Deflating Invariant Subspaces for Rank Structured Pencils

It is known that executing a perfect shifted QR step via the implicit
QR algorithm may not result in a deflation of the perfect shift.
Typically, several steps are required before deflation actually takes place.
This deficiency can be remedied by determining the similarity transformation
via the associated eigenvector. Similar techniques have been
deduced for the QZ algorithm and for the rational QZ algorithm. In this
paper we present a similar approach for executing a perfect shifted
QZ step on a general rank structured pencil instead of a specific rank
structured one, e.g., a Hessenberg--Hessenberg pencil.
For this, we rely on the rank structures present in the transformed matrices. A
theoretical framework is presented for dealing with general rank structured
\rev{pencils} and deflating subspaces. We present the corresponding algorithm allowing} to deflate simultaneously a block
of eigenvalues rather than a single one.
We define the level-rho poles and show that these poles are maintained executing the deflating algorithm.
Numerical experiments illustrate
the robustness of the presented approach showing the importance of using the improved
scaled residual approach.