Local factorization of block tridiagonal matrices and its application to preconditioners

Preconditioners based on incomplete decomposition of the coefficient matrix have been proved to be efficient. But this kind of preconditioner is hard to be parallelized. In this paper, authors consider block tridiagonal matrices. For this kind of matrices, their factorizations have local dependence in some sense, which is analyzed with the help of the evaluation of the actual factors. Then this kind of localization is exploited to construct a type of preconditioners. To analyze this kind of preconditioner, we first give a theorem about the condition number of a preconditioned symmetric M matrix, and then the condition numbers of the preconditioned model matrix are evaluated. With the comparison of these evaluations to the actual ones that are computed with the help of MATLAB, we can conclude that the evaluation is very accurate. Further the condition numbers are small, which means that the constructed preconditioners are effective. To study the efficiency of the preconditioner further, six implementations of the preconditioner are given in this paper. At the same time, there also considers an efficient parallelization, which has the advantage of low communication requirements. Then lots of experiments are done on a cluster of 4 PCs connected with Fast Ethernet to test the provided algorithms. The results show that the preconditioners constructed in this paper are comparable to the known effective ones in serial implementation, but they are more appropriate in parallel computing.