Volume 23, Issue 5
On Hermitian Positive Definite Solution of Nonlinear Matrix Equation $X+A^* X^{-2}A=Q$

Xiao-Xia Guo

J. Comp. Math., 23 (2005), pp. 513-526.

Published online: 2005-10

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  • Abstract

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.

  • Keywords

Nonlinear matrix equation, Hermitian positive definite solution, Sensitivity analysis, Error bound.

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COPYRIGHT: © Global Science Press

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@Article{JCM-23-513, author = {}, title = {On Hermitian Positive Definite Solution of Nonlinear Matrix Equation $X+A^* X^{-2}A=Q$}, journal = {Journal of Computational Mathematics}, year = {2005}, volume = {23}, number = {5}, pages = {513--526}, abstract = {

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.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8836.html} }
TY - JOUR T1 - On Hermitian Positive Definite Solution of Nonlinear Matrix Equation $X+A^* X^{-2}A=Q$ 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 -

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.

Xiao-Xia Guo. (1970). On Hermitian Positive Definite Solution of Nonlinear Matrix Equation $X+A^* X^{-2}A=Q$. Journal of Computational Mathematics. 23 (5). 513-526. doi:
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