Preconditioning Methods For Solving Linear Equations

In the field of natural sciences,the solution of many practical problems to the final general will be attributed to the solution of linear equations.We know that there are two common methods for solving linear equations:one is direct method,the other is iterative method,Iterative methods have their own advan-tages and disadvantages in the process of solving.In the iterative method,the various pretreatment techniques are studied to improve the iterative method into a field of scientific research.This paper mainly studies preconditioning methods for solving linear equations Ax=b,when the size of coefficient matrix A is big or the condition number of it is very large.The preconditioner G is constructed by the iterative methods,such as Newton and Chebyshev.Based on the Harley iterations,a preconditinger sequence G_nalso can obtained,which is approaching a prescribed matrix A.Hence applying G_n,we can obtain the Jacobian iteration to solve the linear equations after preconditioned.The numerical examples are illustrated that the new algorithms have been improved in stability,convergence and accuracy compared with the GMRES and inverse solver in MATLAB.In this paper,the content is divided into four chapters to carry out the introduction:The first chapter mainly introduces the background and the research content of this article,and introduces some related concepts and theorems that this article will cover.The second chapter introduces Newton's method and Chebyshev's method which avoid the inverse operators at first.Then we give the new Harley iterative method based on the correction and analyze their convergence.The third chapter applies G_nto Jacobi iteration and obtains several higher performances iteration methods to solve the linear equations.The fourth chapter applies the methods given in this paper to numerical examples.And compared the results with GMRES method in case one and case two.The results of case three are compared with inverse solver in MATLAB.