TY - JOUR
T1 - On Hermitian Positive Definite Solution of Nonlinear Matrix Equation $X+A^* X^{-2}A=Q$
AU - Xiao-Xia Guo
JO - Journal of Computational Mathematics
VL - 5
SP - 513
EP - 526
PY - 2005
DA - 2005/10
SN - 23
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jcm/8836.html
KW - Nonlinear matrix equation, Hermitian positive definite solution, Sensitivity analysis, Error bound.
AB - <p style="text-align: justify;">Based on the fixed-point theory, we study the existence and the uniqueness of the maximal Hermitian positive definite solution of the nonlinear matrix equation $X+A^*X^{-2}A=Q$, where $Q$ is a square Hermitian positive definite matrix and $A^*$ is the conjugate transpose of the matrix $A$. We also demonstrate some essential properties and analyze the sensitivity of this solution. In addition, we derive computable error bounds about the approximations to the maximal Hermitian positive definite solution of the nonlinear matrix equation $X+A^*X^{-2}A=Q$. At last, we further generalize these results to the nonlinear matrix equation $X+A^*X^{-n}A=Q$, where $n \ge 2$ is a given positive integer.</p>