Two Structure-Preserving-Doubling Like Algorithms to Solve the Positive Definite Solution of the Equation $X-{A^{\rm{H}}}{\overline{X}}^{-1}A=Q$

Abstract

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.