An implicit deflation method for ill-conditioned Sylvester and Lyapunov equations

Abstract

This paper describes an effective algorithm for solving ill-conditioned Sylvester or Lyapunov equations. These equations can be solved by conventional methods for ill-conditioned linear systems. However, such methods are not efficient since they require on the order of n 6 arithmetic operations on order n 2 data. For these ill-conditioned matrix equations, an implicit deflation algorithm is proposed to implement a certain SVD-based minimum norm least-squares approximate solution. The method is practical since it relies only on the ability to solve a Sylvester or Lyapunov equation. Certain practical details are also discussed.