Preconditioning Methods For Solving Linear Equations

This paper mainly studies preconditioning methods for solving linear equa-tions 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 iterations, a preconditinger se-quence Gn also can obtained, which is approaching a prescribed matrix A. Hence applying Gn, we can obtain the Jacobian iteration to solve the linear equations after preconditioned. The numerical examples are illustrated that the new algo-rithms have been improved in stability, convergence and accuracy compared with the GMRES and inverse solver in Matlab.This paper is divided into four chapters:The first chapter mainly introduces the research background, research con-tents, and some concepts and theorems which are involved in this paper.The second chapter introduces Newton’s method and Chebyshev’s method which avoid the inverse operators at first. Then Newton iteration formula which is based on the precondition and the modified are given and an analysis of their convergence is done in this chapter.The third chapter applies Gn to 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.