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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

$X-\sum _{i=1}^{m}{A}_{i}^{*}{X}^{{\delta }_{i}}{A}_{i}=Q\phantom{\rule{1.em}{0ex}}\left(0<|{\delta }_{i}|<1\right)$

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 65H10 Systems of nonlinear equations (numerical methods)
References:
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