中文摘要：

非线性矩阵方程X—A^TX^-1A=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．

英文摘要：

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.