求解非线性矩阵方程X+Σ_(i=1)^(m)A_(i)^(T)X^(-ni)A_(i)=Q的无求逆迭代算法  

An inversion-free iterative method for solving nonlinear matrix equation X+Σ_(i=1)^(m)A_(i)^(T)X^(-ni)A_(i)=Q

作　　者：李涛[1] 彭振赟[1] 王杰 LI Tao;PENG Zhenyun;WANG Jie(School of Mathematics and Computational Science,Guilin University of Electronic Technology,Guilin 541004,China)

机构地区：[1]桂林电子科技大学数学与计算科学学院,广西桂林541004

出　　处：《桂林电子科技大学学报》2021年第1期50-54,共5页Journal of Guilin University of Electronic Technology

基　　金：国家自然科学基金(11961012,11761024);广西自然科学基金(2018GXNSFBA281192,2017GXNSFAA198248);桂林电子科技大学研究生创新项目(2019YCXS084)。

摘　　要：针对从数值角度讨论非线性矩阵方程X+Σ_(i=1)^(m)A_(i)^(T)X^(-ni)A_(i)=Q的最大正定解问题,其中Q为对称正定矩阵,A_(i)∈R^(p×p),m∈N_(+),n_(i)∈N_(+),提出了一个求解该问题的无求逆迭代算法,给出了该算法的迭代格式及收敛性分析。数值实验表明,算法对该问题的求解是有效可行的。In this paper,the maximum positive definite solution of nonlinear matrix equation X+Σ_(i=1)^(m)A_(i)^(T)X^(-ni)A_(i)=Q is discussed numerically,where Q is a symmetric positive definite matrix,A_(i)∈R^(p×p),m∈N_(+),n_(i)∈N_(+).This problem has been widely used in control theory,trapezoidal network,dynamic programming,random filtering and other fields.An inversion-free iterative method for solving this problem is proposed,and its iteration format and convergence analysis are given.Numerical experiments show that the algorithm is effective and feasible for solving this problem.

关 键 词：非线性矩阵方程 最大正定解 无求逆迭代 收敛性分析 

分 类 号：O241.2[理学—计算数学]