Preconditioners Based On Multistage Splittings For Symmetric Positive Definite Linear Systems

Symmetric positive definite (s.p.d.) linear systems play an important role in scientific and engineering computation. In fact, many of those problems are usually reduced into the solutions of some large scale s.p.d. linear systems. Hence, it is an important research topic to design efficient algorithms for solving such a class of linear system in numerical computational field.As we know, direct methods and iterative methods are the main approaches for solving the linear systems. Direct methods are often used to solve linear systems of lower orders. When the orders of the coefficient matrixes are larger, the iterative method is the prime choice. However, the convergence rate of an iterative method for solving a large scale linear system depends on the condition number of the coefficient matrix. Therefore, how to constructing reasonable preconditioners is very important and necessary.In this paper, we studied preconditioners based on multistage splittings for (block) s.p.d. linear systems. Firstly, we construct the multistage splittings of an s.p.d. matricx and deduced the final iterative formula A=MTp-NTpwith the fixed number of nest iterations Then, we letMTP-1as the preconditioner of the conjugate gradient method and prove its validity. For block s.p.d. linear systems, we give the preconditioner based on the block multistage splittings by a similar way. In the numerical experiments, we apply the multistage splitting iterative method, conjugate gradient method, circulant precondintioning conjugate gradient method and multistage splitting preconditioning conjugate gradient method to solve the same system. By the contrast, we find our preconditioners are very effective.This paper is comprised of six chapters which are arranged as follows:The first chapter is an introduction regarding the research background of multistage splitting preconditioning, research situation and the innovation of this paper. The main work was also referred.The preliminaries such as (block) multistage splittings iterations and (preconditioning) conjugate gradient method are proposed in the second chapter. Also, some important theorems, lemmas and corollaries are introduced.In the third chapter, the preconditioner induced by the multistage splittings of the coefficient matrix is proposed for solving the s.p.d. linear systems. Also, the validity of this preconditioner is figured out. We extend the technique of constructing the preconditioners for s.p.d. matrices to the case of block s.p.d. matrices and prove the validity of the corresponding preconditioners in the chapter four.In chapter five, some numerical experiments were given. By applying the multistage splitting iterative method, conjugate gradient method, circulant precondintioning conjugate gradient method and multistage splitting preconditioning conjugate gradient method to solve the same system, It is shown that our preconditioners are very effective.Finally, we summarized the main achieves of this dissertation and discussed the possible research in future.