| Quadratic matrix equation and its related problems arise from many fields, such as physics, mechanism, engineering, optimal control theory, scientific com-puting and so on. The study in the existence of the solutions and the numerical methods are very important from theoretical view point as well as applications. In recent ten years, numerical methods for solving the quadratic matrix equation had become a very hot topic in engineering and computational mathematics. In this thesis, we shall focus on the numerical methods for finding the solutions of two classes of quadratic matrix equations, the unilateral quadratic matrix equa-tion arising from damped-mass system and the nonsymmetric algebraic Riccati equation arising from transport theory of physics.In Chapter2, we are concerned with a unilateral quadratic matrix equation arising from the damped mass-spring system. We first give a sufficient condition for the existence of the solvents to the equation. We then develop a nonsingular M-matrix structure-preserving doubling algorithm (MSD) to calculate the extreme solvents of the equation. Under appropriate conditions, we establish the quadratic convergence of the proposed method. Numerical experiments show that the pro-posed MSD algorithm outperforms Newton’s method with exact line searches and Bernoulli’s method.In Chapter3, we review the cyclic reduction (CR) algorithm for the unilat-eral quadratic matrix equation arising from the overdamped mass-spring system. Unlike the original CR algorithm, we propose a cubic cyclic reduction (CCR) al-gorithm to calculate the extremal solutions of this equation. The CCR algorithm is well defined under the overdamped condition. We also establish its convergence. Our preliminary numerical results show that the CCR algorithm is more effective for finding the extremal solutions than the original CR algorithm especially for the problem near the critical case.In Chapter4, we study the convergence of the cyclic reduction method in the critical case of overdamped system. Guo, Higham and Tisseur has obtained the linear convergence of CR method under the assumption that the partial multiplic-ities of the n-th largest eigenvalue are all equal to2. Moreover, some generated matrix sequences converge to zero matrix. We will give an example to show the convergence of CR method is not the same with Guo et. al.’s result when their assumption on the partial multiplicities is not satisfied. In other words, the related matrix sequences may not necessary converge to zero matrix. We then prove the convergence of CR method for a class of overdamped system in the critical case without the assumption that the partial multiplicities of the n-th largest eigenvalue are all equal to2. The numerical example indicates that the convergence behavior of the CR algorithm is largely dictated by the theory we established.In Chapter5, we consider the nonsymmetric algebraic Riccati equation arising from transport theory. We look at Newton’s method and the fixed-point meth-ods for finding the minimal positive solution of this matrix equation from a new view point. We first rewrite the subproblem of Newton’s method into an equivalent form with some special structure. By the use of the particular structure of the sub-problems, we present a low memory and low complexity version Newton’s method with a factored alternating-direction-implicit iteration. We then extend this idea to some fixed-point methods to develop two low memory fixed-point methods. We also show some nice properties for the eigenvalues of the coefficient matrices of the subproblems. We do some numerical experiments to test the proposed methods and compare their performances with the recently developed NBGS method and fast Newton method. The results show that the proposed methods are highly effi-cient to obtain the minimal positive solution. The low memory and low complexity Newton’s method is particularly efficient for solving large scale Riccati equation arising from transport theory.This dissertation is supported by the major project of the Ministry of Edu-cation of China (309023) and the National Natural Science Foundation of China (11071087).This dissertation is typeset by software LATEX2ε. |
