Hasanov, Vejdi I.; Ivanov, Ivan G. On the matrix equation \(X-A^* X^{-n} A = I\). (English) Zbl 1081.65036 Appl. Math. Comput. 168, No. 2, 1340-1356 (2005). The authors consider the nonlinear matrix equation \[ X-A^{\star}X^{-n}A=I, \] where \(X\) is an unknown matrix, \(I\) is the \(m\times m\) identity matrix and \(n\) is a positive integer. The equations \(X+A^{\star}X^{-1}A=Q\) and \(X-A^{\star}X^{-1}A=Q\) have many applications, and iterative procedures for solving the equation \(X-A^{\star}X^{-1}A=Q\) have been proposed [see C.-H. Guo and P. Lancaster, Math. Comput. 68, 1589–1603 (1999; Zbl 0940.65036) and A. Ferrante and B. C. Levy, Linear Algebra Appl. 247, 359–373 (1996; Zbl 0876.15011 )]. The iterative positive definite solutions and the properties of the equations \(X-A^{\star}X^{-2}A=I,\) and \(X+A^{\star}X^{-2}A=I\) have been discussed by I. G. Ivanov and S. M. El-Sayed [Linear Algebra Appl. 279, 303–316 (1998; Zbl 0935.65041)], and by I. G. Ivanov, V. I. Hasanov and B. V. Minchov [Linear Algebra Appl. 326, No. 1–3, 27–44 (2001; Zbl 0979.15007)]. In this paper, the authors review the existing methods for solving the equation \(X-A^{\star}X^{-n}A=I,\) and derive a sufficient condition for this equation to have a unique positive definite solution. Moreover, the convergence of the iterative methods proposed by S. M. El-Sayed [Comput. Math. Appl. 41, 579–588 (2001; Zbl 0984.65043)] is proved under weaker restrictions for the matrix \(A\). Reviewer: Sonia Pérez Díaz (Madrid) Cited in 7 Documents MSC: 65F30 Other matrix algorithms (MSC2010) 15A24 Matrix equations and identities 65F10 Iterative numerical methods for linear systems Keywords:nonlinear matrix equation; positive definite solution; Cauchy matrix sequence Citations:Zbl 0940.65036; Zbl 0876.15011; Zbl 0935.65041; Zbl 0979.15007; Zbl 0984.65043 PDFBibTeX XMLCite \textit{V. I. Hasanov} and \textit{I. G. Ivanov}, Appl. Math. Comput. 168, No. 2, 1340--1356 (2005; Zbl 1081.65036) Full Text: DOI References: [1] Bhatia, R., Matrix Analysis, Graduate Text in Mathematics, vol. 169 (1997), Springer-Verlag [2] Engwerda, J. C.; Ran, A. C.M.; Rijkeboer, A. L., Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation \(X+A^∗X^{−1}A=Q\), Linear Algebra Appl., 186, 255-275 (1993) · Zbl 0778.15008 [3] Engwerda, J. C., On the existence of a positive definite solution of the matrix equation \(X+A^TX^{−1}A=I\), Linear Algebra Appl., 194, 91-108 (1993) · Zbl 0798.15013 [4] Guo, C.; Lancaster, P., Iterative solution of two matrix equations, Math. Comput., 68, 1589-1603 (1999) · Zbl 0940.65036 [5] Ivanov, I. G.; El-Sayed, S. M., Properties of positive definite solution of the equation \(X+A^∗X^{−2}A=I\), Linear Algebra Appl., 279, 303-316 (1998) · Zbl 0935.65041 [6] Zhan, X., Computing the extremal positive definite solution of a matrix equation, SIAM J. Sci. Comput., 247, 337-345 (1996) [7] Zhan, X.; Xie, J., On the matrix equation \(X+A^TX^{−1}A=I\), Linear Algebra Appl., 247, 337-345 (1996) · Zbl 0863.15005 [8] Ivanov, I. G.; Hasanov, V. I.; Minchev, B. V., On matrix equation \(X\)±\(A^∗X^{−2}A=I\), Linear Algebra Appl., 326, 27-44 (2001) · Zbl 0979.15007 [9] El-Sayed, S. M., Two iteration processes for computing positive definite solution of the equation \(X\)−\(A^∗X^{−n}A=Q\), Comput. Math. Appl., 41, 579-588 (2001) · Zbl 0984.65043 [10] Ferrante, A.; Levy, B. C., Hermition solution of the \(X=Q+ N X^{−1}N^∗\), Linear Algebra Appl., 247, 359-373 (1996) · Zbl 0876.15011 [11] Lancaster, P., Theory of Matrices (1969), Academic Press: Academic Press New York · Zbl 0186.05301 [12] Lio, X.-G.; Gao, H., On the positive definite solutions of the matrix equations \(X^s\)±\(A^TX^{−t}A=I\), Linear Algebra Appl., 368, 83-97 (2003) [13] M. Reurings, Symmetric matrix equations, Ph.D. Thesis, Amsterdam, 2003.; M. Reurings, Symmetric matrix equations, Ph.D. Thesis, Amsterdam, 2003. [14] V. Hasanov, Solutions and perturbation theory of the nonlinear matrix equations, Ph.D. Thessis, Sofia 2003 (in Bulgarian).; V. Hasanov, Solutions and perturbation theory of the nonlinear matrix equations, Ph.D. Thessis, Sofia 2003 (in Bulgarian). [15] Ran, A. C.M.; Reurings, M. C.B., On the matrix equation \(X + A^\ast F(X) A = Q\): solution and perturbation theory, Linear Algebra Appl., 346, 15-26 (2002) · Zbl 1086.15013 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.