  Many Krylov subspace methods for shifted linear systems take advantage of the
invariance of the Krylov subspace under a shift of the matrix. However,
exploiting this fact in the non-Hermitian case introduces restrictions; e.g.,
initial residuals must be collinear and this collinearity must be maintained at
restart. Thus we cannot simultaneously solve shifted systems with unrelated
right-hand sides using this strategy, and all shifted residuals cannot be
simultaneously minimized over a Krylov subspace such that collinearity is
maintained. It has been shown that this renders them generally incompatible
with techniques of subspace recycling [Soodhalter et al. APNUM '14].
  This problem, however, can be overcome. By interpreting a family of shifted
systems as one Sylvester equation, we can take advantage of the known "shift
invariance" of the Krylov subspace generated by the Sylvester operator. Thus we
can simultaneously solve all systems over one block Krylov subspace using FOM
or GMRES type methods, even when they have unrelated right-hand sides. Because
residual collinearity is no longer a requirement at restart, these methods are
fully compatible with subspace recycling techniques. Furthermore, we realize
the benefits of block sparse matrix operations which arise in the context of
high-performance computing applications.
  In this paper, we discuss exploiting this Sylvester equation point of view
which has yielded methods for shifted systems which are compatible with
unrelated right-hand sides. From this, we propose a recycled GMRES method for
simultaneous solution of shifted systems.Numerical experiments demonstrate the
effectiveness of the methods.
