SOR-like Methods Based On Local Symmetric And Skew-symmetric Splitting

Solving saddle point problems for large and sparse systems is applied in many fields, such as constrained optimization,least square problems,image management and so on.Iterative methods often used to solve these problems,Uzawa method and MINRES method are two efficient algorithms.Recent years,in order to solve these problems well many articles have discussed some methods and many iterative methods have been proposed based on Uzawa method,such as a generalized SOR(GSOR) method which has an inevitable relation with Young's in theory background.In 2001,Golub,Wu and Yuan proposed a SOR-like method.In[18],it gives the convergence condition of SOR-like method,optimum parameters and so on,this method has been discussed in many articles,such as[12,24,27,29,34].In this thesis,we consider the SOR-like method and define the symmetric positive definite matrix Q in splitting the matrix.Firstly,we give the local symmetric and skew-syrnmetric matrix splitting,iteration matrix and the corresponding computation.We analyze that only when the saddle point problems take a left multiplication SOR-like method converges,and we get the sufficient and necessary condition of the convergence.Then we discuss the relation between max and min of b/a and the choice of matrix H and Q,and we analyze the choice of optimum parameterωin detail.By using an elementary method the spectral radius of the iteration matrix is fully characterized,hence the explicit formulae for the optimum parameter and the associated optimum spectral radius are given,and the monotony aboutωand b/a of the spectral radius are also given.We discuss and prove the optimum parameter of SOR-like method and spectral radius of the iteration matrix.Finally,we give the numerical experiment and by choosing different Q and b/a we prove that the numerical results have the same conclusions as the theory in this paper.