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For Hermitian positive definite linear systems and eigenvalue problems, theeigCG algorithm is a memory efficient algorithm that solves the linear systemand simultaneously computes some of its eigenvalues. The algorithm is based onthe Conjugate-Gradient (CG) algorithm, however, it uses only a window of thevectors generated by the CG algorithm to compute approximate eigenvalues. Thenumber and accuracy of the eigenvectors can be increased by solving moreright-hand sides. For Hermitian systems with multiple right-hand sides, thecomputed eigenvectors can be used to speed up the solution of subsequentsystems. The algorithm was tested on Lattice QCD problems by solving the normalequations and was shown to give large speed up factors and to remove thecritical slowing down as we approach light quark masses. Here, an extension tothe non-symmetric case based on the two-sided Lanczos algorithm is given. Thenew algorithm is tested on Lattice QCD problems and is shown to give promisingresults. We also study the removal of the critical slowing down and compareresults with those of the eigCG algorithm. We also discuss the case when thesystem is gamma5-Hermitian.

Extending the eigCG algorithm to non-symmetric linear systems with multiple right-hand sides