  We discuss the usage and applicability of deflation methods for the overlap
lattice Dirac operator, focussing on calculating the eigenvalues using a method
similar to the eigCG algorithm used for other Dirac operators. The overlap
operator, which contains several theoretical advantages over other formulations
of lattice Quantum Chromodynamics, is more computationally expensive because it
requires the computation of the matrix sign function. The principle change made
compared to deflation methods for other formulations of lattice QCD is that it
is necessary for best performance to tune or relax the accuracy of the matrix
sign function as the computation proceeds.
  We adapt the eigCG algorithm for two inversion algorithms for overlap
fermions, GMRESR(relCG) and GMRESR(relSUMR). Before deflation, the rate of
convergence of these routines in terms of iterations is similar, but, since the
Shifted Unitary Minimal Residual (SUMR) algorithm only requires one call to the
matrix sign function compared to the two calls required for Conjugate Gradient
(CG), SUMR is usually preferred for single inversions of the Dirac operator. We
construct bounds for the required accuracy of the matrix sign function during
the eigenvalue calculation. For the SUMR algorithm, we use a Galerkin
projection to perform the deflation; while for the CG algorithm, we are able to
use a considerably superior spectral pre-conditioner. The superior performance
of the spectral preconditioner, and its need for less accurate eigenvalues,
almost erodes SUMR's advantage over CG as an inversion algorithm.
  We see factor of three gains for the inversion algorithm from the deflation
on our small test lattices. There is, however, a significant cost in the
eigenvalue calculation because we cannot relax the accuracy of the matrix sign
function as aggressively when calculating the eigenvalues as we do while
performing the inversions.
