A modified matrix sign function method for projected Lyapunov equations

In this paper we discuss the numerical solution of projected generalized Lyapunov equations using the matrix sign function method. Such equations arise in stability analysis and control problems for descriptor systems including model reduction based on balanced truncation. It is known that the matrix sign function method applied to a matrix pencil λE−A converges if and only if λE−A is of index at most two. The convergence is quadratic if E is nonsingular, and it is linear, otherwise. We will propose a modification of the matrix sign function method that converges quadratically for pencils of arbitrary index. Numerical examples will be presented to demonstrate the properties of the modified method.