Browse > Article
http://dx.doi.org/10.4134/BKMS.b170054

ON POSITIVE DEFINITE SOLUTIONS OF A CLASS OF NONLINEAR MATRIX EQUATION  

Fang, Liang (School of Mathematics and Statistics Xidian University)
Liu, San-Yang (School of Mathematics and Statistics Xidian University)
Yin, Xiao-Yan (School of Mathematics and Statistics Xidian University)
Publication Information
Bulletin of the Korean Mathematical Society / v.55, no.2, 2018 , pp. 431-448 More about this Journal
Abstract
This paper is concerned with the positive definite solutions of the nonlinear matrix equation $X-A^*{\bar{X}}^{-1}A=Q$, where A, Q are given complex matrices with Q positive definite. We show that such a matrix equation always has a unique positive definite solution and if A is nonsingular, it also has a unique negative definite solution. Moreover, based on Sherman-Morrison-Woodbury formula, we derive elegant relationships between solutions of $X-A^*{\bar{X}}^{-1}A=I$ and the well-studied standard nonlinear matrix equation $Y+B^*Y^{-1}B=Q$, where B, Q are uniquely determined by A. Then several effective numerical algorithms for the unique positive definite solution of $X-A^*{\bar{X}}^{-1}A=Q$ with linear or quadratic convergence rate such as inverse-free fixed-point iteration, structure-preserving doubling algorithm, Newton algorithm are proposed. Numerical examples are presented to illustrate the effectiveness of all the theoretical results and the behavior of the considered algorithms.
Keywords
nonlinear matrix equations; Sherman-Morrison-Woodbury formula; Positive definite solution; structure-preserving doubling algorithm; fixed-point iteration; Newton iteration;
Citations & Related Records
연도 인용수 순위
  • Reference
1 M. Berzig, X. Duan, and B. Samet, Positive definite solution of the matrix equation $X\;=\;Q\;-\;A*X^{-1}\;A+B*X^{-1}\;B$ via Bhaskar-Lakshmikantham fixed point theorem, Math. Sci. (Springer) 6 (2012), 55-62.
2 J. H. Bevis, F. J. Hall, and R. E. Hartwig, Consimilarity and the matrix equation $A{\overline{X}}\;-\;XB\;=\;C$, in Current trends in matrix theory (Auburn, Ala., 1986), 51-64, North-Holland, New York.
3 J. H. Bevis, F. J. Hall, and R. E. Hartwig, The matrix equation $A{\overline{X}}\;-\;XB\;=\;C$ and its special cases, SIAM J. Matrix Anal. Appl. 9 (1988), no. 3, 348-359.   DOI
4 R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics, 169, Springer-Verlag, New York, 1997.
5 M. T. Chu, On the first degree Fejer-Riesz factorization and its applications to $X\;+\;A*X^{-1}\;A\;=\;Q$, Linear Algebra Appl. 489 (2016), 123-143.   DOI
6 XP.-F. Duan, A.-P. Liao, and B. Tang, On the nonlinear matrix equation $X\;-\;{{\sum}_{i=1}^{m}}A_{i}^{*}X^{{\delta}_{i}}A_{i}\;=\;Q$, Linear Algebra Appl. 429 (2008), no. 1, 110-121.   DOI
7 X.-F. Duan, Q.-W.Wang, and C.-M. Li, Positive definite solution of a class of nonlinear matrix equation, Linear Multilinear Algebra 62 (2014), no. 6, 839-852.   DOI
8 J. C. Engwerda, A. C. M. Ran, and A. L. Rijkeboer, Necessary and suffcient conditions for the existence of a positive definite solution of the matrix equation $X\;+\;A*X^{-1}\;A\;=\;Q$, Linear Algebra Appl. 186 (1993), 255-275.   DOI
9 A. Ferrante and B. C. Levy, Hermitian solutions of the equation $X\;=\;Q\;+\;NX^{-1}\;N*$, Linear Algebra Appl. 247 (1996), 359-373.   DOI
10 C.-H. Guo and P. Lancaster, Iterative solution of two matrix equations, Math. Comp. 68 (1999), no. 228, 1589-1603.   DOI
11 V. I. Hasanov, Positive definite solutions of the matrix equations $X\;{\pm}\;A*X^{-q}A\;=\;Q$, Linear Algebra Appl. 404 (2005), 166-182.   DOI
12 N. Huang and C.-F. Ma, Two structure-preserving-doubling like algorithms for obtaining the positive definite solution to a class of nonlinear matrix equation, Comput. Math. Appl. 69 (2015), no. 6, 494-502.   DOI
13 I. G. Ivanov, V. I. Hasanov, and F. Uhlig, Improved methods and starting values to solve the matrix equations $X\;{\pm}\;A*X^{-1}\;A\;=\;I$ iteratively, Math. Comp. 74 (2005), no. 249, 263-278.   DOI
14 T. Jiang, X. Cheng, and L. Chen, An algebraic relation between consimilarity and similarity of complex matrices and its applications, J. Phys. A 39 (2006), no. 29, 9215-9222.   DOI
15 T. Jiang and M. Wei, On solutions of the matrix equations X - AXB = C and $X\;-\;A{\overline{X}}B\;=\;C$, Linear Algebra Appl. 367 (2003), 225-233.   DOI
16 J. Li and Y. Zhang, Perturbation analysis of the matrix equation $X\;{\pm}\;A*X^{-p}A\;=\;Q$, Linear Algebra Appl. 431 (2009), no. 9, 1489-1501.   DOI
17 B. Meini, Effcient computation of the extreme solutions of $X\;+\;A*X^{-1}A\;=\;Q\;and\;X\;-\;A*X^{-1}A\;=\;Q$, Math. Comp. 71 (2002), no. 239, 1189-1204.   DOI
18 Z.-Y. Li, B. Zhou, and J. Lam, Towards positive definite solutions of a class of nonlinear matrix equations, Appl. Math. Comput. 237 (2014), 546-559.
19 W.-W. Lin and S.-F. Xu, Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations, SIAM J. Matrix Anal. Appl. 28 (2006), no. 1, 26-39.   DOI
20 X.-G. Liu and H. Gao, On the positive definite solutions of the matrix equations $X^{s}\;{\pm}\;A*X^{-t}A\;=\;I_{n}$, Linear Algebra Appl. 368 (2003), 83-97.   DOI
21 Z. Peng, S. M. El-Sayed, and X. Zhang, Iterative methods for the extremal positive definite solution of the matrix equation $X\;+\;A*X^{-{\alpha}}A\;=\;Q$, J. Comput. Appl. Math. 200 (2007), no. 2, 520-527.   DOI
22 A. C. M. Ran and M. C. B. Reurings, On the nonlinear matrix equation X+AF(X)A = Q : solutions and perturbation theory, Linear Algebra Appl. 346 (2002), 15-26.   DOI
23 S. Vaezzadeh, S. M. Vaezpour, and R. Saadati, On nonlinear matrix equations, Appl. Math. Lett. 26 (2013), no. 9, 919-923.   DOI
24 Y. Yao and X.-X. Guo, Numerical methods to solve the complex symmetric stabilizing solution of the complex matrix equation $X\;+\;A^{T}X^{-1}A\;=\;Q$, J. Math. Study 48 (2015), no. 1, 53-65.   DOI
25 X.-X. Yin, S.-Y. Liu, and L. Fang, Solutions and perturbation estimates for the matrix equation $X^{S}\;+\;A*X^{-t}A\;=\;Q$, Linear Algebra Appl. 431 (2009), no. 9, 1409-1421.   DOI
26 B. Zhou, G.-B. Cai, and J. Lam, Positive definite solutions of the nonlinear matrix equation $X\;+\;A^{H}\overline{X}^{-1}\;A\;=\;I$, Appl. Math. Comput. 219 (2013), no. 14, 7377-7391.   DOI
27 J. Yong and X. Y. Zhou, Stochastic Controls, Applications of Mathematics (New York), 43, Springer-Verlag, New York, 1999.
28 L. Zhang, An improved inversion-free method for solving the matrix equation $X\;+\;A*X^{-{\alpha}}A\;=\;Q$, J. Comput. Appl. Math. 253 (2013), 200-203.   DOI