The two-sided Lanczos method is popular for computing a few selected eigentriplets of large non-Hermitian matrices. However, it has been revealed that the Ritz vectors gained by this method may not converge even if the subspaces are good enough and the associated eigenvalues converge. In order to remedy this drawback, a novel method is proposed which is based on the refined strategy, the quasi-refined idea and the Lanczos biothogonalization procedure, the resulting algorithm is presented. The relationship between the new method and the classical oblique projection technique is also established. We report some numerical experiments and compare the new algorithm with the conventional one, the results show that the former is often more powerful than the latter.
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