Volume 29, Issue 3
On the Nonlinear Matrix Equation $\boldsymbol{X} + \boldsymbol{A}^∗ f_1(\boldsymbol{X})\boldsymbol{A} + \boldsymbol{B}^∗ f_2(\boldsymbol{X}) \boldsymbol{B} = \boldsymbol{Q}$

Commun. Math. Res., 29 (2013), pp. 280-288.

Published online: 2021-05

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• Abstract

In this paper, nonlinear matrix equations of the form $\boldsymbol{X} + \boldsymbol{A}^∗ f_1(\boldsymbol{X})\boldsymbol{A} + \boldsymbol{B}^∗ f_2(\boldsymbol{X}) \boldsymbol{B} = \boldsymbol{Q}$ are discussed. Some necessary and sufficient conditions for the existence of solutions for this equation are derived. It is shown that under some conditions this equation has a unique solution, and an iterative method is proposed to obtain this unique solution. Finally, a numerical example is given to identify the efficiency of the results obtained.

• Keywords

nonlinear matrix equation, positive definite solution, iterative method.

15A24

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@Article{CMR-29-280, author = {Haifeng and Sang and and 18967 and and Haifeng Sang and Panpan and Liu and and 18968 and and Panpan Liu and Shugong and Zhang and and 18969 and and Shugong Zhang and Qingchun and Li and and 18970 and and Qingchun Li}, title = {On the Nonlinear Matrix Equation $\boldsymbol{X} + \boldsymbol{A}^∗ f_1(\boldsymbol{X})\boldsymbol{A} + \boldsymbol{B}^∗ f_2(\boldsymbol{X}) \boldsymbol{B} = \boldsymbol{Q}$}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {29}, number = {3}, pages = {280--288}, abstract = {

In this paper, nonlinear matrix equations of the form $\boldsymbol{X} + \boldsymbol{A}^∗ f_1(\boldsymbol{X})\boldsymbol{A} + \boldsymbol{B}^∗ f_2(\boldsymbol{X}) \boldsymbol{B} = \boldsymbol{Q}$ are discussed. Some necessary and sufficient conditions for the existence of solutions for this equation are derived. It is shown that under some conditions this equation has a unique solution, and an iterative method is proposed to obtain this unique solution. Finally, a numerical example is given to identify the efficiency of the results obtained.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19012.html} }
TY - JOUR T1 - On the Nonlinear Matrix Equation $\boldsymbol{X} + \boldsymbol{A}^∗ f_1(\boldsymbol{X})\boldsymbol{A} + \boldsymbol{B}^∗ f_2(\boldsymbol{X}) \boldsymbol{B} = \boldsymbol{Q}$ AU - Sang , Haifeng AU - Liu , Panpan AU - Zhang , Shugong AU - Li , Qingchun JO - Communications in Mathematical Research VL - 3 SP - 280 EP - 288 PY - 2021 DA - 2021/05 SN - 29 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19012.html KW - nonlinear matrix equation, positive definite solution, iterative method. AB -

In this paper, nonlinear matrix equations of the form $\boldsymbol{X} + \boldsymbol{A}^∗ f_1(\boldsymbol{X})\boldsymbol{A} + \boldsymbol{B}^∗ f_2(\boldsymbol{X}) \boldsymbol{B} = \boldsymbol{Q}$ are discussed. Some necessary and sufficient conditions for the existence of solutions for this equation are derived. It is shown that under some conditions this equation has a unique solution, and an iterative method is proposed to obtain this unique solution. Finally, a numerical example is given to identify the efficiency of the results obtained.

Haifeng Sang, Panpan Liu, Shugong Zhang & Qingchun Li. (2021). On the Nonlinear Matrix Equation $\boldsymbol{X} + \boldsymbol{A}^∗ f_1(\boldsymbol{X})\boldsymbol{A} + \boldsymbol{B}^∗ f_2(\boldsymbol{X}) \boldsymbol{B} = \boldsymbol{Q}$. Communications in Mathematical Research . 29 (3). 280-288. doi:
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