Abstract: Using the concepts of element order and order matrix, some practical problems are discussed in which the traditional preconditioning methods ICCG and MICCG are adopted. If the fill-in number is fixed, why the method of ICCG(s,t) becomes the most efficient when (s,t) is successively (1,1), (1,2), (1,3), (2, 4), (3, 5),..? Why the number of iterations didn't decrease when m is larger than 3 for MICCG(m)? Is it possible to improve the fill-in method of MICCG? Is it always true that MICCG is better than ICCG? It tries to give a preliminary discussion on these problems in here. From the way of high order approximate LU decomposition, a method is introduced which improves and systematizes the ICCG and MICCG. An estimation of the condition number of ICCG is given based on the discussion of the order matrix for the error matrix. It is also pointed out that there was a trouble in selecting the parameter for MICCG. A reasonable way to select the parameter is given. Thus the number of iterations of MICCG decreases when the order of MICCG increases.

Key words: diagonally dominant matrix, order matrix, approximate LU decomposition, preconditioned conjugate gradient