Positive definite solutions of the matrix equations \(X\pm A^{\ast}X^{-q} A=Q\). (English) Zbl 1078.15012

Hasanov, Vejdi I.

Linear Algebra Appl. 404, 166-182 (2005).

Nonlinear matrix equations \(X\pm A^*X^{-q} X= Q\) are studied, where the unknown matrix \(X\) is assumed to be positive definite. The real parameter \(q\) satisfies \(0< q\leq 1\). Criteria for existence of solutions and their upper and lower bounds are established. Sufficient conditions are given for uniqueness of a (positive definite) solution of the equation with the plus sign. For the equation with the minus sign, sufficient conditions are given for existence of two different solutions. Iterative methods are developed.

Reviewer: Leiba Rodman (Williamsburg)

MSC:

15A24	Matrix equations and identities
65F30	Other matrix algorithms (MSC2010)

Keywords:

positive definite solution; iterative methods; nonlinear matrix equation

References:

[1]	Anderson, W.N.; Morley, T.D.; Trapp, G.E., Positive solutions to X=A−BX−1B*, Linear algebra appl., 134, 53-62, (1990) · Zbl 0702.15009
[2]	Bhatia, R., Matrix analysis, Graduate texts in mathematics, vol. 169, (1997), Springer Verlag
[3]	Du, Sh.; Hou, J., Positive definite solutions of operator equations Xm+A*X−na=I, Linear and multilinear algebra, 51, 163-173, (2003)
[4]	S.M. El-Sayed, Investigation of the special matrices and numerical methods for the special matrix equation, PhD thesis, Sofia, 1996 (in Bulgarian)
[5]	El-Sayed, S.M.; Ran, A.C.M., On an iteration method for solving a class of nonliner matrix equations, SIAM J. matrix anal. appl., 23, 632-645, (2001) · Zbl 1002.65061
[6]	El-Sayed, S.M.; Ramadan, M.A., On the existence of a positive definite solution of the matrix equation \(X - A^\ast \sqrt[2^m]{X^{- 1}} A = I\), Int. J. comp. math., 76, 331-338, (2001) · Zbl 0972.65030
[7]	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
[8]	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−1A=Q, Linear algebra appl., 186, 255-275, (1993) · Zbl 0778.15008
[9]	Ferrante, A.; Levy, B.C., Solutions of the equation X=Q+NX−1N*, Linear algebra appl., 247, 359-373, (1996) · Zbl 0876.15011
[10]	Furuta, T., Operator inequalities associated with Hölder-McCarthy and Kantorovich inequalities, J. inequal. appl., 6, 137-148, (1998) · Zbl 0910.47014
[11]	Guo, C.; Lancaster, P., Iterative solution of two matrix equations, Math. comput., 68, 1589-1603, (1999) · Zbl 0940.65036
[12]	V.I. Hasanov, Positive definite solutions of a nonlinear matrix equation, in: Mathematics and Education in Mathematics, Proceedings of Twenty Eighth Spring Conference of the Union of Bulgarian Mathematicians, 1999, pp. 107-112
[13]	V.I. Hasanov, Solutions and perturbation theory of the nonlinear matrix equations, PhD thesis, Sofia, 2003 (in Bulgarian)
[14]	Ivanov, I.; Hasanov, V.; Uhlig, F., Improved methods and starting values to solve the matrix equations X±A*X−1A=I iteratively, Math. comput., 74, 263-278, (2005) · Zbl 1058.65051
[15]	Ivanov, I.G.; Minchev, B.V.; Hasanov, V.I., Positive definite solutions of the equation \(X - A^\ast \sqrt{X^{- 1}} A = I\), (), 113-116
[16]	Liu, X-G.; Gao, H., On the positive definite solutions of the equation Xs±ATX−ta=I, Linear algebra appl., 368, 83-97, (2003)
[17]	Meini, B., Efficient computation of the extreme solutions of X+A*X−1A=Q and X−A*X−1A=Q, Math. comput., 71, 1189-1204, (2002) · Zbl 0994.65046
[18]	Ran, I.C.M.; Reurings, M.C.B., On the nonlinear matrix equation \(X + A^\ast \mathcal{F}(X) A = Q\): solution and perturbation theory, Linear algebra appl., 346, 15-26, (2002)
[19]	Zhan, X., Computing the extremal positive definite solution of a matrix equation, SIAM J. sci. comput., 219, 330-345, (1996)
[20]	Zhan, X.; Xie, J., On the matrix equation X+ATX−1A=I, Linear algebra appl., 247, 337-345, (1996) · Zbl 0863.15005