几类矩阵方程特殊解的计算

发布时间：2018-02-12 10:51

本文关键词： 非线性矩阵方程(方程组) Hermitian 正定解不动点迭代免逆迭代最小二乘解　出处：《青岛科技大学》2017年硕士论文　论文类型：学位论文

【摘要】：非线性矩阵方程的求解问题是近年来数值代数领域讨论的重要课题之一,它在最优控制理论、梯形网络、动态规划、随机过滤等领域均有广泛的应用.迭代法是求解非线性矩阵方程常用的方法,但在采用迭代法求解非线性矩阵方程时,经常会出现解的收敛速度缓慢、计算量大的问题.近年来,我们较多采用不动点迭代法和免逆迭代法求解非线性矩阵方程,其中免逆迭代法大大地简化了计算的复杂度.基于Kronecker积的性质,首先得到了非线性矩阵方程X + A*(Im(?)X-C)~(-t)A = Q(t0)存在Hermitian正定解的充分必要条件;其次,运用有界序列的收敛原理,分别提出了求解方程的不动点迭代法和免逆迭代法;最后,通过数值例子验证了这两种迭代方法的有效性.我们也考虑非线性矩阵方程X~s+A*X~(-t_1) A+B*X~(-t_2)B = I(s,t_1,t_20).首先得到方程存在Hermitian正定解的一些新的条件和唯一 Hermitian正定解存在的充分条件,并通过对s,t_1,t_2取值范围的讨论,给出了方程解的存在区间;其次,构造了求解方程的不动点迭代法;最后,通过数值例子验证了迭代方法是行之有效的.进而,我们研究了非线性矩阵方程组首先得到方程组存在正定解的条件;其次,提出了求解方程组的不动点迭代法;最后,通过数值例子验证了迭代方法的有效性.最后,我们研究了非线性矩阵方程组分别运用最速下降法和Newton法求解方程组的最小二乘解,并且通过具体的数值例子验证了 Newton法的有效性.
[Abstract]:The problem of solving the nonlinear matrix equation is one of the important issues discussed in numerical algebra field in recent years, it is in the optimal control theory, ladder networks, dynamic programming, stochastic filtering etc. were applied widely. The iterative method is a method of solving the nonlinear matrix equation, but in using the iterative method of solving the nonlinear matrix equation, often appear the convergence speed is slow, the problem of large amount of calculation. In recent years, we are using fixed point iteration method and free inverse iteration method for solving nonlinear matrix equation, the free inverse iteration method greatly simplifies the computational complexity. Based on the properties of product Kronecker, has been the first X + A* (Im (nonlinear matrix equation X-C?)) ~ (-t) A = Q (T0) Hermitian positive definite solution of the necessary and sufficient condition; secondly, based on the convergence principle of bounded sequence, are proposed respectively fixed point iteration method for solving equations and free inverse iteration On behalf of the law; finally, through numerical examples verify the effectiveness of the two kinds of iterative methods. We also consider the nonlinear matrix equation X~s+A*X~ (-t_1) A+B*X~ (-t_2) B = I (s, t_1, t_20). We obtain sufficient conditions for Hermitian positive definite solutions of some new conditions and only Hermitian positive definite solutions of the problems based on the equation, s, t_1, t_2 discussed the range interval equations are presented; secondly, we construct fixed point iteration method for solving equations; finally, a numerical example shows the iterative method is effective. Then, we study the nonlinear matrix equations firstly obtained equations the existence of positive definite solutions the conditions; secondly, the fixed point iteration method for solving equations; finally, through numerical examples verify the effectiveness of the iterative method. Finally, we study the nonlinear matrix equations using the steepest descent method and Newton The method is used to solve the least square solution of the equation group, and the validity of the Newton method is verified by a specific numerical example.

【学位授予单位】：青岛科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.6

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