  Algebraic Riccati equations are encountered in many applications of control
and engineering problems, e.g., LQG problems and $H^\infty$ control theory. In
this work, we study the properties of one type of discrete-time algebraic
Riccati equations. Our contribution is twofold. First, we present sufficient
conditions for the existence of a unique positive definite solution. Second, we
propose an accelerated algorithm to obtain the positive definite solution with
the rate of convergence of any desired order. Numerical experiments strongly
support that our approach performs extremely well even in the almost critical
case. As a byproduct, we provide show that this method is capable of computing
the unique negative definite solution, once it exists.
