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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.

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