非对称代数Riccati方程数值算法的若干研究

发布时间：2018-06-25 05:10

本文选题：非对称代数Riccati方程 + 最小非负解　；参考：《福建师范大学》2015年硕士论文

【摘要】：在科学计算和工程应用中经常需要求解非对称代数Riccati方程的最小非负解.当方程中矩阵的规模越大时,数值迭代方法会更有效.目前,许多专家和学者已经提出了许多具有良好的性质并且十分有效的数值方法.对于这些算法仍然可以通过某些思想和技巧,使其改善和完善,因此具有丰富的内容可以研究.本文提出了三种新的数值方法来求解非对称代数Riccati方程的最小非负解,主要内容如下：绪论,概述了求解非对称代数Riccati方程的发展和研究现状,简单介绍一些已有的求解非对称代数Riccati方程的经典数值解法,并介绍了一些有关的基本引理.第一章,提出了线性隐式迭代算法(LI)来求解非对称代数Riccati方程最小非负解.接着利用萨曼斯基技巧提出修正线性隐式迭代算法(MLI).在适当的条件下,证明了LI和MLI迭代算法的单调收敛性.数值实验表明LI和MLI迭代算法是可行并且有效的.第二章,在交替线性隐式迭代算法的基础上构造了新交替线性隐式迭代法来求解非对称代数Riccati方程的最小非负解,进一步提高了算法的收敛速度.在适当条件下证明了算法的单调收敛性,并利用广义的卡莱变换理论得出最优参数且估计了该算法的渐进收敛因子.最后在理论和数值实验中证明了该算法比已有的交替线性化隐式迭代法更有效.第三章,牛顿法是求解Riccati方程的有效算法.将广义参数迭代法用于求解每一牛顿步的Sylvester方程,控制每一牛顿迭代步的内迭代,得到广义非精确牛顿迭代法.接着证明该算法的收敛性.通过广义卡莱变换的理论得出算法中参数的最优值.数值实验表明所提出的算法是可行且有效的.第四章,对本文的工作进行了总结,并对未来有关非对称代数Riccati方程的研究工作提出展望和一些设想.
[Abstract]:In scientific calculation and engineering applications, the minimum nonnegative solution of the asymmetric algebraic Riccati equation is often needed. The numerical iterative method is more effective when the matrix size in the equation is larger. At present, many experts and scholars have put forward many numerical methods which have good properties and are very effective. These algorithms can still be improved and perfected by some ideas and techniques, so they have rich contents to study. In this paper, three new numerical methods are proposed to solve the minimum nonnegative solutions of asymmetric algebraic Riccati equations. The main contents are as follows: introduction, the development and research status of solving asymmetric algebraic Riccati equations are summarized. This paper briefly introduces some classical numerical methods for solving asymmetric algebraic Riccati equations, and introduces some basic Lemma. In chapter 1, a linear implicit iterative algorithm (Li) is proposed to solve the minimum nonnegative solution of the asymmetric algebraic Riccati equation. Then a modified linear implicit iterative algorithm (MLI) is proposed by using the Samansky technique. Under suitable conditions, the monotone convergence of Li and MLI iterative algorithms is proved. Numerical experiments show that Li and MLI iterative algorithms are feasible and effective. In chapter 2, a new alternating linear implicit iterative method is constructed to solve the minimum nonnegative solution of the asymmetric algebraic Riccati equation based on the alternating linear implicit iterative algorithm, which further improves the convergence rate of the algorithm. The monotone convergence of the algorithm is proved under proper conditions, and the optimal parameters are obtained by using the generalized Kaley transform theory and the asymptotic convergence factor of the algorithm is estimated. Finally, theoretical and numerical experiments show that the proposed algorithm is more effective than the existing alternating linearization implicit iteration method. In chapter 3, Newton method is an effective algorithm for solving Riccati equation. The generalized parameter iteration method is used to solve the Sylvester equation of each Newton step and the generalized inexact Newton iterative method is obtained by controlling the inner iteration of each Newton iteration step. Then the convergence of the algorithm is proved. The optimal value of the parameters in the algorithm is obtained by the theory of generalized Kalay transform. Numerical experiments show that the proposed algorithm is feasible and effective. In the fourth chapter, the work of this paper is summarized, and the future research work of asymmetric algebraic Riccati equation is prospected and some tentative ideas are put forward.
【学位授予单位】：福建师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.8

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