A unification of unitary similarity transforms to compressed representations

In this paper a new framework for transforming arbitrary matrices to
compressed representations is presented. The framework provides a
generic way of transforming a matrix via unitary similarity
transformations to e.g.\ Hessenberg, Hessenberg\-/like and combinations
of both. The new algorithms are deduced, based on the $QR$-factorization
of the original matrix. Based on manipulations with Givens
transformations, all the algorithms consists of eliminating the
correct set of Givens transformations, resulting in a matrix obeying
the desired structural constraints.

Based on this new reduction procedure we investigate further
correspondences such as irreducibility, unicity of the reduction
procedure and the link with (rational) Krylov methods.


The unitary similarity transform to Hessenberg\-/like form as presented here, differs
significantly from the one presented in earlier work. Not only does it
use less Givens transformations to obtain the desired structure, also
the convergence to rational Ritz values is not observed in the
standard way.