# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The iterative method for solving nonlinear matrix equation $X^{s} + A^{*}X^{-t}A = Q$. (English) Zbl 1131.65039
The author provides necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the nonlinear matrix equation $X^s+A^*X^{-t}A=Q$, where $Q$ is an Hermitian positive definite matrix, $A^*$ is the conjugate transpose of the matrix $A$, and $s,\,t$ are positive integers. In order to compute Hermitian positive definite solutions, an iterative method is derived. In addition, the author also provides a perturbation bound for the Hermitian positive definite solutions. Moreover some numerical examples are also presented.

##### MSC:
 65F30 Other matrix algorithms 65F10 Iterative methods for linear systems 15A24 Matrix equations and identities
Full Text:
##### References:
 [1] Anderson, W. N.; Morley, T. D.; Trapp, G. E.: Positive solutions X=A - X - 1B$\ast$. Linear algebra appl. 134, 53-62 (1990) · Zbl 0702.15009 [2] Bhatia, R.: Matrix analysis. Graduate texts in mathematics (1997) [3] Chen, X.; Li, W.: On the matrix equation X+A$\ast X - 1A=P$: solution and perturbation analysis. Math. num. Sin. 27, 303-310 (2005) [4] El-Sayed, S. M.; Petkov, M. G.: Iterative methods for nonlinear matrix equations X$\pm a\ast$X - 2A=I. Linear algebra appl. 403, 45-52 (2005) · Zbl 1074.65057 [5] El-Sayed, S. M.; Ran, A. C. M.: On an iteration method for solving a class of nonlinear matrix equations. SIAM J. Matrix anal. Appl. 23, 632-645 (2001) · Zbl 1002.65061 [6] Engwerda, J. C.: On the existence of a positive definite solution of the matrix equation X+ATX - 1A=I. Linear algebra appl. 194, 91-108 (1993) · Zbl 0798.15013 [7] Engwerda, J. C.; Ran, A. Cm.; Rijkeboer, A. L.: Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+A$\ast X - 1A=Q$. Linear algebra appl. 186, 255-275 (1993) · Zbl 0778.15008 [8] Ferrante, A.; Levy, B. C.: Solutions of the equation X=Q+NX - 1N$\ast$. Linear algebra appl. 247, 359-373 (1996) · Zbl 0876.15011 [9] Guo, X.: On Hermitian positive definite solution of nonlinear matrix equation X+A$\ast X - 2A=Q$. J. comput. Math. 23, 513-526 (2005) · Zbl 1081.15008 [10] Guo, C.; Lancaster, P.: Iterative solution of two matrix equations. Math. comput. 68, 1589-1603 (1999) · Zbl 0940.65036 [11] Hasanov, V. I.; Ivanov, I. G.: Solutions and perturbation estimates for the matrix equations X$\pm a\ast$X - na=Q. Appl. math. Comput. 156, 513-525 (2004) · Zbl 1063.15012 [12] Hasanov, V. I.: Positive definite solutions of the matrix equations X$\pm a\ast$X - qa=Q. Linear algebra appl. 404, 166-182 (2005) · Zbl 1078.15012 [13] Ivanov, I. G.; Hasanov, V. I.; Minchev, B. V.: On matrix equation X $\pm a\ast$X - 2A=Q. Linear algebra appl. 326, 27-44 (2001) · Zbl 0979.15007 [14] Ivanov, I. G.; El-Sayed, S. M.: Properties of positive definite solutions of the equation X$\pm a\ast$X - 2A=I. Linear algebra appl. 279, 303-316 (1998) · Zbl 0935.65041 [15] Liu, X. G.; Gao, H.: On the positive definite solutions of the equation xs$\pm$ATX - ta=I. Linear algebra appl. 368, 83-97 (2003) [16] Meini, B.: Efficient computation of the extreme solutions of X +A$\ast X - 1A=Q$ and X - A$\ast X - 1A=Q$. Math. compt. 71, 1189-1204 (2002) [17] Wang, J.; Zhang, Y.; Zhu, B.: The Hermitian positive definite solutions of matrix equation X+A$\ast X - qa=I(q>0)$. Math. num. Sin. 26, 61-72 (2004) [18] Xu, S. F.: Perturbation analysis of the maximal solution of the matrix equation X+A$\ast X - 1A=P$. Linear algebra appl. 336, 61-70 (2001) · Zbl 0992.15013 [19] Yang, Y.; Duan, F.; Zhao, X.: On solutions for the matrix equation xs+A$\ast X$ - ta=Q. Proc. of the seventh int. Conf. on matrix theory and its applications in China, 21-24 (2006) [20] Zhan, X.; Xie, J.: On the matrix equation X+ATX - 1A=I. Linear algebra appl. 247, 337-345 (1996) · Zbl 0863.15005 [21] Zhang, Y.: On Hermitian positive definite solutions of matrix equation X+A$\ast X - 2A=I$. Linear algebra appl. 372, 295-304 (2003) · Zbl 1035.15017