Abstract: 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.

Key words: symmetric positive definite matrix, M matrix, condition number, incompIete factorization, preconditioner