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In this paper we discuss the results of the parallel implementation of a new scheme for the solution of large and dense linear systems: an implicit version of the well known Gauss-Jordan method. We present a complete computational analysis of the method and we explain the superiority of implicit schemes on the basis of the number of accesses to the problem data.A new pivoting strategy for the implicit methods is presented. We prove that this strategy gives better results than the classical partial pivoting used with Gaussian Elimination.