The Research On Preconditioning Methods For The Solution Of The Large Discrete Equations

In our life,many problems can be transformed into large discrete eqations.It is very meaningful to solve this kind of equations.Many mathematicians are trying to find better methods for solving them.Some different iterative methods can solve discrete problems well.Krylov subspace iterative methods are better in solving problems than traditional iterative ones for thev have more development.Though iterative methods have more advantages in solving discrete equations,they are restricted bv condition number.discontinuity point.etc.The precondition methods are better ones we seek for.In this paper.we briefly introduce fundamental theory and construction strategy of preconditional methods.The general problem of finding an efficient preconditioner is to identify a linear K(the preconditioner)with the properties that:(1)K is a good approximation to A in some sense.(2 )The cost of the construction of K is not prohibitive.(3)The svstem Ky=z is much easier to solve than the original systems.This kind of preconditioner is what we need.Then we present the construction of preconditioner.It is divided into left-preconditioner right-preconditioner and two-sided precontditioner.Different forms of pre-conditions of different effects can be achieved.Besides,this paper presents some preconditional methods used often presently. i.e.incomplete LU factorization,Sparse Approximate Inverse,Hybird technique. polynominal reconditioning,preconditiong by domains.The theory,specific arithmetics and some numerical experimental results of these methods are given in the paper.We also bring forward comparisons between them.Incomplete LU factorization is an effective preconditioned method put forward on the basis of LU factorization and for keeping sparse form of original matrix.It is easy to construct and keeps sparse form of original matrix.Then the matrix is easy to solve after precondition.The paper describes arithmetic and detailed analysis of incomplete LU factorization.Hybrid technique is an arithmetic that combine direct method and iterative one.Its computation efficiencv has been proved in some papers bv numerical experiment. It is applicable to run paralM and it has been proved by numerical trials that even in serial computation mode,this method can get better approximationIn order to make the condition number of matrix smaller and approximate to identity matrix as far as possible after precondition.We construct preconditioner and the idea of sparse approximation inverse.There are three forms when we construct right approximation inverse and two-sided approximation inverse.The construct of left approximation inverse is similar to that of right approximation inverse and two-sided approximation inverse.Polynomial preconditioning is applicable to run parallel.This paper only presents terminals and arithmetics for Chebyshev polynomials.Polynomial preconditioning is a method of minimizing condition number.So it makes the condition number better.We introduce preconditioning by domains.It is a special method,for we construct it according to the properties of the problem itself.We give the introduction of different preconditioning by domains and prove that each kind of this method is equal to a preconditioner.Finally,a variable angle for the five pairs of matrix preconditioning ideas, and give numerical examples to verity.