矩阵方程X^(m)-A^(*)X^(-s)A+B^(*)X^(-t)B=Q的Hermite正定解

Hermite Positive Definite Solution of the Matrix Equation Xm-A*X-sA+B*X-tB=Q

非线性矩阵方程X^(m)-A^(*)X^(-s)A+B^(*)X^(-t)B=Q广泛应用于计算物理和最优控制等领域,由于参数与正负混杂项的存在,使得方程的求解较为困难.本文在一定条件下讨论该方程的Hermite正定解的迭代方法.首先通过矩阵变换把原问题转化为一个等价的矩阵方程,然后利用系数矩阵及其偏序构建出方程解的存在区间及三种迭代格式,并根据每种迭代序列的特点,利用解的残差范数和迭代序列的单调有界性,证明了所给迭代均收敛到原方程的Hermite正定解,同时获得解的误差估计式.最后运用两个数值算例验证所给方法的有效及可行性.

The nonlinear matrix equation X^(m)-A^(*)X^(-s)A+B^(*)X^(-t)B=Q form computational physics and optimal control, etc., the presence of parameters and positive and negative hybrid terms, it is difficult to solve the equation. In this paper, the iterative methods of the Hermite positive definite solution of the equation are discussed under certain conditions. First, the original problem is transformed into an equivalent matrix equation by matrix transformation. Then, the coefficient matrix and its partial order are used to construct the existence interval of the solution of the equation and three iterative schemes. According to the characteristics of each iterative sequence, the residual norm of the solution and the monotonic boundedness of the iterative sequence is used to prove that the given iteration converges to the Hermite positive definite solution of the original equation, and the error estimation formula of the solution is obtained. Finally, two numerical examples demonstrate the effectiveness and feasibility of the proposed method.