Convergence Of The Stationary Chebyshev Acceleration Methods

Many problems of scientific computation will eventually be transformed into problems of solving linear systems. In the literature, Young gives the convergence of semi-iterative method when the iteration matrix of linear equation Ax = b is symmetrical (at this time the eigenvalues of iteration matrix are real numbers). Semi-iterative, or Chebyshev semi-iterative method is a popular and effective method for solving linear equations, and it can greatly improve the convergence rate of matrixes. The purpose of this paper is to make stationary Chebyshev semi-iterative method, reduces the amount of calculation, and receives asymptotically the same convergence rate to the non- stationary method.Firstly, we elaborated the concept of iteration, reviewed several different iterative methods and format, and showed development ideas and structure of non-stationary iterative in order to make good theoretical basis for the introduction of stationary Chebyshev accelerated iterative.Secondly, we discussed the stationary Chebyshev accelerated iterative method for solving linear equations. The method could be got by stationary Chebyshev accelerated method, and the article gave the relationships between the eigenvalues of iterative matrixes. Also, we proposed four theorems and their proofs, where we paved the way for proposing and proving convergence theorems.Then, we summarized the convergence theorem of stationary Chebyshev accelerated iterative method. Also, we made some new convergence theorems and their proofs in order to consummate the theoretical basis of the iterative method. By comparison of convergence rate between stationary Chebyshev accelerated iterative method and the nonstationary one through numerical examples, we found that both of them were equivalent ,but it could greatly reduce the calculation task of nonstationary Chebyshev iterative method.Finally, we put forward the parallel algorithm of stationary Chebyshev accelerated iterative method. At the beginning, we reviewed the basic knowledge of parallel algorithms, then we analyzed iterative formula of the stationary accelerated Chebyshev iteration, at the end we gave parallel algorithm of the stationary accelerated Chebyshev iteration for thought.