Iterative Solution Of Large Sparse Linear Equations

Large-scale linear systems arise widely in domains of science and engineering such as solutions of PDEs with high orders, computational electromagnetics, fluid mechanics and optimization problems. Solving large-scale linear systems plays a key role in scientific and engineering computing. Some special matrices and numerical characteristics related to iteration solutions of linear systems, convergence and comparison theorems of matrix splittings, preconditioning techniques for iteration solutions of saddle point problems and the complex linear systems are deeply studied in this thesis. This thesis consists of four parts:Brief introductions of the background, the current situation and the history of the saddle point problems and the complex system are given.We further study the spectral properties of the HSS splitting preconditioner for saddle point problems, which is introduced by Simoncini and Benzi. Some new upper bounds are given.We extend the block diagonal preconditioners. We study the spectral characteristics of the preconditioners, show that all eigenvalues of the preconditioned matrices are strongly clustered. Two preconditioners based on augmentation are introduced to the solution of large saddle point-type systems with singular (1, 1) blocks.Modified shifted Laplace preconditioners are introduced to the solution of complex linear systems, which are frequently indefinite and large, it is difficult to solve iteratively. We study the spectral characteristics of the preconditioners, show that all eigenvalues of the preconditioned matrices are strongly clustered. Finally, numerical experiments are also reported for illustrating the efficiency of the presented preconditioners.