一类双变量Riccati矩阵方程组对称解的迭代算法  

An Iterative Method of Finding Symmetric Solutions to a Kind of Two-variable Riccati Matrix Equations

作　　者：朱寿升[1] 张凯院[1] 

机构地区：[1]西北工业大学应用数学系,西安710072

出　　处：《工程数学学报》2014年第1期93-102,共10页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(11071196)~~

摘　　要：基于求线性矩阵方程组约束解的修正共轭梯度法,讨论了由Nash均衡对策导出的一类双矩阵变量Riccati矩阵方程组(R-MEs)对称解的数值计算问题.提出用牛顿算法将R-MEs的对称解问题转化为双矩阵变量线性矩阵方程组的对称解或者对称最小二乘解问题,并采用修正共轭梯度法解决后一计算问题,建立了求R-MEs对称解的新型迭代算法.新型迭代算法仅要求R-MEs有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定.数值算例表明,新型迭代算法是有效的.Based on the modified conjugate gradient method for computing constrained solu-tions to linear matrix equations, a new effective iterative method is discussed, which is used to find the symmetric solutions to a kind of two-variable Riccati matrix equations （R-MEs） associated with the Nash equilibrium strategies. First, the Newton’s method is applied to the R-MEs for computing the symmetric solutions. Then the symmetric solutions or symmetric least-square solutions to the linear matrix equations are derived. Moreover, the modified con-jugate gradient method is used to solve the derived linear matrix equations. Finally, a new effective iterative method is established to find the symmetric solutions to the R-MEs. The proposed iterative method can get a symmetric solution as long as the R-MEs have symmetric solutions. It does not require the uniqueness of the problem solution and has no additional limit with respect to the coefficient matrix. Numerical examples show that the new iterative method is effective.

关 键 词：Riccati矩阵方程组 对称解 牛顿算法 修正共轭梯度法 迭代算法 

分 类 号：O241.7[理学—计算数学]