矩阵方程X-A^(*)(R+B^(*)XB)^(-t)A=Q的Hermite正定解及其扰动分析

The Hermite positive definite solution and perturbation analysis for the matrix equation X-A^(*)(R+B^(*)XB)^(-t)A=Q

研究了非线性矩阵方程X-A^(*)(R+B^(*)XB)^(-t)A=Q(0<t≤1)的Hermite正定解及其扰动问题.给出矩阵方程有解的充分条件和必要条件,利用矩阵偏序和不动点定理讨论了该矩阵方程Hermite正定解的包含区间和存在性,得到取值范围并进一步精确.构造出计算该矩阵方程唯一Hermite正定解的迭代方法并推导出一阶扰动界.最后通过数值算例验证了所给迭代方法的有效性和可行性.

The Hermitian positive definite solution and its perturbation problem of the nonlinear matrix equation X-A^(*)(R+B^(*)XB)^(-t)A=Q(1<t≤1)are studied.The sufficient conditions and necessary conditions for the existence of the solution of the matrix equation are given.By using matrix partial order and fixed point theorem,the inclusion interval and existence of the Hermite positive definite solution of the matrix equation are discussed,the value range is obtained and further accurate.An iterative method for calculating the unique Hermite positive definite solution of the matrix equation is constructed and the first order perturbation bound are derived.Finally,numerical examples are given to verify the effectiveness and feasibility of the proposed method.