{"title":"On the Positive Definite Solutions of Nonlinear Matrix Equation","authors":"Tian Baoguang, Liang Chunyan, Chen Nan","volume":87,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":580,"pagesEnd":583,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/9997855","abstract":"
In this paper, the nonlinear matrix equation is investigated. Based on the fixed-point theory, the boundary and the existence of the solution with the case r>-δi<\/sub> are discussed. An algorithm that avoids matrix inversion with the case -1<-δi<\/sub><0 is proposed.<\/p>\r\n","references":"[1]\tV.I. Hasanov. \"Positive definite solutions of the matrix equations \u201d. Linear Algebra Appl. 2005, 404,pp. 166-182.\r\n[2]\tI.G. Ivanov. \"On positive definite solutions of the family of matrix equations \u201d. Compute. Appl. Math. 2006, 193 ,pp. 277-301.\r\n[3]\tX.Y. Yin, S.Y. Liu.\"Positive definite solutions of the matrix equations \u201d. Computers and Mathematics with Applications. 2010, 59,pp. 3727-3739.\r\n[4]\tX. Duan, A. Liao, B. Tang. \"On the nonlinear matrix equation \u201d. Linear Algebra Appl. 2008,429,pp.110-121. \r\n[5]\tX. Duan, A. Liao. \"On the existence of Hermitian positive definite solutions of the matrix equation \u201d. Linear Algebra Appl. 2008, 429,pp. 673-687.\r\n[6]\tX.Y. Yin, S.Y. Liu, L. Fang. \"Solutions and perturbation estimates for the matrix equations \u201d. Linear Algebra Appl. 2009, 431 ,pp. 409-1421.\r\n[7]\tC.H. Guo. \"Convergence rate of an iterative method for a nonlinear matrix equation\u201d. SIAM J. Matrix Anal . Appl. 2001, 23,pp. 295-302\uff0e\r\n[8]\tZ.Y. Peng, Salah M. El-Sayed, X.L. Zhang.\"Iterative methods for the extremal positive definite solution of the matrix equation \u201d. Journal of Computational and mathematics. 2007, 200,pp. 520-527.\r\n[9]\tM. Monsalve, M. Raydan. \"A new inversion-free method for a rational matrix equation\u201d. Linear Algebra Appl. 2010, 433,pp. 64-71. \r\n[10]\tA.M. Sarhan, Naglaa M.El-Shazly, Enas M. Shehata. \"On the existence of extremal positive definite solutions of the nonlinear matrix equation \u201d. Mathematical and Computer Modeling. 2010, 51,pp. 1107-1117.\r\n[11]\tR. Bhatia. Matrix analysis. Graduate Texts in Mathematics, Spring, Berlin, 1997.\r\n[12]\tX. Zhan. \"Computing the extremal positive definite solutions of a matrix equation\u201d. SIAM J. Sci. Comput. 1996, 17,pp.1167-1174.\r\n","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 87, 2014"}