  The use of block Krylov subspace methods for computing the solution to a
sequence of shifted linear systems using subspace recycling was first proposed
in [Soodhalter, SISC 2016], where a recycled shifted block GMRES algorithm
(rsbGMRES) was proposed. Such methods use the equivalence of the shifted system
to a Sylvester equation and exploit the shift invariance of the block Krylov
subspace generated from the Sylvester operator. This avoids the need for
initial residuals to span the same subspace and allows for a viable restarted
Krylov subspace method with recycling for solving sequences of shifted systems.
  In this paper we propose to develop these types of methods using unprojected
Krylov subspaces. In doing so we show how one can overcome the difficulties
associated with developing methods based on projected Krylov subspaces such as
rsbGMRES, while also allowing for practical methods to fit within a well known
residual projection framework. In addition, unprojected methods are known to be
advantageous when the projector is expensive to apply, making them of
significant interest for High-Performance Computing applications. We develop an
unprojected rsbFOM and unprojected rsbGMRES. We also develop a procedure for
extracting shift dependent harmonic Ritz vectors over an augmented block Krylov
subspace for shifted systems yielding an approach for selecting a new recycling
subspace after each cycle of the algorithm. Numerical experiments demonstrate
the effectiveness of our methods.
