The Hermitian Positive Definite Solution of the Nonlinear Matrix Equation

References

[1] Bhatia R., Matrix analysis, Grad. Texts in Math. 169, Springer-Verlag, New York, 1997.10.1007/978-1-4612-0653-8Search in Google Scholar

[2] Chen M. S. and Xu S. F., Perturbation analysis of the Hermitian positive definite solution of the matrix equation X−A∗X−2A=I$X-A^{*}X^{-2}A=I$, Linear Algebra Appl., 394 (2005), 39–51.10.1016/j.laa.2004.05.013Search in Google Scholar

[3] Dehghan M. and Hajarian M., The general coupled matrix equations over generalized bisymmetric matrices, Linear Algebra Appl., 432 (2010), 1531–1552.10.1016/j.laa.2009.11.014Search in Google Scholar

[4] Dehghan M. and Hajarian M., An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices, Appl. Math. Model., 34 (2010), 639–654.10.1016/j.apm.2009.06.018Search in Google Scholar

[5] Dehghan M. and Hajarian M., The reflexive and anti-reflexive solutions of a linear matrix equation and systems of matrix equations, Rocky Mountain J. Math., 40 (2010), 825–848.10.1216/RMJ-2010-40-3-825Search in Google Scholar

[6] Dehghan M. and Hajarian M., Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations, Appl. Math. Model., 35 (2011), 3285–3300.10.1016/j.apm.2011.01.022Search in Google Scholar

[7] Dehghan M. and Hajarian M., On the generalized bisymmetric and skew-symmetric solutions of the system of generalized Sylvester matrix equations, Linear. Multilinear. A., 59 (2011), 1281–1309.10.1080/03081087.2010.524363Search in Google Scholar

[8] Duan X. F. and Liao A. P., On Hermitian positive definite solution of the matrix equation X−∑i=1mAi∗XrAi=Q$X-\sum^{m}_{i=1}A^{*}_{i}X^{r}A_{i}=Q$, J. Comput. Appl. Math., 229 (2009), 27–36.10.1016/j.cam.2008.10.018Search in Google Scholar

[9] Duan X. F., A. Liao P. and Tang B., On the nonlinear matrix equation X−∑i=1mAi∗XδiAi=Q$X-\sum^{m}_{i=1}A^{*}_{i}X^{\delta_{i}}A_{i}=Q$, Linear Algebra Appl., 429 (2008), 110–121.10.1016/j.laa.2008.02.014Search in Google Scholar

[10] Duan X. F. and Liao A. P., On the existence of Hermitian positive definite solutions of the matrix equation Xs+A∗X−tA=Q$X^{s}+A^{*}X^{-t}A=Q$, Linear Algebra Appl., 429 (2008), 673-687.10.1016/j.laa.2008.03.019Search in Google Scholar

[11] El-Sayed S. M. and Ramadan M. A., On the existence of a positive definite solution of the matrix equation X+A∗X−12mA=I$X+A^{*}\sqrt[2m]{X^{-1}}A=I$, Int. J. Comput. Math., 76 (2001), 331–338.10.1080/00207160108805029Search in Google Scholar

[12] El-Sayed S. M. and Ran A. C. M., On an iterative method for solving a class of nonlinear matrix equations, SIAM J. Matrix Anal. Appl., 23 (2001), 632–645.10.1137/S0895479899345571Search in Google Scholar

[13] El-Shazly N. M. A., A theoretical and numerical study of solving a class nonlinear matrix equations, Ph.D. Thesis, Menoufia University, Faculty of Science, Mathmatics Department, 2006.Search in Google Scholar

[14] Hasanov V. I., Solutions and perturbation theory of nonlinear matrix equations, Ph.D. Thesis, Faculty of Mathematics and Informatics, Shoumen University, Shoumen, Bulgaria, 2003.Search in Google Scholar

[15] Hasanov V. I., Positive definite solutions of the matrix equations X±ATX−qA=Q$X\pm A^{T}X^{-q}A=Q$, Linear Algebra Appl., 404 (2005), 166–182.10.1016/j.laa.2005.02.024Search in Google Scholar

[16] Ivanov I. G., Hasanov V. I. and Uhilg F., Improved methods and starting values to solve the matrix equations X±A∗X−1A=I$X\pm A^{*}X^{-1}A=I$ iteratively, Math. Comput., 74 (2004), 263–278.10.1090/S0025-5718-04-01636-9Search in Google Scholar

[17] Ran A. C. M. and Reurings M. C. B., A nonlinear matrix equation connected to interpolation theory, Linear Algebra Appl., 379 (2004), 289–302.10.1016/S0024-3795(03)00541-XSearch in Google Scholar

[18] Yang Y. T., The Iterative Method for Solving Nonlinear Matrix Equation Xs+A∗X−tA=Q$X^{s} + A^{*}X^{-t}A = Q$, Appl. Math. Comput., 188 (2007), 46–53.10.1016/j.amc.2006.09.085Search in Google Scholar

[19] Liu A. J. and Chen G. L., On the Hermitian positive definite solutions of nonlinear matrix equation Xs+∑i=1mAi∗X−tiAi=Q$X^{s}+\sum^{m}_{i=1}A^{*}_{i}X^{-t_{i}}A_{i}=Q$, Appl. Math. Comput., 243 (2014), 950–959.Search in Google Scholar

[20] Li J. and Zhang Y. H., Notes on the Hermitian Positive Definite Solutions of a Matrix Equation, J. Appl. Math., 2014 (2014), Article ID 128249, http://dx.doi.org/10.1155/2014/128249.http://dx.doi.org/10.1155/2014/128249Search in Google Scholar

[21] Shi X. Q., Liu F. S., Umoh H. and Gibson F., Two kinds of nonlinear matrix equations and their corresponding matrix sequences, Linear. Multilinear. A, 52 (2004), 1–15.10.1080/0308108031000112606Search in Google Scholar

[22] Peng Z. Y., S. El-Sayed M. and Zhang X. L., Iterative methods for the extremal positive definite solution of the matrix equation X+A∗X−αA=Q$X+A^{*}X^{-\alpha}A=Q$, Appl. Math. Comput., 200 (2007), 520–527.10.1016/j.cam.2006.01.033Search in Google Scholar

[23] Ramadan M. A., On the existence of extremal positive definite solutions of a kind of matrix, Int. J. Nonlinear Sci. Numer. Simul., 6 (2) (2005), 115–126.10.1515/IJNSNS.2005.6.2.121Search in Google Scholar

[24] Ramadan M. A., Necessary and sufficient conditions to the existence of positive definite solution of the matrix equation X+ATX−2A=I$X+A^{T} X^{-2}A=I$, Int. J. Comput. Math., 82 (7) (2005), 865–870.10.1080/00207160412331336107Search in Google Scholar

[25] Ramadan M.A., N.M. El-Shazly, On the matrix equation X+ATX−12mA=I$X+A^{T} \sqrt[2m]{X^{-1}}A=I$, Appl. Math. Comput., 173 (2006), 992–1013.10.1016/j.amc.2005.04.030Search in Google Scholar

[26] Zhan X., Computing the extremal positive definite solutions of a matrix equation, SIAM J. Sci. Comput., 17 (1996), 1167–1174.10.1137/S1064827594277041Search in Google Scholar

[27] Liu X. G. and Gao H., On the positive definite solutions of the matrix equation Xs±ATX−tA=In$X^{s}\pm A^{T}X^{-t}A=I_{n}$, Linear Algebra Appl., 368 (2003), 83–97.10.1016/S0024-3795(02)00661-4Search in Google Scholar

[28] Sarhan A. M., N. El-Shazly M., E. Shehata M., On the existence of extremal positive definite solutions of the nonlinear matrix equation Xr+∑i=1mAi∗XδiAi=I$X^{r}+\sum^{m}_{i=1}A^{*}_{i}X^{\delta_{i}}A_{i}=I$, Math. Comput. Model., 51 (2010), 1107–1117.10.1016/j.mcm.2009.12.021Search in Google Scholar

[29] Horn R. A. and Johnson C. R., Matrix analysis, Cambridge University Press, 2005.Search in Google Scholar

[30] Guo D., Analysis Nonlinear Functional, Shandong Sci. and Tech. Press, Jinan, 2001 (in Chinese).Search in Google Scholar

[31] Guo D. and Lakshmikantham V., Nonlinear problems in abstract cones, Notes and reports in mathematices, Science and Engineering, vol. 5, Academic Press, Inc., 1988.Search in Google Scholar

[32] Hashemi B. and Dehghan M., Results concerning interval linear systems with multiple right-hand sides and the interval matrix equation AX=B$AX=B$, J. Comput. Appl. Math., 235 (2011), 2969–2978.10.1016/j.cam.2010.12.015Search in Google Scholar