矩阵方程X—A*X^qA=I(0＜q＜1)Hermite正定解的扰动分析

PERTURBATION ANALYSIS OF THE HERMITIAN POSITIVE DEFINITE SOLUTIONS OF THE MATRIX EQUATION X-A*X^qA=I(0<q<1)

首先证明了非线性矩阵方程X-A~*X^qA=I(0<q<1)有唯一的正定解.讨论了方程唯一解的扰动界,并且说明了方程是适定的.给出了解的条件数的显式表达式.并且估计出矩阵方程近似解的向后误差.利用数值例子验证了以上结果.

Consider the nonlinear matrix equation X-A^＊X^qA=I（0〈q〈1） with 0 〈 q 〈 1. it shows that there exists a uique positive definite solution to the equation. A perturbation bound for the unique solution to the equation is derived, and shows that the equation is well-posed. Explicit expressions of the condition number for the unique positive definite solution are obtained, and the backward error of an approximate solution to the unique positive definite solution is evaluated.The results are illustratd by numerical examples.