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 respect to performance on different examples.
A deflation strategy is also proposed to overcome possible breakdown in the
recurrence.
A new implicitly-restarted Krylov subspace method
for real symmetric/skew-symmetric generalized eigenvalue problems
is presented. The new method improves and generalizes the SHIRA method
to the case where the skew symmetric matrix is singular.
It computes a few eigenvalues and eigenvectors of the matrix pencil
close to a given target point. Several applications from control theory are
presented and the properties of the new method are illustrated by benchmark
examples.