  We study the use of Krylov subspace recycling for the solution of a sequence
of slowly-changing families of linear systems, where each family consists of
shifted linear systems that differ in the coefficient matrix only by multiples
of the identity. Our aim is to explore the simultaneous solution of each family
of shifted systems within the framework of subspace recycling, using one
augmented subspace to extract candidate solutions for all the shifted systems.
The ideal method would use the same augmented subspace for all systems and have
fixed storage requirements, independent of the number of shifted systems per
family. We show that a method satisfying both requirements cannot exist in this
framework.
  As an alternative, we introduce two schemes. One constructs a separate
deflation space for each shifted system but solves each family of shifted
systems simultaneously. The other builds only one recycled subspace and
constructs approximate corrections to the solutions of the shifted systems at
each cycle of the iterative linear solver while only minimizing the base system
residual. At convergence of the base system solution, we apply the method
recursively to the remaining unconverged systems. We present numerical examples
involving systems arising in lattice quantum chromodynamics.
