The authors consider the positive definite solutions of the nonlinear matrix equation
this equation arises in the analysis of ladder networks, in dynamic programming, control theory, stochastic filtering and statistics.) For
the equation has been first considered in S. M. El-Sayed
and A. C. M. Ran
[SIAM J. Matrix Anal. Appl. 23, 632–645 (2001; Zbl 1002.65061
)]. In the present paper necessary and sufficient conditions for the existence of a solution are derived. An iterative algorithm for obtaining the positive definite solutions is proposed, and an error bound is found.