Towards positive definite solutions of a class of nonlinear matrix equations

This paper is concerned with the positive definite solutions of the nonlinear matrix equation (NME) X+^AHX̄^-1A= ^In where A is a given complex matrix. We show that such an NME has a positive definite solution if and only if an auxiliary standard NME in the form of Y+(^ĀA)HY-¹(ĀA)=Q has. Furthermore, the relationship regarding the maximal positive definite solution and the minimal positive definite solution between these two NMEs are established. Three iterative algorithms are proposed to solve the considered NME. The first method is directly related to the considered original equation but involves matrix inversion; the second method avoids the computation of matrix inversion; while the third method is an accelerated version of the first one. Convergence performances of these three kinds of algorithms are analyzed in detail. Numerical examples are worked out to validate the effectiveness of the proposed algorithms.