Numerical Analysis Of Special Matrices And Preconditioning Techniques For Iteration Solutions Of Saddle Point Problems

Large-scale linear systems arise widely in domains of science and engineering such as solutions of PDEs with high orders, computational electromagnetics, fluid mechanics, reservoir modeling 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 are deeply studied in this thesis. This thesis consists of four parts with six chapters.Part one (Chapter two) is to study two classes of special matrices: nonsingular H-matrices and generalized H-matrices. Based onα-diagonal dominance of matrices, we derive a new brief criterion for nonsingular H-matrices, which gives a new way to study the new criteria in future. Several equivalent propositions, sufficient or necessary conditions for generalized .H-matrices are obtained. Meanwhile, we give a new construction of generalized .H-matrices, which partly answers Nabben's open problem.Part two (Chapter three) contributes to estimates of numerical characteristics of matrices. We first present a new class of real nonsingular matrix (MC-matrices) containing C-matrices. As an application of M C-matrices, we obtain an exclusion interval of real eigenvalues of a real matrix, which, furthermore, is used to localize the real eigenvalues of stochastic matrices precisely. Moreover, we achieve inclusion intervals for real parts of eigenvalues of real matrices and brief bounds of real eigenvalues of matrices whose off-diagonal elements are all nonnegative. Then, some lower bounds of numerical radius of matrices and the smallest eigenvalues of the Hadamard product of nonsingular M-matrices and inverse M-matrices are derived. All results presented in this part are tighter than the corresponding existing ones.Part three (Chapter four) is devoted to investigating convergence and comparison theorems of splittings of matrices. We firstly give some sufficient conditions for the convergence of single splittings of non-Hermitian positive definite matrices. Secondly, with the help of nonnegative matrix and Hermitian positive definite matrix theories, convergence theorems of double splittings of Hermitian positive definite matrices and monotone matrices are presented. Furthermore, convergence regions of Jcaobi and Gauss-Seidel double SOR methods for a nonsingular H-matrix are established. Finally, comparison theorems for double splittings and parallel chaotic multisplittings of matrices are also obtained, which provide theoretical base for the choice of iteration methods.Part four (Chapter five) contributes to preconditioning techniques for iteration solutions of saddle point problems. A sufficient condition is firstly established such that a nonsymmetric saddle point matrix is diagonalizable with real and positive eigenvalues. This condition is weaker than the corresponding earlier conditions. Secondly, we deeply study the PBP preconditioners, in particular, the regularized preconditioners. The regions containing real and non-real eigenvalues of the PBP preconditioned matrix are obtained. All eigenvalues of the PBP preconditioned matrix are real positive and the non-standard Conjugate Gradient algorithm can be used if certain conditions are satisfied. The model problems of Stokes equations and Maxwell equations show that regularized preconditioners are robust. Thirdly, we give some spectral properties of PSS preconditioners for generalized saddle point problems, which overcome the defect that the earlier results are only concerned with the saddle point problems with zero (2,2) blocks. It is shown that all eigenvalues of the PSS preconditioned matrix form two tight clusters, one is near (0,0) and the other is near (2,0) when the iteration parameter approaches to zero from above. The model problems of Stokes equations and Oseen equations show that the 'optimal' iteration parameter is usually between 0 and 1. Finally, we present block triangular augmentation-free and Schur complement-free preconditioners for the discretization of time-harmonic Maxwell equations in mixed form. The cost of their construction and application is almost the same to that of block diagonal augmentation-free and Schur complement-free preconditioners. However, theoretical analysis and numerical experiments show that the quality of the presented block triangular preconditioners is better than that of the corresponding block diagonal preconditioners.