The Study On Large Linear Systems Incomplete Factorizations Preconditioners

As we all know, solving large linear equations has been an important researchtopic in numerical linear algebra. We often encounter large-scale sparse matrices inmany engineering problems. Therefore, the efficiency on solving large sparse equationshas become particularly important. Prior to solution of large spares linear equations, weoften preprocess the equation coefficient matrix firstly. The pretreatment method of theincomplete decomposition preconditioning has already received wide attention. Notonly can it keep the sparse coefficient matrix sparsely, but it is able to reduce therequirement of computation and storage a lot. This paper mainly studies the solution ofsparse linear equations and particularly constructs an effective incompletedecomposition preconditioned.Firstly, the paper introduces the iterative method including classic iterative method,modern iterative method and the definitions of some special matrices. Then, it brieflyintroduces the pretreatment technologies such as the incomplete decompositionpreconditioning and sparse approximate inverse preconditioning. Furthermore, theincomplete decomposition preconditioning method MILUT(p,τ)is improved to get anew algorithm MILUT(p,τ)It can be gotten from the numerical experimentation thatthe new algorithm for sparse matrix decomposition achieves a good result. Finally, thepaper introduces several common IC decomposition algorithms and analyses thestability of two sparse patterns, the first case of which is the static spares patterndetermined prior to the process of factorization, while the second is the dynamic sparespattern which is unpredictable during the factorization.For general sparse matrix, whether symmetrical or not, the preconditioned isusually constructed by decomposing the matrix. We introduce a parameter to control thenumber of the decomposition matrix non-zero elements in each line in order to ensurethat the sparse nature of matrices is not destroyed after decomposition. In this paper,based on MILUT(p,τ)algorithm, parameters are introduced to control the number ofthe decomposition matrix non-zero elements for two times. MMILUT(p,τ)algorithmcan be gotten from the improved MILUT(p,τ)algorithm, and the result of its numerical experiment is good.