Multistage Iteration Method For Symmetric Positive Definite Matrix

Symmetric positive definite matrix as a class of common matrix, not only widely used in mathematics many branches, such as numerical linear algebra, differential and integral equations, mathematical programming, mathematical statistics and network flow optimization, etc., but it is also widely used in engineering, automatic control, image storage issues as well as economic and scientific management sciences, theory and structure algorithm is also widely studied. Solution of symmetric positive definite equations divided into direct and iterative method, symmetric positive definite equa-tions as a special class of linear equations by classical iterative method for solving the classical iterative methods including Jacobi, Gauss-Seidel, SOR, SSOR method, the convergence rate of the classical iterative method depends on the spectral radius of the coefficient matrix, for many problems, using an iterative method the convergence rate of particularly slow or does not converge. Based on the above-mentioned drawbacks, classical iterative method is less used directly for solving large linear equations. Classi-cal iterative method has the advantage of easy-to-parallel, it does not directly used for large-scale linear equations in the algorithm to construct classical iterative method to construct parallel algorithm is very suitable for solving large special linear equations in parallel.In t his paper the multistage iterative methods for the solutions of symmetric pos-itive definite linear systems are studied, the convergence is proved and the numerical examples are demonstrated. Those methods are especially suitable for parallel com-putation, and can be viewed as extensions of classical iterative methods or as parallel preconditioners for conjugate gradient methods.The content of this text is composed of six chapters.In Chapter1, we give a brief introduction of the background of the current re-search, the main results of the paper and the innovation of the paper.In Chapter2, we mainly introduce the symbols, the definition and basic properties used in this article.In Chapter3. we use block Jacobi separatist to construct a multi-stage iterative algorithm which is the core algorithm of this article for solving symmetric positive definite matrix.The convergence of the multi-stage iterative algorithm which constructed in chap- ter3has been to analyzed and to proved it in Chapter4.In chapter5. we give numerical examples for solving symmetric positive definite matrix, multi-stage iterative algorithm, and the experimental results are discussed.In the last chapter, in order to get a better result of the operation, we mainly to improve the algorithm which constructed in chapter3, and prove its convergence and give the numerical examples.