methods for Solving Large Lyapunov Equations

Krylov subspace methods for projected Lyapunov equations

We consider the numerical solution of projected Lyapunov equations using Krylov subspace iterative methods. Such equations play a fundamental role in balanced truncation model reduction of descriptor systems. We present generalizations of the extended block and global Arnoldi methods to projected Lyapunov equations and compare these methods with the alternating direction implicit method with re...

Krylov Subspace Type Methods for Solving Projected Generalized Continuous-Time Lyapunov Equations

In this paper we consider the numerical solution of projected generalized continuous-time Lyapunov equations with low-rank right-hand sides. The interest in this problem stems from stability analysis and control problems for descriptor systems including model reduction based on balanced truncation. Two projection methods are proposed for calculating low-rank approximate solutions. One is based ...

Krylov Subspace Methods for Large-Scale Constrained Sylvester Equations

We consider the numerical approximation to the solution of the matrix equation A1X+XA2 −Y C = 0 in the unknown matrices X, Y , under the constraint XB = 0, with A1, A2 of large dimensions. We propose a new formulation of the problem that entails the numerical solution of an unconstrained Sylvester equation. The spectral properties of the resulting coefficient matrices call for appropriately des...

Krylov subspace methods for solving linear systems

Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations

We consider linear matrix equations where the linear mapping is the sum of a standard Lyapunov operator and a positive operator. These equations play a role in the context of stochastic or bilinear control systems. To solve them efficiently one can fall back on known efficient methods developed for standard Lyapunov equations. In the present paper we describe a direct and an iterative method ba...