一类非线性矩阵方程的Hermite正定解  

Hermite Positive Definite Solution of a Class of Nonlinear Matrix Equation

作　　者：熊昊 黄敬频[1] 张姗姗 XIONG Hao;HUANG Jing-pin;ZHANG Shan-shan(College of Mathematics and Physics,Guangxi University for Nationalities,Nanning 530006,China)

机构地区：[1]广西民族大学数学与物理学院,广西南宁530006

出　　处：《数学的实践与认识》2022年第8期202-210,共9页Mathematics in Practice and Theory

基　　金：国家自然科学基金(11661011);广西民族大学研究生创新项目(gxun-chxps202071)。

摘　　要：利用两个Hermite正定矩阵的Thompson度量方法和闭凸集上单调算子不动点定理,给出非线性矩阵方程X-A^(*)(R+BXB^(*))^(-t)A=Q(0<t≤1)存在Hermite正定解的必要和充分条件,以及解的存在区间.建立求解该方程的迭代格式,并证明了迭代收敛性,同时应用Thompson度量和系数矩阵的特征值刻划了解的误差估计式.最后通过数值算例验证所给方法的有效性及可行性.In this paper,by using the Thompson metric method of two Hermite positive definite matrices and thefixed point theorem of monotone operators on closed convex sets,we give some necessary and sufficient conditions for the existence of the Hermite positive definite solution of the nonlinear complex matrix equation X−A^(∗)(R+BX B^(∗))^(−t)A=Q(0<t≤1)and existence interval of the solution.The iterative scheme for solving the equation is established,and the iterative convergence is proved.In the meantime,the properties of the Thompson metric and the eigenvalue of the coefficient matrix are used to characterize the error estimation formula of the solution.Finally,the effectiveness and feasibility of the proposed method are verified by numerical examples.

关 键 词：非线性矩阵方程 Thompson度量 HERMITE正定解 不动点迭代 误差估计 

分 类 号：O241.6[理学—计算数学]