The direct-projection methods

In this paper, the direct-projection method given by Benzi and Meyer (1995) is derived by slightly different way - Basic Solutions of corresponding ho mogeneous system. The idea of the method differs from the idea of the Gauss ian Elimination (or LU decomposition). The method works for every nonsingul ar coefficient matrix in the absence of rounding error. The Gaussian Elimin ation is explained by the method. The corresponding numerical algorithms of the method are also given, The method is applied to find the preconditione r for least squares problems and solvers of singular systems. The numerical experiments illustrate that the method has better numerical stability than the Gaussian Elimination.