Preconditioning For Some Linear Systems And Hadamard Products Of Matrices

Solutions of large-scale linear systems are deeply involved in many areas of science and engineering, such as high-order partial differential equations, computational fluid dynamics, electromagnetics, constrained optimization and linear complementarity prob-lems. Research of fast iterative methods for solving large-scale systems has important theoretical significance and practical applications. This dissertation deeply studies iter-ative solutions and preconditioning technologies of M-(H-)matrices linear systems and saddle point systems. Meanwhile, the estimate of the spectral radius of Hadamard prod-ucts of matrices are studied. Three main kinds of problems have been studies and resolved as follows:1. Preconditioning for M-(H-)matrices linear systemsThis part consists of two chapters, preconditioning technologies for M-(H-)matrices linear systems are studied. In the second chapter, preconditioning technologies for M-matrices linear systems are researched. According to the structure of M-matrices, some new preconditioners are presented and the convergence analysis of the preconditioned Gauss-Seidel iterative methods is given. Meanwhile, using the matrix-splittings and com-parison theorems, we prove theoretically that the preconditioned iterative methods have better convergence speed. Numerical experiments are used to illustrate our results.In the third chapter, preconditioned iterative methods for H-matrices linear systems are researched. Firstly, generalized preconditioners for H-matrices linear systems are given. According to the H-splitting and the H-compatible splitting theory, the conver-gence analysis of the preconditioned matrices and the convergence interval of parameters are given.2. Preconditioning for saddle point systemsIn the forth chapter, preconditioning technologies for saddle point systems are re-seached. Based on the positive stable preconditioner and the indefinite preconditioner, preconditioning technologies for saddle point problem arising form a discretization of the Maxwell's equations are discussed. Block diagonal and block triangular preconditioners with parameter are given. For nonsymmetric saddle point problem, augmentation block diagonal and block triangular preconditioners with parameter are given. Meanwhile, the analysis of the spectrum of the preconditioned saddle point matrix and the choice of the parameter are given. Numerical experiments show that the quality of the presented block preeonditioners is better than that of the corresponding block preconditioners.3. Hadamard products involving nonnegative matrices and M-matricesIn the fifth chapter, the estimate of the spectral radius of Hadamard products of non-negative matrices and M-matrices is discussed. For two nonnegative matrices, using the properties of the Hadamard products of matrices and the estimate of the spectral radius of nonnegative matrices, un upper bounds for the spectral radius of Hadamard products of two nonnegtive matrices are given. For two M-matrices, using the properties of the Fan products of matrices and the ovals of Cassini of eigenvalues, the new lower bounds of minimum eigenvalue of Fan product of two M-matrices are given. Some bounds improve the resulting results.