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

Duan, Xuefeng; Liao, Anping; Tang, Bin

Linear Algebra Appl. 429, No. 1, 110-121 (2008).

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.

Reviewer: Qing-Wen Wang (Shanghai)

MSC:

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

Keywords:

nonlinear matrix equation; positive definite solution; iterative method; numerical examples

Citations:

Zbl 1025.15018

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