The Study Of Numerical Methods For The Solutions Of Several Classes Of Matrix Equations

The solutions of large matrix equations are central to many numerical simulations in science and engineering, such as the nuclear power industry, petroleum industry, the design and computer analysis of circuits, PDEs, and image processing, which is also often the most time-consuming part of a computation. Therefore, deriving the efficient iterative algorithms for solving large matrix equations is an important project in scientific and engineering computation field.In this dissertation, I study the fast iterative algorithms for solving the solutions of3classes of matrix equations, which can be recognized as symmetric algebraic Riccati equations, nonsymmetric algebraic Riccati equations and the matrix equation AXB=C. The symmetric algebraic Riccati equations arise in the areas of control theory, monograph theory, etc. The nonsymmetric algebraic Riccati equations arise mainly in transport theory, Markov chains and practical probability theory. The matrix equation AXB=C has very important implementations in the areas of signal and image processing as a special case of the coupled Sylvester equations ΣAijXjBij=C(i=1,...,m). These three classes of matrix equations have become hot shots in the research of recent years. A lot of methods of solving the equations have been discovered and put into practical implements.I not only use different skills in accelerating and expanding the existing methods for solving these three classes of equations, the latest iterative thoughts have been considered in this dissertation as well. Concretely speaking, in this dissertation I firstly consider the smallest positive solutions of nonsymmetric algebraic Riccati equations arising in transport theory and lay out a new modified Newton method. This method can get the smallest positive solutions through computing the smallest positive solutions of the matrix-vector equations X=T(?)(uvT). Secondly, I get the largest positive semi-definite solutions of symmetric algebraic Riccati equations by a class of inexact Newton methods. These methods use doubling iterations to solve the Lyapunov equations arising in each outer Newton iteration and get monotone convergence by controlling the number of inner iterations. Last I get a new iterative method to compute the numerical solutions of AXB=C. This method is based on the Hermitian and skew-Hermitian splitting of coefficient matrices, it is an expanding of the HSS iterative method of solving Ax=b. This dissertation includes five chapters, which is organized as follows:Firstly, the research background and research status are given, as well as the preliminary knowledge. Furthermore, the main contents of this paper are briefed.In the second chapter, the smallest positive solutions of nonsymmetric algebraic Riccati equations arising in transport theory are computed. According to the structure of this kind of algebraic Riccati equations and their unique parametric form, I can get the smallest positive solutions by computing the smallest positive solutions of the vector equations X=T(?)(uvT). A new modified Newton method is laid out and it is faster in convergence rate and has similar computational complexity compared to the general Newton method.In Chapter3, the numerical solutions of symmetric Riccati equations are solved by inexact Newton method. I get a class of new iterative methods with Newton iterations as their outer iterations and doubling iterations as their inner iterations. The monotone convergence of these methods is proved through controlling the leap-out conditions of inner iterations.In Chapter4, a new iterative method of solving AXB=C is laid out. This method is an expanding of HSS method of solving Ax=b. I give the sufficient condition of convergence for this method and the best selections of the parameters.Finally, the research work of this dissertation is summarized and the possible research lines are discussed.