  We present two minimum residual methods for solving sequences of shifted
linear systems, the right-preconditioned shifted GMRES and shifted recycled
GMRES algorithms which use a seed projection strategy often employed to solve
multiple related problems. These methods are compatible with general
preconditioning of all systems, and when restricted to right preconditioning,
require no extra applications of the operator or preconditioner. These seed
projection methods perform a minimum residual iteration for the base system
while improving the approximations for the shifted systems at little additional
cost. The iteration continues until the base system approximation is of
satisfactory quality. The method is then recursively called for the remaining
unconverged systems. We present both methods inside of a general framework
which allows these techniques to be extended to the setting of flexible
preconditioning and inexact Krylov methods. We present some analysis of such
methods and numerical experiments demonstrating the effectiveness of the
algorithms we have derived.
