Preconditioning Techniques And Numerical Methods For Two Kinds Of Linear System Of Equations

Large sparse linear system of equations Ax=b and AX=B remain the key ingredient for problems from both scientific computation and industrial applications, whose solutions have a great impact upon the original problems. For this reason, contriving efficient solvers for these two kinds of linear systems maintains one of the most-studied research area in linear algebra as well as other related disciplines. The state-of-the-art Krylov subspace methods often extract satisfactory solutions of the two linear systems with moderate cost and storage. Due to restrictions like low-dimension of the chosen subspace, Krylov subspace methods are likely to suffer from slow convergence. A common strategy to overcome this drawback is to precondition the original linear system and solve the resulting one with some appropriate Krylov subspace methods.This thesis is concerned with the efficient methods for solving Ax=b and AX=B (with flexible or polynomial preconditioning techniques) and some corresponding theoretical results. The main work can be summarized as follows:1. CMRH is an effective method for solving Ax=b. In practice, CMRH often performs analogously to GMRES but with less computational overhead and storage per step. To reduce the computational cost and storage, it is a common practice to restart CMRH which, however, may lead to slow convergence or even stagnation. The combination of flexible preconditioning and CMRH has been rarely addressed in the literature. We use the flexible preconditioning to improve the original CMRH method and obtain a flexible CMRH variant. We also compare this flexible variant with its prototype through the residual norm. Numerical examples indicate that the new method converges faster than CMRH with less CPU time.2. As an extension of CMRH, the global CMRH method based on the global Hessenberg process can be employed to solve AX=B. As done in CMRH, we often restart the global CMRH method periodically to reduce the cost and storage per iteration, which may slow down the convergence. To ameliorate this drawback, we exploit the global CMRH method itself and obtain a low-degree polynomial preconditioner that in turn is used to accelerate the original method. Such choice is justified for the square case. Numerical examples show that the resulting method is superior to its original counterpart.3. The global generalized Hessenberg method (Gl-GH) based on the global generalized Hessenberg process is another candidate for solving AX=B, which includes the global CMRH and GMRES methods as special cases. The practice of combining this method with variable preconditioning has not been covered in the literature. We use the flexible preconditioning to improve the global generalized Hessenberg method and derive a new variant FGl-GH. Moreover, two flexible methods (FGl-CMRH and FGl-GMRES) has also been investigated in detail. The-oretical results that relate the residual norm between FGl-GH and GI-GH, between FGl-CMRH and FGl-GMRES have also been given. Numerical experiments show that the performance of Gl-GH can be greatly improved by FGl-GH.