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Iterative methods for the extremal positive definite solution of the matrix equation \(X+A^{*}X^{-\alpha}A=Q\). (English) Zbl 1118.65029
The authors propose two algorithms that avoid matrix inversion for every iteration, called inversion free variant of the basic point iteration. The first algorithm computes the maximal positive definite solution \(X_{+}\) of the nonlinear equation \(X+A^{\ast }X^{-\alpha }A=Q,\) where \(A\) is a nonsingular matrix, \(Q\) is a Hermitian positive definite matrix and \(\alpha \in (0,1].\) The second algorithm computes the minimal positive definite solution \(X_{-}\) of the same nonlinear equation with the case \(\alpha \in [ 1,\infty )\). Convergence theorems are provided. Some numerical examples are added to illustrate the convergence features.

MSC:
65F30 Other matrix algorithms (MSC2010)
65F10 Iterative numerical methods for linear systems
15A24 Matrix equations and identities
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[1] Anderson, W.N.; Morley, T.D.; Trapp, G.E., Positive solutions to \(X = A - \mathit{BX}^{- 1} B^*\), Linear algebra appl., 134, 53-62, (1990) · Zbl 0702.15009
[2] Bhatia, R., Matrix analysis, graduate texts in mathematics, (1997), Spring Berlin
[3] S.M. El-Sayed, Investigation of the special matrices and numerical methods for the special matrix equation, Ph.D. Thesis, Sofia, 1996 (in Bulgarian).
[4] El-Sayed, S.M., Two sided iteration methods for computing positive definite solutions of a nonlinear matrix equation, J. austral. math, soc. ser. B., 44, 1-8, (2003)
[5] El-Sayed, S.M.; Al-Dbiban, A.M., On positive definite solutions of nonlinear matrix equation, Appl. math. comput., 151, 533-541, (2004) · Zbl 1055.15022
[6] El-Sayed, S.M.; Al-Dbiban, A.M., A new inversion free iteration for solving the equation \(X + A^* X^{- 1} A = Q\), J. comput. appl. math., 181, 148-156, (2005) · Zbl 1072.65060
[7] El-Sayed, S.M.; El-Alem, M., Some properties for the existence of a positive definite solution of matrix equation \(X + A^* X^{- 2^m} A = I\), Appl. math. comput., 128, 99-108, (2002) · Zbl 1031.15015
[8] El-Sayed, S.M.; Petkov, M.G., Iterative methods for nonlinear matrix equations \(X + A^* X^{- \alpha} A = I\), Linear algebra appl., 403, 45-52, (2005) · Zbl 1074.65057
[9] El-Sayed, S.M.; Ran, A.C.M., On an iteration methods for solving a class of nonlinear matrix equations, SIAM. J. matrix anal. appl., 23, 632-645, (2001) · Zbl 1002.65061
[10] Engwerda, J.C., On the existence of a positive definite solution of the matrix equation \(X + A^T X^{- 1} A = I\), Linear algebra appl., 194, 91-108, (1993) · Zbl 0798.15013
[11] Ferrante, A.; Levy, B.C., Hermitian solution of the matrix \(X = Q - \mathit{NX}^{- 1} N^*\), Linear algebra appl., 247, 359-373, (1996)
[12] Furuta, T., Operator inequalities associated with holder – mccarthy and Kantorovich inequalities, J. inequal. appl., 6, 137-148, (1998) · Zbl 0910.47014
[13] Guo, C.H.; Lancaster, P., Iterative solution of two matrix equations, Math. comput., 228, 1589-1603, (1999) · Zbl 0940.65036
[14] 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 Mathematics, 1999, pp. 107-112.
[15] V.I. Hasanov, Solutions and perturbation theory of the nonlinear matrix equations, Ph.D. Thesis, Sofia, 2003 (in Bulgarian).
[16] Hasanov, V.I., Positive definite solutions of the matrix equations \(X \pm A^* X^{- q} A = Q\), Linear algebra appl., 404, 166-182, (2005) · Zbl 1078.15012
[17] Hasanov, V.; Ivanov, I.G., Solutions and perturbation estimates for the matrix equations \(X \pm A^* X^{- n} A = Q\), Appl. math. comput., 156, 513-525, (2004) · Zbl 1063.15012
[18] Ivanov, I.G.; El-Sayed, S.M., Properties of positive definite solutions of the equation \(X + A^* X^{- 2} A = I\), Linear algebra appl., 279, 303-316, (1998) · Zbl 0935.65041
[19] Ivanov, I.G.; Hasanov, V.; Minchev, B.V., On matrix equations \(X \pm A^* X^{- 2} A = I\), Linear algebra appl., 326, 27-44, (2001) · Zbl 0979.15007
[20] Lancaster, P.; Rodman, L., Algebraic Riccati equations, (1995), Oxford Science Publishers Oxford · Zbl 0836.15005
[21] M. Parodi, La localisation des valeurs caracterisiques des matrices etses applications, Gauthiervillars, Paris, 1959.
[22] Ramadan, M.A.; El-Shazly, N.M., On the matrix equation \(X + A^T \sqrt[2^m]{X^{- 1}} A = I\), Appl. math. comput., 173, 992-1013, (2006) · Zbl 1089.65037
[23] Zhan, X., Computing the extremal positive definite solutions of a matrix equation, SIAM J. sci. comput., 17, 1167-1174, (1996) · Zbl 0856.65044
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