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The conjugate gradient method has been suggested as a better alternative to direct methods for the solution of certain large sparse linear systems A x = b, where A is symmetric and positive de nite. E ciency considerations often require that the conjugate gradient method be accelerated by preconditioning a linear transformation of A. One of the most widely used preconditioners is based on the incomplete LU factors of A. P ositive de nite preconditioner matrices assure convergence. However, thedoi:10.1109/59.630475 fatcat:3r4gr33rp5dhhiazxxjjfvvbby