Abstract
In this paper, we consider a class of nonlinear matrix equation of the type \(X+\sum _{i=1}^mA_i^{*}X^{-q}A_i-\sum _{j=1}^nB_{j}^{*}X^{-r}B_j=Q\), where \(0<q,\,r\le 1\) and Q is positive definite. Based on the Schauder fixed point theorem and Bhaskar–Lakshmikantham coupled fixed point theorem, we derive some sufficient conditions for the existence and uniqueness of the positive definite solution to such equations. An iterative method is provided to compute the unique positive definite solution. A perturbation estimation and the explicit expression of Rice condition number of the unique positive definite solution are also established. The theoretical results are illustrated by numerical examples.
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The authors wish to thank the anonymous referees for providing valuable comments and suggestions which improved this paper.
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This work is supported by National Science Foundation of China (No. 61373174), Natural Science Foundation of Shaanxi Province (No. 2014JQ1021), and the Fundamental Research Funds for the Central Universities (No. K5051370007).
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Fang, L., Liu, S. & Yin, X. Positive definite solutions and perturbation analysis of a class of nonlinear matrix equations. J. Appl. Math. Comput. 53, 245–269 (2017). https://doi.org/10.1007/s12190-015-0966-7
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DOI: https://doi.org/10.1007/s12190-015-0966-7
Keywords
- Nonlinear matrix equation
- Positive definite solution
- Perturbation analysis
- Coupled fixed point theorem
- Condition number