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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
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References:
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