矩阵方程X-A^TX^(-1)A=Q的牛顿迭代解法  

Newton's Iterative Method for Solving the Matrix Equation X-A^TX^(-1)A=Q

作　　者：程可欣 彭振赟 杜丹丹 肖宪伟 

机构地区：[1]桂林电子科学大学数学与计算科学学院广西高校数据分析与计算重点实验室,桂林541004

出　　处：《工程数学学报》2016年第1期63-72,共10页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(11261014;11101100);广西区科研创新项目(2014137)~~

摘　　要：非线性矩阵方程X-A^TX^(-1)A=Q在控制理论、动态规划、插值理论和随机滤波等领域中具有广泛的应用.本文给出了该矩阵方程的等价形式并利用牛顿法对该等价矩阵方程进行求解.通过定义一类用牛顿法求根时产生的矩阵序列与用牛顿法求解矩阵方程时产生的矩阵序列相同的矩阵函数,证明了由牛顿迭代法求解矩阵方程时产生的矩阵序列包含在具有唯一解的闭球内,并收敛到闭球内的唯一解.给出了该方程近似解与真解的误差估计式,并给出了说明牛顿算法对该方程求解有效性的数值例子.Nonlinear matrix equation X - A^TX^-1A - Q has been widely applied to control theory, dynamic programming, interpolation theory and stochastic filtering. In this paper, an equivalent form of this equation is derived, and the Newton＇s iterative method is applied to solving this equivalent equation. By defining a class of matrix functions which have the property that the matrix sequence generated by the Newton＇s method to compute its root is the same as that generated by the Newton＇s method to solve the nonlinear matrix equation, we prove that the matrix sequence generated by the Newton＇s method to solve the nonlinear matrix equation is included in the closed ball which has an unique solution to the matrix equation. It is also convergent to the unique solution in that closed ball. The error estimate of the approximate solution with the true solution is derived, and a numerical example to illustrate the efficiency of Newton＇s method is also given.

关 键 词：非线性矩阵方程 牛顿迭代法 收敛性定理 

分 类 号：O241.7[理学—计算数学]