||Incomplete factorization preconditioning is one of the most efficient and rebust preconditioning techniques in the conjugate gradient method for solving systems of linear equations. To improve the performance of the incomplete factorization preconditioners for difficult problems, extra fill-ins are often used. In this thesis, we study the effect of using different ordering schemes and more accurate incomplete factorization preconditioners in the solution process. It is found that the incomplete factorization preconditioners that drop fill-in by its magnitude are often better than preconditioners that drop fill-in by the size of their residual. Furthermore, the relative preformance of different orderings may change as the accuracy of the preconditioners improves.