Numerical solution of large scale Lyapunov equations using Krylov subspace methods

The authors consider several methods for calculating low rank approximate solutions to large scale Lyapunov equations of the form AP+PA´+BB´=0. The interest in this problem stems from model reduction where the task is to approximate high-dimensional models by ones of lower order. The two recently developed Krylov subspace methods exploited are the Arnoldi method and the generalized minimum residual method (GMRES). Exact expressions for the approximation errors incurred are derived in both cases. The numerical solution of the low-dimensional linear matrix equation arising from the GMRES method is discussed, and an algorithm for its solution is proposed. Problems are considered in which B has more than one column with the use of block Krylov schemes