Some Preconditioning Techniques For Solving Linear Systems

Systems of simultaneous linear equations play a major role in various areas such as mathematics, physics, statistics, engineering and social sciences. In particular, approx-imating the solution of linear partial differential equations in physics finally leads to a set of simultaneous algebraic linear equations. Therefore, it has been being an impor-tant problem in numerical linear algebra area to solve linear systems. As is well-known, the numerical methods for solving systems of linear equations involve direct methods and iterative methods. Direct methods are characteristic of fewer computations but compli-cated computation scheme, thus, which are suitable for systems with lower order. The real-world problems are always of large and sparse linear equations, for which direct meth-ods are helpless, while iterative methods can make up the shortage of the direct methods with the advantage of simple scheme and fewer storages required. Therefore, the iter-ative methods are usually used in large and sparse cases. Numerous results have come out for various problems and cases with the classical iterative methods such as Jacobi, Gauss-Seidel, SOR, Krylov subspace and many modified methods.For improving the effectiveness of the iterative methods, and fastening the convergent rate, preconditioning techniques are used. Preconditioning techniques refers to trans-forming the original linear system into another equivalent system with more favorable properties for iterative solution. A preconditioner is a matrix that effects such a transfor-mation. Preconditioning techniques attempts to improve the properties of the coefficient matrix, for example, spectral properties, such that the computation efficiency can be im-proved when the iterative method applied to the preconditioned linear system, and the faster convergent rate can be reached.Firstly, for M-matrix linear system, which appears in many science computing prob-lems such as discretioning the partial differential equations, finite Markov chain and so on, a left preconditioner was proposed in1987by Milaszewicz. Since then, many scholars attempts to modifying and improving it, and there are some preconditioners are proposed. The characteristics of these class of preconditioners are easy to construction, it is formu- lated by the few elements of the coefficient matrix, has the sparse structure and preserve the properties of the M-matrix. The preconditioners for this line applied the original lin-ear system such that some elements of the coefficient matrix to be eliminated, and made the associated Jacobi、Gauss-Seidel、SOR method converge faster that the original ones. In this thesis, some modified left preconditioners for M-matrix linear systems are pro-posed. The convergence and comparison theorems, established in the thesis, showed that the proposed preconditioners are efficient when Gauss-Seidel method is used for solving the preconditioned linear system, and the converge rate are improved. Furthermore, by es-tablishing the comparison theorem of the double splitting iterative method, we confirmed that some of the proposed preconditioners are efficient for the double splitting iterative method. Meanwhile, the proposed preconditioners are applied to the H-matrix linear systems, convergent theorem and numerical experiments shows the efficiency.Secondly, some comparison theorems for the double splittings of different monotone matrices are established. As an application of the comparison results, we confirm some modified left preconditioners for M-matrix linear systems also efficient for double splitting iteration methods.Finally, we consider the preconditioned GAOR methods and preconditioned MAOR methods for solving a class of2×2block linear systems, which appeared in generalized least squares problems. New type of preconditioners are proposed. Theoretical as well as numerical results showed that the new preconditioner are more efficiency for improving the convergent rate of the GAOR and MAOR methods.