The Study Of Iterative Methods And Preconditioning Methods For Linear Algebraic Systems

Computation mathematics is involved in many applications of science and engineering. It plays a vital role in aeromechanics, life sciences, resources exploration and materials designs. By means of modern computers, physical models can be established according to mathematical theories and will give rise to linear algebraic systems. By solving the resulted linear algebraic systems, we can obtain useful results which are required in physical model problems. Consuming time and computational work used to solve the linear algebraic systems often occupy the whole process, which can be more than80%. So more and more people pay attention to find effective numerical solutions of linear algebraic systems based on large-scale scientific and engineering computing. Research of methods for solving large-scale sparse systems of linear algebraic systems becomes one of the key issues of large-scale scientific and eigineering computing and such research has important theoretic significance and practical applications. Solutions of large-scale sparse linear algebraic systems are always iterative methods, so convergence and convergence rate of iterative methods are deeply studied by many experts and scholars. The author mainly studies some iterative methods of special matrices related to iteration solutions of large-scale sparse linear algebraic systems, especially convergence properties and comparison theories of matrix splitting methods have been investigated and preconditioning techniques for special linear algebraic systems have been also studied. The main results and innovations are as follows:1. For Z-matrices linear algebraic systems, the preconditioned SOR iterative method and the preconditioned Gauss-Seidel iterative method are researched. Construct the new preconditioned matriced and give theoretical analysis and comparisons of convergence and divergence rates. The numerical example proves that the preconditioned SOR iterative method and the preconditioned Gauss-Seidel iterative method are effective.2. M-matrices is a kind of positive definite matrices with many elegant properties and the iterative solvers of such linear algebraic systems are always charming for lots of researchers. How to use these properties to construct a more effective preconditioning matrix is very attractive. For M-matrices linear algebraic systems, the new preconditioning matrix was introduced, and the preconditioned AOR iterative method is proposed. We proved that if the coefficient matrix of the original system is an M-matrix, then the coefficient matrix of the preconditioning system is also an M-matrix. The convergence theorems of the new method were proposed. Finally. numerical example shows that the convergence rate for the new iterative method is better than the corresponding classical AOR method.3. For H-matrices linear algebraic system, we construct two new preconditioned matrices, set up two kinds of corresponding the Gauss-Seidel of iterative format and determine the convergence conditions of the new methods. By virtue of the special properties of H-Matrices, we give the comparisons of convergence and divergence rates between the preconditioned iterative methods and classic iterative methods. Because the preconditioned matrices involve the parameters, we give the suitable restrictions on the parameters. With MATLAB programming language, the numerical experiments show the validity of the new algorithms.4. Using preconditioning technology, we construct the double parameters preconditioned generalized accelerative over-relaxation iterative method and the multi-parameters preconditioned generalized acceleration over-relaxation iterative method, improve theoretical achievements of the generalized acceleration over-relaxation iteration method. For the linear weighted least squares problem, we establish the iterative format, give the convergence theorem and the suitable restrictions on the parameters. The application range of the generalized acceleration over-relaxation iteration method is expanded. Comparison theorem and numerical example show the established iterative methods are better than the existing algorithms in the convergence range or convergence speed.