Large Linear Systems Of Equations Is Not Completely Break Down Pre-conditions Of Methods Of Research

For the sparse matrix A , the preconditioner which generates from complete decomposition usually does not have the same sparse as the matrix A, generally much dense. So, in order to gain the two ends that preconditioner keep sparse and have sufficient effect, incomplete decomposition approach has been proposed.The preconditioner that comes from the incomplete Cholesky factorization has been shown to be a very effective preconditioner for a wide variety of large systems of linear equations. The preconditioner L which is generated form incomplete Cholesky factorization Algorithm 3-4 in this paper retains the number of nonzero of each column between nk and nk + p. The specific number of nonzero elements which is retained in each column is defined by the optimal parameterτ. So, the memory of preconditioner is constrained. From the numerical experiment, it can conclude that it seems that the optimal parameterτis the best when it chooses the countdown of the order of magnitude of the Frobenius norm of the matrix and that the number of PCG iteration of the new algorithm in this paper is same as the number of PCG iteration of Algorithm 3-2 of Lin and Moré, but the memory is less than Algorithm 3-2.From the relation between the moment of sparse pattern S generated and the whole process of incomplete factorization, sparse pattern S can be divided into two cases. The first case is the static sparse pattern which is determined prior to the process of factorization, the second case is the dynamic sparse pattern which can not be predetermined is determined during the factorization. For the second sparse pattern, this paper proved the stability when the matrix A which is to be broken down is a symmetrical M ? matrix. At the same time, from the process of proof we can find that, at the same parameter p and the same number of PCG iteration, Algorithm 3-4 is at least as stable as the Algorithm 3-2 of Lin and Moré.