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On the nonlinear matrix equation X- i=1 m A i * X δ 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- i=1 m A i * X δ i A i =Q(0<|δ 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:
15A24Matrix equations and identities
65F30Other matrix algorithms
65H10Systems of nonlinear equations (numerical methods)
References:
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