FURTHER DEVELOPMENTS OF ICCG AND MICCG

Using the order matrix of the LU factors of A obtained from the 5-point discretization of the model self-adjoint elliptic problem, a P-order fill-in is suggested to give the incomplete LU decompositions, where P = 0, 1, 2,.... This strategy produces a series of incomplete LU factors which has the identical sparse structure with the series of ICCG(s, t) preconditioner for (s, t) = (1, 1), (1, 2), (1, 3), (2, 4); (3, 5),.... The order matrices of the corresponding error matrices are given and some of their properties are discussed. According to the modified P-order truncated elimination, a simplified factorisation procedure is presented for the P-order ICCG method to reduce the factorisation costs. A similar procedure is also presented for the P-order MICCG method. Numerical results show that the simplification does not impair the convergence rate. It is demonstrated that sometimes a stagnation may happen in the numbers of MICCG iterations. This is caused by the parameter 5 used in the method. A rational manner to select the value of 5 is presented. Meanwhile, an upper bound for the condition number of the iteration matrix of the P-order ICCG method is given. Two examples are tested to investigate the efficiency of the P-order ICCG method when P increases.