Iterative refinement method for the approximate factorization of a matrix inverse
Anders M. N. Niklasson · Physical Review B · 2004
An iterative refinement technique for the sparse approximate inverse factor $Z$ of a symmetric matrix $S$, where $Z{Z}^{T}\ensuremath{\approx}{S}^{\ensuremath{-}1}$ is proposed. Its efficiency is demonstrated for the approximate inverse square root and Cholesky factorization of the overlap matrix $S$ in the generalized eigenvalue problem occurring in linear scaling electronic structure theory using a localized, nonorthogonal basis set. A version of the refinement method useful for local perturbations of the overlap matrix is also proposed. In addition to its applications in electronic structure theory, the scheme may be useful for iterative refinement of preconditioners.