Block Iterative Methods For Linear And Fuzzy 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, approximating 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 important 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 complicated iterative 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 methods 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 iterative 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, AOR and many modified methods. But since the emergence of many unknown problems wants new approaches and methods to solve those which can't be solved or be well solved by known methods, we need to study the iterative methods for linear systems continuously.Fuzzy mathematics has made a rapid progress since Zadeh first presented the concept of fuzzy set in the foundation thesis "Fuzzy Sets" published in 1965, and by now has been regarded as a self-contained branch of mathematics. Fuzzy mathematics has many applications in engineering, pattern recognition, automation, economics, finance and so on. Most of them may boil down to finding the solution of fuzzy linear systems in the end. Consequently, as the role that general linear systems act in solving real-world problems, fuzzy linear systems are in the key position for solving fuzzy problems and play actions stronger and stronger. This requires us to investigate the numerical methods for systems of fuzzy linear equations. When the numbers of variables become much greater, iterative methods will be considered as first choices.The thesis presents some further work of iterative methods for linear and fuzzy linear systems.First, by augmenting the system to a block 4×4 consistent system and splitting the augmented coefficient matrix by subproper splitting, we consider the SSOR, AOR and GSOR methods for linear least squares problems with the iterative schemes and theconvergent conditions. The numerical examples show that the methods are effective and practical.Secondly, we study the SSOR methods, so-called SSOR-like methods, for saddle point problems by introducing a symmetric and nonsingular preconditioning matrix Q and provide the convergent intervals and the inexplicit expressions of the optimal parameters. Subsequently, the numerical experiments for solving the weighted least squares problem and the Stokes equation show that the SSOR-like methods are better than SOR-like methods presented by Golub et al. for some preconditioners.Finally, we discuss the Jacobi, Gauss-Seidel and SSOR methods for an n x n fuzzy linear system whose coefficients matrix is crisp and the right-hand side column is an arbitrary fuzzy number vector by using the embedding method, accompanied with the analysis of the convergence and numerical examples. We also discuss the solutions of the general m x n fuzzy linear system and inconsistent fuzzy linear system.