  We investigate the application of Krylov space methods to the solution of
shifted linear systems of the form (A+\sigma) x - b = 0 for several values of
\sigma simultaneously, using only as many matrix-vector operations as the
solution of a single system requires. We find a suitable description of the
problem, allowing us to understand known algorithms in a common framework and
developing shifted methods basing on short recurrence methods, most notably the
CG and the BiCGstab solvers. The convergence properties of these shifted
solvers are well understood and the derivation of other shifted solvers is
easily possible. The application of these methods to quark propagator
calculations in quenched QCD using Wilson and Clover fermions is discussed and
numerical examples in this framework are presented. With the shifted CG method
an optimal algorithm for staggered fermions is available.
