三维问题的局部块分解预条件

利用块三对角矩阵的嵌套局部块分解构造了一个不完全分解预条件子,并考虑了其修正型变种,分析了两者的存在性及若干性质.针对标准七点差分矩阵,给出了预条件后的实际条件数.结果表明,采用局部块分解预条件时条件数与矩阵阶数的2/3次幂成正比,而采用修正型预条件时条件数与矩阵阶数的立方根成正比.最后考虑了预条件的高效实现并在主频为550MHz、内存为256M的微机上作了若干数值实验,并与其它较有效的预条件方法进行了比较.

Preconditioners to Three-dimensional Problems Based on Local Block Decomposition

A preconditioner of incomplete factorization types and a modified version are provided for the 3-D problems with the help of local block decomposition of a block tridiagonal matrix recursively. Then the existence of both preconditioners is focused on. For the seven point matrix discreted from the 3-D Laplace operator with the seven point difference technique, the actual condition numbers are computed and the results show that for the non-modified preconditioner, the condition number is proportional to N^2／3, where N is the order of the matrix. For the modified version, the condition number is proportional to the cube root of the order of the matrix thereafter. Finally, highly efficient implementation is considered and several effective experiments are done for these preconditioners on personal computers with main frequency of 550MHz and memory of 256M.