  For Hermitian positive definite linear systems and eigenvalue problems, the
eigCG algorithm is a memory efficient algorithm that solves the linear system
and simultaneously computes some of its eigenvalues. The algorithm is based on
the Conjugate-Gradient (CG) algorithm, however, it uses only a window of the
vectors generated by the CG algorithm to compute approximate eigenvalues. The
number and accuracy of the eigenvectors can be increased by solving more
right-hand sides. For Hermitian systems with multiple right-hand sides, the
computed eigenvectors can be used to speed up the solution of subsequent
systems. The algorithm was tested on Lattice QCD problems by solving the normal
equations and was shown to give large speed up factors and to remove the
critical slowing down as we approach light quark masses. Here, an extension to
the non-symmetric case based on the two-sided Lanczos algorithm is given. The
new algorithm is tested on Lattice QCD problems and is shown to give promising
results. We also study the removal of the critical slowing down and compare
results with those of the eigCG algorithm. We also discuss the case when the
system is gamma5-Hermitian.
