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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.
A generalization of the method of Chu, Liu and Mehrmann
for the computation of the Hamiltonian real Schur form is presented.
The new method avoids some of the difficulties that may arise when
a Hamiltonian matrix has tightly clustered groups of eigenvalues.
A detailed analysis of the method is presented and several numerical examples demonstrate the superior behavior of the method.