Krylov Subspace Methods For Large Sparse Systems Of Equations

Many scientific and engineering computational problems have been changed into a large sparse linear equation Ax= b,such as difference format of partial differential equations,stiffness matrix obtained from discretion by using finite element method etc.Due to the scale of the problem are often very large,the iterative method becomes one of the most commonly used methods for solving large sparse linear equations Ax=b.The basic idea of iterative method is as follows:Starting from some approximation of the solution,we construct an infinite series to approximate the exact solutions(generally solution would not be the exact solution within several step).Compared with the direct method,iterative method can maintain the sparsity of the matrix,simple calculation,the advantages of easy programming,and in many cases,rapid convergence,thus can effectively solve large sparse equations.In this paper,we will review some famous iterative methods of Krylov subspace such as IGCG,CGS,and BICGSTAB and discuss at what kind of situation the breakdown will occur,then select several algorithms to solve large-scale sparse linear equations,obtained from discretization of partial differential equations by using the method of finite difference, and compare their iteration times and speed of convergence.Five chapters are folded in this paper.In chapter one,we simply introduce the background,motivation and purpose of Krylov subspace methods.In chapter two,we give the definition and some theorems of Krylov subspace,and review some relations between subspace and matrix to discuss the development of Krylov subspace.In chapter three,we introduce some well-known methods of Krylov subspace.In chapter four,we give some numerical experiment to demonstrate the advantages of these methods.In chapter five,we give a conclusion of this paper.