Abstract and Applied Analysis

Volume 2013 (2013), Article ID 979832, 11 pages

http://dx.doi.org/10.1155/2013/979832

Research Article

## Perturbation Analysis of the Nonlinear Matrix Equation

School of Mathematics and Statistics, Shandong University, Weihai 264209, China

Received 15 March 2013; Accepted 7 May 2013

Academic Editor: Vejdi I. Hasanov

Copyright © 2013 Jing Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- B. L. Buzbee, G. H. Golub, and C. W. Nielson, “On direct methods for solving Poisson's equations,”
*SIAM Journal on Numerical Analysis*, vol. 7, pp. 627–656, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. N. Anderson Jr., G. B. Kleindorfer, P. R. Kleindorfer, and M. B. Woodroofe, “Consistent estimates of the parameters of a linear system,”
*Annals of Mathematical Statistics*, vol. 40, pp. 2064–2075, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. N. Anderson, T. D. Morley, and G. E. Trapp, “The cascade limit, the shorted operator and quadratic optimalcontrol,” in
*Linear Circuits, Systems and Signal Processsing: Theory and Application*, C. I. Byrnes, F. C. Martin, and R. E. Saeks, Eds., pp. 3–7, North-Holland, New York, NY, USA, 1988. View at Google Scholar - R. S. Bucy, “A priori bounds for the Riccati equation,” in
*Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability*, vol. 3 of*Probability Theory*, pp. 645–656, University of California, Berkeley, Calif, USA, 1972. View at MathSciNet - D. V. Ouellette, “Schur complements and statistics,”
*Linear Algebra and its Applications*, vol. 36, pp. 187–295, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Pusz and S. L. Woronowicz, “Functional calculus for sesquilinear forms and the purification map,”
*Reports on Mathematical Physics*, vol. 8, no. 2, pp. 159–170, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Zabczyk, “Remarks on the control of discrete-time distributed parameter systems,”
*SIAM Journal on Control and Optimization*, vol. 12, pp. 721–735, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Berzig, X. F. 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,”
*Mathematical Sciences*, vol. 6, article 27, 2012. View at Google Scholar - J. Cai and G. L. Chen, “On the Hermitian positive definite solution of nonlinear matrix equation ${X}^{s}+{A}^{*}{X}^{-t}A=Q$,”
*Applied Mathematics and Computation*, vol. 217, pp. 2448–2456, 2010. View at Google Scholar - X. Duan and A. Liao, “On the existence of Hermitian positive definite solutions of the matrix equation ${X}^{s}+{A}^{*}{X}^{-t}A=Q$,”
*Linear Algebra and its Applications*, vol. 429, no. 4, pp. 673–687, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Duan, C. Li, and A. Liao, “Solutions and perturbation analsis for the nonlinear matrix equation $X+{\sum}_{i=1}^{m}{A}_{i}^{*}{X}^{-1}{A}_{i}=I$,”
*Applied Mathematics and Computation*, vol. 218, no. 8, pp. 4458–4466, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - J. C. Engwerda, “On the existence of a positive definite solution of the matrix equation $X+{A}^{T}{X}^{-1}A=I$,”
*Linear Algebra and its Applications*, vol. 194, pp. 91–108, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Ferrante and B. C. Levy, “Hermitian solutions of the equation $X=Q+N{X}^{-1}{N}^{*}$,”
*Linear Algebra and its Applications*, vol. 247, pp. 359–373, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C.-H. Guo and P. Lancaster, “Iterative solution of two matrix equations,”
*Mathematics of Computation*, vol. 68, no. 228, pp. 1589–1603, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. I. Hasanov, “Positive definite solutions of the matrix equations $X\pm {A}^{*}{X}^{-q}A=Q$,”
*Linear Algebra and its Applications*, vol. 404, pp. 166–182, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. I. Hasanov and I. G. Ivanov, “On two perturbation estimates of the extreme solutions to the equations $X\pm {A}^{*}{X}^{-1}A=Q$,”
*Linear Algebra and its Applications*, vol. 413, no. 1, pp. 81–92, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. I. Hasanov, I. G. Ivanov, and F. Uhlig, “Improved perturbation estimates for the matrix equations $X\pm {A}^{*}{X}^{-1}A=Q$,”
*Linear Algebra and its Applications*, vol. 379, pp. 113–135, 2004, Tenth Conference of the International Linear Algebra Society. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y.-M. He and J.-H. Long, “On the Hermitian positive definite solution of the nonlinear matrix equation $X+{\sum}_{i=1}^{m}{A}_{i}^{*}{X}^{-1}{A}_{i}$,”
*Applied Mathematics and Computation*, vol. 216, no. 12, pp. 3480–3485, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. G. Ivanov and S. M. El-sayed, “Properties of positive definite solutions of the equation $X+{A}^{*}{X}^{-2}A=I$,”
*Linear Algebra and its Applications*, vol. 279, no. 1–3, pp. 303–316, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. G. Ivanov, V. I. Hasanov, and B. V. Minchev, “On matrix equations $X\pm {A}^{*}{X}^{-2}A=I$,”
*Linear Algebra and its Applications*, vol. 326, no. 1–3, pp. 27–44, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Li and Y. H. Zhang, “The Hermitian positive definite solution and its perturbation analysis for the matrix equation $X-{A}^{*}{X}^{-1}A=Q$,”
*Mathematica Numerica Sinica*, vol. 30, no. 2, pp. 129–142, 2008 (Chinese). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Li and Y. Zhang, “Perturbation analysis of the matrix equation $X-{A}^{*}{X}^{-q}A=Q$,”
*Linear Algebra and its Applications*, vol. 431, no. 9, pp. 1489–1501, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X.-G. Liu and H. Gao, “On the positive definite solutions of the matrix equations ${X}^{s}\pm {A}^{T}{X}^{-t}A={I}_{n}$,”
*Linear Algebra and its Applications*, vol. 368, pp. 83–97, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-G. Sun and S.-F. Xu, “Perturbation analysis of maximal solution of the matrix equation $X+{A}^{*}{X}^{-1}A=P$. II,”
*Linear Algebra and its Applications*, vol. 362, pp. 211–228, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - H. Xiao and J. T. Wang, “On the matrix equation $X-{A}^{*}{X}^{-p}A=Q\hspace{0.17em}\hspace{0.17em}(p>1)$,”
*Chinese Journal of Engineering Mathematics*, vol. 26, no. 2, pp. 305–309, 2009. View at Google Scholar · View at MathSciNet - S.-F. Xu, “Perturbation analysis of the maximal solution of the matrix equation $X+{A}^{*}{X}^{-1}A=P$,”
*Linear Algebra and its Applications*, vol. 336, pp. 61–70, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Yin, S. Liu, and L. Fang, “Solutions and perturbation estimates for the matrix equation ${X}^{s}+{A}^{*}{X}^{-t}A=Q$,”
*Linear Algebra and its Applications*, vol. 431, no. 9, pp. 1409–1421, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Yin, S. Liu, and T. Li, “On positive definite solutions of the matrix equation $X+{A}^{*}{X}^{-q}A=Q\hspace{0.17em}\hspace{0.17em}(0<q\le 1)$,”
*Taiwanese Journal of Mathematics*, vol. 16, no. 4, pp. 1391–1407, 2012. View at Google Scholar · View at MathSciNet - X. Zhan, “Computing the extremal positive definite solutions of a matrix equation,”
*SIAM Journal on Scientific Computing*, vol. 17, no. 5, pp. 1167–1174, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Zhan and J. Xie, “On the matrix equation $X+{A}^{T}{X}^{-1}A=I$,”
*Linear Algebra and its Applications*, vol. 247, pp. 337–345, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Zhang, “On Hermitian positive definite solutions of matrix equation $X+{A}^{*}{X}^{-2}A=I$,”
*Linear Algebra and its Applications*, vol. 372, pp. 295–304, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y.-H. Zhang, “On Hermitian positive definite solutions of matrix equation $X-{A}^{*}{X}^{-2}A=I$,”
*Journal of Computational Mathematics*, vol. 23, no. 4, pp. 408–418, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jia and D. Gao, “Perturbation estimates for the nonlinear matrix equation $X-{A}^{*}{X}^{q}A=Q(0<q<1)$,”
*Journal of Applied Mathematics and Computing*, vol. 35, no. 1-2, pp. 295–304, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Duan, A. Liao, and B. Tang, “On the nonlinear matrix equation $X-{\sum}_{i=1}^{m}{A}_{i}^{*}{X}^{{\delta}_{i}}{A}_{i}=Q$,”
*Linear Algebra and its Applications*, vol. 429, no. 1, pp. 110–121, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Zhan,
*Matrix Inequalities*, vol. 1790 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - J. R. Rice, “A theory of condition,”
*SIAM Journal on Numerical Analysis*, vol. 3, no. 2, pp. 287–310, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet