In this paper we improve the BIF algorithm which computes simultaneously the LU factors (direct factors) of a given matrix, and their inverses (inverse factors). This algorithm was introduced in [R. Bru, J. Marín, J. Mas and M. Tůma, SIAM J. Sci. Comput., 30 (2008), pp. 2302– 2318]. The improvements are based on a deeper understanding of the Inverse Sherman-Morrison (ISM) decomposition and they provide a new insight into the BIF decomposition. In particular, it is shown that a slight algorithmic reformulation of the basic algorithm implies that the direct and inverse factors influence numerically each other even without any dropping for incompleteness. Algorithmically, the nonsymmetric version of the improved BIF algorithm is formulated. Numerical experiments show very high robustness of the incomplete implementation of the algorithm used for preconditioning nonsymmetric linear systems
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