Two Structure-Preserving-Doubling Like Algorithms to Solve the Positive Definite Solution of the Equation $$X-{A^{\rm{H}}}{\overline{X}}^{-1}A=Q$$
Tóm tắt
In this paper, we study the nonlinear matrix equation
$$X-{A^{\rm{H}}}{\overline{X}}^{-1}A=Q$$
, where
$$A,Q \in {{\mathbb {C}}}^{n\times n}$$
, Q is a Hermitian positive definite matrix and
$$X \in {{\mathbb {C}}}^{n\times n}$$
is an unknown matrix. We prove that the equation always has a unique Hermitian positive definite solution. We present two structure-preserving-doubling like algorithms to find the Hermitian positive definite solution of the equation, and the convergence theories are established. Finally, we show the effectiveness of the algorithms by numerical experiments.
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