## On the nonlinear matrix equation $$X - \sum_{i=1}^{m}A_{i}^{*}X^{\delta _{i}}A_{i} = Q$$.(English)Zbl 1148.15012

Based on fixed point theorems for monotone and mixed monotone operators in a normal cone, the authors prove that the nonlinear matrix equation
$X - \sum_{i=1}^{m}A_{i}^{*}X^{\delta _{i}}A_{i} = Q\quad (0 < | \delta_{i}| < 1)$
always has a unique positive definite solution. A conjecture is solved, which was proposed by X.-G. Liu and H. Gao [ibid. 368, 83–97 (2003; Zbl 1025.15018)]. A multi-step stationary iterative method is proposed to compute the unique positive definite solution. Numerical examples show that this iterative method is feasible and effective.

### MSC:

 15A24 Matrix equations and identities 65F30 Other matrix algorithms (MSC2010) 65H10 Numerical computation of solutions to systems of equations

Zbl 1025.15018
Full Text:

### References:

 [1] Chen, M.S.; Xu, S.F., Perturbation analysis of the Hermitian positive definite solution of the matrix equation $$X - A^\ast X^{- 2} A = I$$, Linear algebra appl., 394, 39-51, (2005) · Zbl 1063.15010 [2] El-Sayed, S.M.; Ran, Andre C.M., On an iterative method for solving a class of nonlinear matrix equations, SIAM J. matrix anal. appl., 23, 632-645, (2001) · Zbl 1002.65061 [3] Ferrante, A.; Levy, B.C., Hermitian solutions of the equation $$X = Q + N^\ast X^{- 1} N$$, Linear algebra appl., 247, 359-373, (1996) · Zbl 0876.15011 [4] Guo, D., Existence and uniqueness of positive fixed point for mixed monotone operators with applications, Appl. anal., 46, 91-100, (1992) · Zbl 0792.47053 [5] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, notes and reports in mathematices, () [6] Guo, C.H.; Lancaster, P., Iterative solution of two matrix equations, Math. comput., 68, 1589-1603, (1999) · Zbl 0940.65036 [7] Guo, D., Nonlinear functional analysis, (2001), Shandong Sci. and Tech. Press Jinan, (in Chinese) [8] Gao, D.J.; Zhang, Y.H., On the Hermitian positive definite solutions of the matrix equation $$X - A^\ast X^q A = Q(q > 0)$$, Math. numer. sinica, 29, 73-80, (2007), in Chinese · Zbl 1121.15302 [9] V.I. Hasanov, Solutions and perturbation theory of nonlinear matrix equations, PhD Thesis, Sofia, 2003. [10] Hasanov, V.I., Positive definite solutions of the matrix equations $$X \pm A^{\operatorname{T}} X^{- q} A = Q$$, Linear algebra appl., 404, 166-182, (2005) · Zbl 1078.15012 [11] Ivanov, I.G.; Hasanov, V.I.; Uhilg, F., Improved methods and starting values to solve the matrix equations $$X \pm A^\ast X^{- 1} A = I$$ iteratively, Math. comput., 74, 263-278, (2004) [12] Ivanov, I.G., On positive definite solutions of the family of matrix equations $$X + A^\ast X^{- n} A = Q$$, J. comput. appl. math., 193, 277-301, (2006) · Zbl 1096.15003 [13] Liu, X.G.; Gao, H., On the positive definite solutions of the matrix equation $$X^s \pm A^{\operatorname{T}} X^{- t} A = I_n$$, Linear algebra appl., 368, 83-97, (2003) [14] Liao, A.P., On positive definite solutions of the matrix equation $$X + A^\ast X^{- n} A = I$$, Numer. math. - A J. chin. univ., 26, 156-161, (2004) · Zbl 1065.15017 [15] M.C.B. Reurings, Symmetric matrix equation, PhD Thesis, Vrije Universiteit, Amsterdan, 2003, ISBN 90-9016681-5. [16] Ran, Andre C.M.; Reurings, M.C.B., A nonlinear matrix equation connected to interpolation theory, Linear algebra appl., 379, 289-302, (2004) · Zbl 1039.15007 [17] Shi, X.Q.; Liu, F.S.; Umoh, H.; Gibson, F., Two kinds of nonlinear matrix equations and their corresponding matrix sequences, Linear multilinear algebra, 52, 1-15, (2004) · Zbl 1057.15016 [18] Zhan, X., Matrix inequalities, (2002), Springer-Verlag Berlin
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.